Quantum Condensed Matter Field Theory - Internet Archive€¦ · [11] C. Kittel, Quantum Theory of Solids, Wiley (1963). Not to be confused with the other famous text by the same

Quantum Condensed MatterField Theory

ii

Preface

The aim of this course is to provide a self-contained introduction to the basic tools andconcepts of many-body quantum mechanics and quantum field theory, motivated by phys-ical applications, and including the methods of second quantisation, the Feynman pathintegral and functional field integral. The course synopsis is outlined below. Items in-dicated by a † will either be covered in lectures (depending on time) or will be used asadditional source material for problem sets and supervision. The italicised items representparticular mathematical concepts:

. Collective Excitations: From Particles to Fields: Linear harmonic chainand free scalar field theory; functional analysis ; quantisation of the classical field;phonons; †relation to quantum electrodynamics; concepts of broken symmetry, col-lective modes, elementary excitations and universality. [3]

. Second Quantisation: Fock states; creation and annihilation operators for bosonsand fermions; representations of one- and two-body operators; canonical transforma-tions ; Applications to phonons; the interacting electron gas; Wannier states, strongcorrelation and the Mott transition; quantum magnetism and spin wave theory; spinrepresentations; †spin liquids; the weakly interacting Bose gas. [6]

. Path Integral Methods: Propagators and construction of the Feynman Pathintegral; Gaussian functional integration and saddle-point analyses; relation to semi-classics and statistical mechanics; harmonic oscillator and the single well; doublewell, instantons, and tunneling; †metastability and the fate of the false vacuum. [7]

. Many-Body Field Integral: Bose and Fermi coherent states; Grassmann al-gebra; coherent state Path integral; quantum partition function; Applications toBogoluibov theory of the weakly interacting Bose gas and superfluidity; Cooperinstability and the BCS condensate; Ginzburg-Landau phenomenology and the con-nection to classical statistical field theory, †Gauge theory and the Anderson-Higgsmechanism; †Resonance superfluidity in ultracold atomic gases and the BEC to BCScrossover, †Peierls instability. [8]

Quantum Condensed Matter Field Theory

iii

Course Objectives

From analytical dynamics and fluid mechanics, to electrodynamics and quantum mechan-ics, lectures can often leave an impression that to each problem in physics a specific andformal exact solution is at hand. Such misconceptions are often reinforced by the allure ofsophisticated analytical machinery developed in courses devoted to mathematical meth-ods. However, the limitations of a ‘first-principles’ or ‘microscopic approach’ is nowheremore exposed than in the study of strongly interacting classical and quantum many-particle systems. The aim of this course is to introduce modern methods of theoreticalphysics tailored to the description of collective phenomena where microscopic (and, often,perturbative) approaches fail. The fundamental concepts on which we rely are (broken)symmetries, collective modes, elementary excitations, and universality. The foundationof our approach will be functional methods of classical and quantum field theory.

To introduce the notion and significance of the quantum field, the first few introductorylectures involve the construction and quantisation of a classical continuum field theorystarting from a discrete model of lattice vibrations. By the end of the course, we willsee that this system provides a platform to describe the elementary excitations of spin-waves in a quantum antiferromagnet, excitations in a weakly interacting Bose gas, andthe relativistic scalar field!

In the study of quantum many-body phenomena in both high energy and condensedmatter physics, second quantisation provides a basic and common language. In the nextfew lectures, a formal introduction to this operator method is consolidated by applicationsto both fermionic and bosonic systems. Beginning with a study of the strongly interactingelectron gas, we exploit the second quantisation to expose an instability towards theformation of an electron “solid phase” — out of which a magnetic state emerges. Thisapplication in turn motivates the investigation of the hydrodynamic or spin-wave spectrumof the quantum Heisenberg spin (anti)ferromagnet. We then close this section with adiscussion of the weakly interacting dilute Bose gas.

As preparation for the field theory of the many-body system, the functional fieldintegral method will be introduced and developed within the framework of the Feynmanpath integral. Emphasis will be given to the connection of the path integral to classicalLagrangian mechanics through the semi-classical expansion, as well as the relation to thequantum and classical statistical partition function through the Euclidean time action.The example of a single well and the instanton approach to the double well will be exploredin lectures. Further applications to metastability and macroscopic quantum tunneling willbe discussed depending on time.

In the study of both high energy and condensed matter physics, methods of quantumstatistical field theory play a central role. Although modern field theory applications inthe respective fields have developed to a high degree of specialisation, a common originis shared. The aim of the remaining lectures is to introduce the subject of quantum andstatistical field theory placing emphasis on generic concepts. Introducing the bosonic andfermionic coherent state, the first two lectures are concerned with the microscopic deriva-tion of the coherent state path integral. The latter is applied to the weakly interactingBose gas and the phenomenon of superfluidity. Continuing this theme, we then explorethe pair instability of the interacting electron gas and the formation of the supercon-


iv

ducting BCS condensate. Here, the connection between the e↵ective BCS action and theGinzburg-Landau theory of phase transitions and critical phenomena will be emphasised.

Problem Sets

The Problem sets represent an integral part of the course providing the means to reinforcekey ideas as well as practice techniques. Problems indicated by a † symbol are regardedas particularly challenging.

Books

Several texts cover the introduction to second quantisation, path integrals and quantumfield theory. However, one can draw great benefit by studying a variety of di↵erent texts.The bibliography below includes many books, some explicitly referenced in these lecturenotes, others that I have found useful in preparing the course, and still others that arefrequently mentioned but which I find less useful. A note has been included concerningtheir relevance and accessibility. Those books which would seem to be of particular usehave been denoted by a “”. You will also, no doubt, find books not included in the thislist which are both relevant and useful...

Alongside the literature, detailed lecture notes have been prepared to supplement thecourse. Although these lecture notes will include additional commentaries and examplesnot covered in the course, they are intended to complement the material contained in thelectures, and they will not form part of the examinable material for the course.


Bibliography

[1] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinskii, Methods of Quantum FieldTheory in Statistical Physics, Dover Publications, Inc., 1975. Advanced, historicallysignificant, but somewhat out of date (only diagrammatics), and often overly concise.

[2] A. Altland and B. D. Simons, Condensed Matter Field Theory, CUP Second Edition(2010). Originally inspired by the preparation of this course, this text includes material thatreaches far beyond these lectures.

[3] N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt-Saunders InternationalEds. (1983). Classical reference on solid state physics.

[4] A. Auerbach, Interacting Electrons and Quantum Magnetism, Spinger-Verlag, NewYork (1994). Tutorial discussion on applications of second quantisation and the path integralto problems in quantum magnetism.

[5] P. W. Anderson, Concepts in Solids: Lectures of the Theory of Solids, World-Scientific, Singapore (1997). Exellent introduction to the concepts of elementary excitationsin condensed matter physics.

[6] P. W. Anderson, Basic Notions in Condensed Matter Physics, Benjamin (1984).Inspirational but not tutorial — a highly subjective view of modern condensed matter physicsfrom one of the most influential figures.

[7] S. Coleman, Aspects in Symmetry — Selected Erice Lectures (CUP), 1985. Chapter5. has an excellent introduction to the subject of instantons — including applications toparticle physics and cosmology — highly recommended.

[8] A. Das, Field Theory: A Path Integral Approach, World Scientific Publishing, (1993).A good, cheap, introduction to the path integral approach.

[9] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, (1965). A classic text: inevitably excludes a lot of modern material but stillan excellent introduction to the Feynman path integral.

[10] R. P. Feynman, Statistical Mechanics, Benjamin, New York, (1972). Another classictext.

[11] C. Kittel, Quantum Theory of Solids, Wiley (1963). Not to be confused with the otherfamous text by the same author — good introduction to second quantisation and the physicsof the weakly interacting electron gas.


vi BIBLIOGRAPHY

[12] N. Nagaosa, Quantum Field Theory in Condensed Matter Physics, Sringer 1999.An excellent and useful modern text. Short on detail but plenty of examples taken fromcondensed matter physics. See also the second volume in the series.

[13] J. W. Negele and H. Orland, Quantum Many-Particle Systems, Addison-Wesley Pub-lishing, 1988. Covers several core topics in this course including second quantisation andpath integrals — high on detail, short on examples.

[14] D. Pines and P. Nozieres, The Theory of Quantum Liquids — Neutral Fermi Liq-uids, Addison-Wesley Publishing, NY (1989). A classic text on solid state physics. Volume2 is also interesting and relevant.

[15] V. N. Popov, Functional Integrals and Collective Excitations, CUP, (1987). Advancedand concise introduction to path integral methods generally in condensed matter physics —useful for much of the course.

[16] L. S. Schulman, Techniques and Applications of Path Integration, John Wiley &Sons, 1981. Excellent introduction to the Feynman path integral.

[17] A. M. Tsvelik, Quantum Field Theory in Condensed Matter Physics, (CUP) 1995.A modern and entertaining discussion of applications of quantum field theory in condensedmatter physics. Advanced, but useful as a source of inspiration.


Contents

1 From Particles to Fields 11.1 Free scalar field theory: phonons . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Classical chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Quantum Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.3 Quasi-Particle Interpretation of the Quantum Chain . . . . . . . . . 10

1.2 †Quantum Electrodynamics (QED) . . . . . . . . . . . . . . . . . . . . . . 121.3 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.1 Questions on Collective Modes and Field Theories . . . . . . . . . . 141.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Second Quantisation 192.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Applications of Second Quantisation . . . . . . . . . . . . . . . . . . . . . 25

2.2.1 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.2 Interacting Electron Gas . . . . . . . . . . . . . . . . . . . . . . . . 262.2.3 Tight-binding theory and the Mott transition . . . . . . . . . . . . 272.2.4 Quantum Spin Chains . . . . . . . . . . . . . . . . . . . . . . . . . 322.2.5 Bogoliubov theory of the weakly interacting Bose gas . . . . . . . . 37

2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.4.1 Questions on the Second Quantisation . . . . . . . . . . . . . . . . 422.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Feynman Path Integral 513.1 The Path Integral: General Formalism . . . . . . . . . . . . . . . . . . . . 523.2 Construction of the Path Integral . . . . . . . . . . . . . . . . . . . . . . . 53

3.2.1 Path Integral and Statistical Mechanics . . . . . . . . . . . . . . . . 613.2.2 Semiclassics from the Feynman path integral . . . . . . . . . . . . . 633.2.3 Construction Recipe of the Path Integral . . . . . . . . . . . . . . . 67

3.3 Applications of the Feynman Path Integral . . . . . . . . . . . . . . . . . . 683.3.1 Quantum Particle in a Well . . . . . . . . . . . . . . . . . . . . . . 693.3.2 Double Well Potential: Tunneling and Instantons . . . . . . . . . . 713.3.3 †Tunneling of Quantum Fields: ‘Fate of the False Vacuum’ . . . . . 79

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.5 Problem set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85


viii CONTENTS

3.6 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4 Functional Field Integral 914.1 Construction of the many–body path integral . . . . . . . . . . . . . . . . 92

4.1.1 Coherent states (Bosons) . . . . . . . . . . . . . . . . . . . . . . . . 924.1.2 Coherent States (Fermions) . . . . . . . . . . . . . . . . . . . . . . 95

4.2 Field integral for quantum partition function . . . . . . . . . . . . . . . . . 1004.2.1 Partition function of non–interacting gas . . . . . . . . . . . . . . . 103

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.4 Problem set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.5 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5 Broken Symmetry and Collective Phenomena 1115.1 Mean-field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.2 †Plasma theory of weakly interacting electron gas . . . . . . . . . . . . . . 1125.3 Bose–Einstein Condensation and Superfluidity . . . . . . . . . . . . . . . . 123

5.3.1 Bose–Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . 1235.3.2 The Weakly Interacting Bose Gas . . . . . . . . . . . . . . . . . . . 1275.3.3 Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.4 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.4.1 Mean-field theory of superconductivity . . . . . . . . . . . . . . . . 1365.4.2 Superconductivity from the path integral . . . . . . . . . . . . . . . 1385.4.3 Gap Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.4.4 †Superconductivity: Anderson-Higgs Mechanism . . . . . . . . . . . 1435.4.5 Statistical Field Theory: Ferromagnetism Revisited . . . . . . . . . 144

5.5 Problem set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.6 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153


Chapter 1

From Particles to Fields

The aim of this section is to introduce the language and machinery of classical and quan-tum field theory through its application to the problem of lattice vibrations in a solid. Indoing so, we will become acquainted with the notion of symmetry breaking, universality,elementary excitations and collective modes — concepts which will pervade much of thecourse.

1.1 Free scalar field theory: phonons

As a grossly simplified model of a (one-dimensional) quantum solid consider a chain ofpoint particles of mass m (atoms) which are elastically connected by springs with springconstant ks (chemical bonds) (see Fig. 1.1). The aim of this chapter will be to construct

ks

(I-1)a Ia (I+1)a

RI-1 I M

Figure 1.1: Toy model of a 1D solid – a chain of elastically bound massive point particles.

an e↵ective quantum field theory of the vibrations of the one-dimensional solid. However,before doing so, we will first consider its classical behaviour. Analysing the classical casewill not only tell us how to quantise the system, but also get us acquainted with somebasic methodological concepts of field theory in general.

1.1.1 Classical chain

The classical Lagrangian of the N -atom chain is given by

L = T V =NX

n=1

m

2x2

n ks

2(xn+1

xn a)2

, (1.1)


2 CHAPTER 1. FROM PARTICLES TO FIELDS

where the first term accounts for the kinetic energy of the particles whilst the seconddescribes their coupling.1 For convenience, we adopt periodic boundary conditions such

Joseph-Louis Lagrange 1736-1813: A mathe-matician who excelled in all fields of analysis, numbertheory, and celestial mechanics. In 1788 he publishedMecanique Analytique, which summarised all of thework done in the field of mechanics since the time ofNewton, and is notable for its use of the theory of dif-ferential equations. In it he transformed mechanicsinto a branch of mathematical analysis.

that xN+1

= Na + x1

.Anticipating that the ef-fect of lattice vibrationson the solid is weak (i.e.long-range atomic orderis maintained) we will as-sume that (a) the n-thatom has its equilibriumposition at xn na (with a the mean inter-atomic distance), and (b) that the deviationfrom the equilibrium position is small (|xn(t) xn| a), i.e. the integrity of the solid ismaintained. With xn(t) = xn + n(t) (N+1

= 1

) the Lagrangian (1.1) takes the form,

L =NX

n=1

m

22

n ks

2(n+1

n)2

.

Typically, one is not concerned with the behaviour of a given system on ‘atomic’ lengthscales. (In any case, for such purposes, a modelling like the one above would be much too

RI

I

(x)

Figure 1.2: Continuum limit of harmonic chain.

primitive!) Rather, one is interested inuniversal features, i.e. experimentally ob-servable behaviour that manifests itself onmacroscopic length scales. For example,one might wish to study the specific heatof the solid in the limit of infinitely manyatoms (or at least a macroscopically largenumber, O(1023)). Under these conditions,microscopic models can usually be substan-tially simplified. In particular it is of-ten permissible to subject a discrete latticemodel to a continuum limit, i.e. to neglect the discreteness of the microscopic entitiesof the system and to describe it in terms of e↵ective continuum degrees of freedom.

In the present case, taking a continuum limit amounts to describing the lattice fluc-tuations n in terms of smooth functions of a continuous variable x (Fig. 1.2). Clearlysuch a description makes sense only if relative fluctuations on atomic scales are weak (forotherwise the smoothness condition would be violated). Introducing continuum degreesof freedom (x), and applying a first order Taylor expansion,2 we define

n ! a1/2(x)x=na

, n+1

n ! a3/2@x(x)x=na

,NX

n=1

! 1

a

Z L

0

dx,

1In realistic solids, the inter-atomic potential is, of course, more complex than just quadratic. Yet, for“weak coupling”, the harmonic (quadratic) contribution plays a dominant role. For the sake of simplicitywe, therefore, neglect the e↵ects caused by higher order contributions.

2Indeed, for reasons that will become clear, higher order contributions to the Taylor expansion do notcontribute to the low-energy properties of the system where the continuum approximation is valid.


1.1. FREE SCALAR FIELD THEORY: PHONONS 3

where L = Na (not to be confused with the Lagrangian itself!). Note that, as defined, thefunctions (x, t) have dimensionality [Length]1/2. Expressed in terms of the new degreesof freedom, the continuum limit of the Lagrangian then reads

L[] =

Z L

0

dx L(@x, ), L(@x, ) =m

22 ksa2

2(@x)2, (1.2)

where the Lagrangian density L has dimensions [energy]/[length]. (Here, and hereafter,we will adopt the shorthand convention O @tO.) The classical action associated withthe dynamics of a certain configuration is defined as

S[] =

Zdt L[] =

Zdt

Z L

0

dx L(@x, ) (1.3)

We have thus succeeded in abandoning the N -point particle description in favour of oneinvolving continuous degrees of freedom, a (classical) field. The dynamics of the latteris specified by the functionals L and S which represent the continuum generalisationsof the discrete classical Lagrangian and action, respectively.

. Info. In the physics literature, mappings of functions into the real or complex numbersare generally called functionals. The argument of a functional is commonly indicated in angularbrackets [ · ]. For example, in this case, S maps the ‘functions’ @x(x, t) and (x, t) to the realnumber S[].

——————————————–Although Eq. (1.2) specifies the model in full, we have not yet learned much about its

Sir William Rowan Hamilton 1805-1865: A mathematician credited with thediscovery of quaternions, the first non-commutative algebra to be studied. Healso invented important new methods inMechanics.

actual behaviour. To extract con-crete physical information fromthe action we need to deriveequations of motion. At firstsight, it may not be entirely clearwhat is meant by the term ‘equa-tions of motion’ in the context ofan infinite dimensional model. The answer to this question lies in Hamilton’s extremalprinciple of classical mechanics:

Suppose that the dynamics of a classical point particle with coordinate x(t) is de-scribed by the classical Lagrangian L(x, x), and action S[x] =

RdtL(x, x). Hamilton’s

extremal principle states that the configurations x(t) that are actually realised are thosethat extremise the action, viz. S[x] = 0. This means that for any smooth curve, y(t),

lim!0

1

(S[x + y] S[x]) = 0, (1.4)

i.e. to first order in , the action has to remain invariant. Applying this condition, onefinds that it is fulfilled if and only if x(t) obeys Lagrange’s equation of motion (afamiliar result left here as a revision exercise)

d

dt(@xL) @xL = 0 (1.5)



In Eq. (1.3) we are dealing with a system of infinitely many degrees of freedom, (x, t).Yet Hamilton’s principle is general, and we may see what happens if (1.3) is subjected to anextremal principle analogous to Eq. (1.4). To do so, we must implement the substitution(x, t) ! (x, t) + (x, t) into Eq. (1.3) and demand that the contribution first order in vanishes. When applied to the specific Lagrangian (1.2), a substitution of the ‘varied’field leads to

S[ + ] = S[] +

Zdt

Z L

0

dxm ksa

2 @x @x

+ O(2).

Integrating by parts and demanding that the contribution linear in vanishes, one obtainsZdt

Z L

0

dxm ksa

2@2

x

= 0.

(Notice that the boundary terms associated with both t abnd x vanish identically – thinkwhy.) Now, since (x, t) was defined to be an arbitrary smooth function, the integralabove can only vanish if the term in parentheses is globally vanishing. Thus the equationof motion takes the form of a wave equation

m@2

t ksa2@2

x

= 0 (1.6)

The solutions of Eq. (1.6) have the general form +

(x + vt) + (x vt) where v =ap

ks/m, and ± are arbitrary smooth functions of the argument. From this we candeduce that the low energy elementary excitations of our model are lattice vibrationspropagating as sound waves to the left or right at a constant velocity v (see Fig. 1.3). Ofcourse, the trivial behaviour of our model is a direct consequence of its simplistic definition— no dissipation, dispersion or other non-trivial ingredients. Adding these refinementsleads to the general classical theory of lattice vibrations (see, e.g., Ref. [3]).

x=-t

+–

x=t

Figure 1.3: Schematic illustrating typical left and right moving excitations of the classicalharmonic chain.

. Info. Functional Analysis: Before proceeding further, let us briefly digress and revisitthe derivation of the equations of motion (1.6). Although straightforward, the calculation wasneither ecient, nor did it reveal general structures. In fact, what we did — expanding explicitlyto first order in the variational parameter — had the same status as evaluating derivativesby explicitly taking limits: f 0(x) = lim!0

(f(x + ) f(x))/. Moreover, the derivation madeexplicit use of the particular form of the Lagrangian, thereby being of limited use with regard to ageneral understanding of the construction scheme. Given the importance attached to extremalprinciples in all of field theory, it is worthwhile investing some e↵ort in constructing a moreecient scheme for general variational analysis of continuum theories. In order to carry out



this programme we first need to introduce a mathematical tool of functional analysis, viz. theconcept of functional di↵erentiation.

In working with functionals, one is often concerned with how a given functional behavesunder (small) variations of its argument function. In order to understand how answers to thesetypes of questions can be systematically found, it is helpful to temporarily return to a discreteway of thinking, i.e. to interpret the argument f of a functional F [f ] as the limit N ! 1 ofa discrete vector f = fn f(xn), n = 1, . . . N, where xn denotes a discretisation of thesupport of f (cf. Fig. 1.2 $ f). Prior to taking the continuum limit, N ! 1, f has the statusof a N -dimensional vector and F [f ] is a function defined over N -dimensional space. After thecontinuum limit, f becomes a function itself and F [f ] becomes a functional.

Now, within the discrete picture it is clear how the variational behaviour of functions is tobe analysed, e.g. the condition that, for all and all vectors g, the linear expansion of F [f + g]ought to vanish, is simply to say that the total derivative, rF [f ], at f has to be zero. Inpractice, one often expresses conditions of this type in terms of a certain basis. For example, ina Cartesian basis of N unit vectors, en, n = 1, . . . , N ,

F [f + g] = F [f ] + NX

n=1

(@fn

F [f ])gn + · · · , @fn

F [f ] lim!0

1

(F [f + en] F [f ]) . (1.7)

The total derivative of F is zero, if and only if 8n, @fn

F = 0.Taking the continuum limit of such identities will lead us to the concept of functional

di↵erentiation, a central tool in all areas of field theory. In the continuum limit, sums runningfrom 1 to N become integrals. The nth unit vector en becomes a function that is everywherevanishing save at one point where it equals 1, i.e. en ! x, where the function x(y) (xy).3

Thus, the continuum limit of Eq. (1.7) reads

F [f + g] = F [f ] +

Zdx

F [f ]

f(x)g(x) + O(2)

F [f ]

f(x) lim

!0

1

(F [f + x] F [f ]) . (1.8)

Here the second line represents the definition of the functional derivative, i.e. the generalisationof a conventional partial derivative to infinitely many dimensions. Experience shows that ittakes some time to get used to the concept of functional di↵erentiation. However, after somepractice it will become clear that this operation is not only extremely useful but also as easyto handle as conventional partial di↵erentiation. In particular, all rules known from ordinarycalculus (product-rule, chain-rule, etc.) immediately generalise to the functional case (as followsstraightforwardly from the way the functional derivative has been introduced). For example,the generalisation of the standard chain rule,

@fn

F [g[f ]] =Xm

@gm

F [g]g=g[f ]

@fn

gm[f ]

reads

F [g[f ]]

f(x)=

Zdy

F [g]

g(y)

g=g[f ]

g(y)[f ]

f(x). (1.9)

3If you find the singularity of the continuum version of the unit-vector dicult to accept, rememberthat the limit

Pn

hen

|fi = fn

!Rdy

x

(y)f(y) = f(x) enforces x

(y) = (x y).



Furthermore, given some functional F [f ], we can Taylor expand it as

F [f ] = F [0] +

Zdx

1

F [f ]

f(x1

)f(x

1

) +

Zdx

1

Zdx

2

1

2

2F [f ]

f(x2

)f(x1

)f(x

1

)f(x2

) + · · ·

Some basic definitions underlying functional di↵erentiation as well as their finite dimensionalcounterparts are summarised in the following table:

entity discrete continuumf vector function

F [f ] multi-dimensional function functionalCartesian basis en x

‘partial derivative’ @fn

F [f ]F [f ]

f(x)

After this preparation, let us re-examine the extremal condition for a general action S[x]by means of functional di↵erentiation. As follows from the definition of the functional deriva-tive (1.7), the action is extremal, if and only if 8x(t),

S[x]

x(t)=

Zdt0

L(x(t0), x(t0))

x(t)= 0.

Employing the definition of the action in terms of the Lagrangian and applying the chain rule(1.9), we find

Zdt0

L(x(t0), x(t0))

x(t)=

Zdt0h@L(x(t0), x(t0))

@x(t0)

(t0 t)z | @x(t0)

@x(t)+@L(x(t0), x(t0))

@x(t0)

dt0(t0 t)z |

@x(t0)

@x(t)

i,

From this result a rearrangement by integration obtains the familiar Euler-Lagrange equations

S[x]

x(t)=

@L

@x(t) d

dt

@L

@x(t)

= 0.

It is left as a straightforward exercise to show that the general equations of motion of aclassical continuum system with Lagrangian density L(, @x, ) is given by

S[]

(x, t)=

@L@

d

dt

@L@

d

dx

@L

@(@x)

(1.10)

Eq. (1.10) represents the generalisation of Lagrange’s equation of motion of point mechanics toclassical field theory (1.5). The particular application to the equations of motion of the simplephonon model (1.2) are illustrative of a general principle. All field theoretical models — bethey classical or quantum — are represented in terms of certain actions whose extremal fieldconfigurations play a fundamental role.

——————————————–After this digression, let us return to the discussion of the original model (1.2). As

mentioned above, the classical vibrational physics of solids can be formulated in terms ofmodels like (1.2) and its generalisations. On the other hand it is known (e.g. from theexperimental study of specific heat [3]) that various aspects of the physics of lattices arenon-classical and necessitate a quantum mechanical description. Hence, what is called foris an extension of the classical field theory to a quantum field theory.



1.1.2 Quantum Chain

The first question to ask is a conceptual one: how can a model like (1.2) be quantised ingeneral? As a matter of fact there exists a standard procedure of quantising Lagrangiancontinuum theories which closely resembles the quantisation of point particle mechanics.The first step is to introduce canonical momenta conjugate to the continuum degrees offreedom (coordinates), , which will later be used to introduce canonical commutation re-lations. The natural generalisation of the definition pn @x

n

L of point particle mechanicsto a continuum suggests

(x) @L@

(1.11)

or, more concisely, = @˙L. In common with , the canonical momentum, , is a

continuum degree of freedom. At each space point it may take an independent value.From the Lagrangian, we can define the Hamiltonian,

H[, ] Z

dx H[, ], H[, ] L[]

where H represents the Hamiltonian density. (All the quantities appearing in H areto be expressed in terms of and .) In particular, applied to the lattice model (1.2),

H(, ) =1

2m2 +

ksa2

2(@x)2.

where = m.In this form, the Hamiltonian can be quantised according to the rules: (i) promote the

fields (x) and (x) to operators: 7! , 7! , and (ii) generalise the canonical com-mutation relations of single-particle quantum mechanics, [pm, xn] = i~mn, according tothe relation4

[(x), (x0)] = i~(x x0) (1.12)

Operator-valued functions like and are generally referred to as quantum fields.Employing these definitions, we obtain the quantum Hamiltonian density

H[, ] =1

2m2 +

ksa2

2(@x)2. (1.13)

The Hamiltonian above represents a quantum field theoretical formulation of the problembut not yet a solution. In fact, the development of a spectrum of methods for the analysisof quantum field theoretical models will represent a major part of this lecture course. Atthis point the objective is merely to exemplify how physical information can be extractedfrom models like (1.13).

4Note that the dimensionality of both the quantum and classical continuum fields is compatible withthe dimensionality of the Dirac -function, [(x x0)] = [Length]1.



As with any function, operator-valued functions can be represented in a variety offorms. In particular they can be subjected to Fourier expansion,

k

k

1

L1/2

Z L

0

dx eikx

(x)(x)

,

(x)(x)

=1

L1/2

Xk

e±ikx

k

k

, (1.14)

whereP

k represents the sum over all Fourier coecients indexed by quantised coordinatesor “quasi-momenta” k = 2m/L, m 2 Z. (Do not confuse the momenta k with the‘operator momentum’ !) Note that the real classical field (x) quantises to a Hermitianquantum field (x) implying that k = †

k (and similarly for k) — exercise. In theFourier representation, the transformed field operators obey the canonical commutationrelations (exercise)

[k, k0 ] = i~kk0

When expressed in the Fourier representation, making use of the identity

Z L

0

dx (@)2 =Xk,k0

(ikk)(ik0k0)

k+k0,0z | 1

L

Z L

0

dx ei(k+k0)x=Xk

k2kk

Xk

k2|k|2!

together with a similar relation forR L

0

dx 2, the Hamiltonian assumes the “near-diagonal”form5

H =Xk

1

2mkk +

ksa2

2k2kk

. (1.15)

In this form, the Hamiltonian can be identified as nothing more than a superpositionof independent harmonic oscillators.6 This result is actu-ally not dicult to understand (see figure): Classically, thesystem supports a discrete set of wave excitations, each in-dexed by a wave number k = 2m/L. (In fact, we could haveperformed a Fourier transformation of the classical fields (x)and (x) to represent the Hamiltonian function as a superpo-sition of classical harmonic oscillators.) Within the quantumpicture, each of these excitations is described by an oscilla-tor Hamiltonian with a k-dependent frequency. However, itis important not to confuse the atomic constituents, also os-cillators (albethey coupled), with the independent collectiveoscillator modes described by H.

5As a point of notation, when expressed in terms of a complete orthonormal basis |mi, a generalHamiltonian can be expressed as a matrix, H

mn

= hm|H|ni. In the eigenbasis |↵i, the Hamiltonian issaid to be diagonalised, viz. H

↵

= h↵|H|i = E↵

↵

. In the present case, when expressed in the Fourierbasis, the matrix elements correlate only k with k.

6The only di↵erence between (1.15) and the canonical form of an oscillator Hamiltonian H = p2/2m+m!2x2/2 is the presence of the sub-indices k and k (a consequence of †

k

= k

). As we will showshortly, this di↵erence is inessential.



The description above, albeit perfectly valid, still su↵ers from a deficiency: Our anal-ysis amounts to explicitly describing the low-energy excitations of the system (the waves)in terms of their microscopic constituents (the atoms). Indeed the di↵erent contributionsto H keeps track of details of the microscopic oscillator dynamics of individual k-modes.However, it would be much more desirable to develop a picture where the relevant excita-tions of the system, the waves, appear as fundamental units, without explicit account ofunderlying microscopic details. (As with hydrodynamics, information is encoded in termsof collective density variables rather than through individual molecules.) As preparationfor the construction of this improved formulation of the system, let us temporarily focuson a single oscillator mode.

. Info. Revision of the quantum harmonic oscillator: Consider a standard harmonicoscillator (HO) Hamiltonian

H =p2

2m+

m!2

2x2 .

The few first energy levels n = ~!n+ 1

2

and the associated Hermite polynomial eigenfunctions

are displayed schematically in Fig. 1.4. In quantum mechanics, the HO has, of course, the statusof a single-particle problem. However, the fact that the energy levels are equidistant suggestsan alternative interpretation: One can think of a given energy state n as an accumulation ofn elementary entities, or quasi-particles, each having energy ~!. What can be said aboutthe features of these new objects? First, they are structureless, i.e. the only ‘quantum number’identifying the quasi-particles is their energy ~! (otherwise n-particle states formed of the quasi-particles would not be equidistant). This implies that the quasi-particles must be bosons. (Thesame state ~! can be occupied by more than one particle — see Fig. 1.4.)

!

Figure 1.4: Low-lying energy levels/states of the harmonic oscillator Hamiltonian.

This idea can be formulated in quantitative terms by employing the formalism of ladderoperators in which the operators p and x are traded for the pair of Hermitian adjoint operatorsa

pm!2~ (x+

im! p), a

† p

m!2~ (x i

m! p). Up to a factor of i, the transformation (x, p) ! (a, a†)is canonical, i.e. the new operators obey the canonical commutation relation

[a, a†] = 1 . (1.16)

More importantly, the a-representation of the Hamiltonian is very simple, viz.

H = ~!a†a+

1

2

, (1.17)

as can be checked by direct substitution. Suppose that we had been given a zero eigenvaluestate |0i of the operator a: a|0i = 0. As a direct consequence, H|0i = (~!/2)|0i, i.e. |0i is



identified as the ground state of the oscillator.7 The complete hierarchy of higher energy statescan now be generated by setting |ni (n!)1/2 (a†)n|0i.

. Exercise. Using the canonical commutation relation, verify that H|ni = ~!(n+ 1/2)|niand hn|ni = 1.

So far, we have succeeded merely in finding yet another way of constructing eigenstates of thequantum HO problem. However, the real advantage of the a-representation is that it naturallya↵ords a many-particle interpretation. Temporarily forgetting about the original definition ofthe oscillator, let us declare |0i to represent a ‘vacuum’ state, i.e. a state with no particlespresent. Next, imagine that a†|0i is a state with a single featureless particle (the operator a†

does not carry any quantum number labels) of energy ~!. Similarly, (a†)n|0i is considered asa many-body state with n particles, i.e. within the new picture, a† is an operator that createsparticles. The total energy of these states is given by ~! (occupation number). Indeed, it isstraightforward to verify that a†a|ni = n|ni, i.e. the Hamiltonian basically counts the numberof particles. While, at first sight, this may look unfamiliar, the new interpretation is internallyconsistent. Moreover, it fulfils our objective: it allows an interpretation of the excited states ofthe HO as a superposition of independent structureless entities.

The representation above illustrates the capacity to think about individual quantum prob-lems in complementary pictures. This principle finds innumerable applications in moderncondensed matter physics. To get used to it one has to realize that the existence of di↵erentinterpretations of a given system is by no means heretic but, rather, is consistent with the spiritof quantum mechanics. Indeed, it is one of the prime principles of quantum theories that there isno such thing as ‘the real system’ which underpins the phenomenology. The only thing that mat-ters is observable phenomena. For example, we will see later that the ‘fictitious’ quasi-particlestates of oscillator systems behave as ‘real’ particles, i.e. they have dynamics, can interact, bedetected experimentally, etc. From a quantum point of view there is actually no fundamentaldi↵erence between these objects and the ‘real’ particles.

——————————————–

1.1.3 Quasi-Particle Interpretation of the Quantum Chain

WIth this background, we may return to the harmonic chain and transform the Hamilto-nian (1.15) to a form analogous to (1.17) by defining the ladder operators8

ak r

m!k

2~

k +

i

m!kk

, a†

k r

m!k

2~

k i

m!kk

,

where !k = v|k|, and v = a(ks/m)1/2 denotes the classical sound wave velocity. With thisdefinition, applying the commutation relations, one finds that the ladder operators obey

7... as can be verified by explicit construction: Switching to a real space representation, thesolution of the equation [x + ~@

x

/(m!)]hx|0i = 0 obtains the familiar ground state wavefunction

hx|0i =p

2~/(m!)em!x

2/2~.

8As for the consistency of these definitions, recall that †k

= k

and †k

= k

. Under these conditionsthe second of the definitions below indeed follows from the first upon taking the Hermitian adjoint.



the commutation relations (characteristic of Bose particles)

[ak, a†k0 ] =

i

2~

i~kk0z | [k, k0 ] [k, k0 ]

= kk0 , (1.18)

[ak, ak0 ] =i

2~

[k, k0 ] + [k, k0 ]

= 0, [a†

k, a†k0 ] = 0.

With this definition, one finds that the Hamiltonian assumes the diagonal form

H =Xk

~!k

a†kak +

1

2

(1.19)

Equations (1.18) and (1.19) represent the final result of our analysis: The HamiltonianH takes the form of a sum a set of harmonic oscillators with characteristic frequencies!k. In the limit k ! 0 (i.e. long wavelength), one finds !k ! 0; excitations with thisproperty are said to be massless.

An excited state of the system is indexed by a set nk = (n1

, n2

, . . .) of quasi-particleswith energy !k. Physically, the quasi-particles of the harmonic chain are identified withthe phonon modes of the solid. A comparison with measured phonon spectra (Fig. 1.5)reveals that, at low momenta, !k |k| in agreement with our simplistic model (evenin spite of the fact that the spectrum was recorded for a three-dimensional solid withnon-trivial unit cell — universality!).

Figure 1.5: By applying energy and momentum conservation laws, one can determine thespectrum of the phonons from neutron scattering. The figure shows part of the phonon spectrumfor plutonium. The measurement is sensitive to both longitudinal (L) and transverse (T) acousticphonons. Notice that for small momenta, the dispersion is linear. (Figure from Joe Wong,Lawrence Livermore National Laboratory.)



1.2 †Quantum Electrodynamics (QED)

. Additional Example: As a second and important example of an analogous quantum fieldtheory, consider electrodynamics. In this lecture course quantum electrodynamics (QED) willplay comparatively little role. Nonetheless it is worthwhile to mention it briefly because

. QED is historically the oldest and still most successful field theory. (The QED result forthe anomalous magnetic moment of the electron agrees with experiment up to a precisionof O(106)!)

. Quantised electromagnetic fields (! photons) play a significant role in many areas ofcondensed matter physics.

In the following short discussion we merely wish to illustrate the basic principle of field quantisa-tion — in particular the parallels to the quantisation scheme employed in the previous example.For the evaluation of the resulting quantised theory we refer the reader to the literature. Anexcellent exposition of QED and its applications can be found, e.g., in Ryder’s text on QuantumField Theory.

The starting point of the quantisation scheme is again a classical variational principle. Inother words we start out from a formulation where the classical physics of electromagnetic fieldsis derived from a Lagrangian function. As shown within the framework of classical relativisticelectrodynamics the source-free Maxwell equations can be generated from the action

S[A] =

Zd4xL[A], L = 1

4FµF

µ ,

where Fµ = @µA @Aµ is the electromagnetic field tensor, and A = (,A)T is the 4-vectorpotential (c = 1). The Lagrangian above has the property of being (a) gauge invariant, and (b)exhibiting the solutions of the free Maxwell equations

@µFµ = 0

as its extremal field configurations.9

To work with the Lagrangian density L one needs to specify a gauge. (As a parentheticalremark we mention that the necessity to gauge fix is in fact a source of notorious dicultiesin gauge field theories in general. However, these problems are of little concern for the presentdiscussion.) Here we chose the so-called radiation or Coulomb gauge = 0, r · A = 0, therebyreducing the number of independent components of A from four to two. The next step towards aquantised theory is again to introduce canonical momenta. In analogy to section 1.1.2 we defineµ = @

˙Aµ

L which leads to

0 = @˙A0

L = 0, i = @˙Ai

L = @0Ai @iA0 = Ei,

where E is the electric field.10

9To see this, one applies the usual variational principle, S[A]A(x) = 0.

10The definitions above di↵er from the analogous equation (1.11) in so far as (a) the fields carry anadditional discrete index i = 1, . . . 4 — they are ‘vector’ rather than ‘scalar’ fields — and (b) that theindices appear as upper and sometimes as lower indices, where upper and lower indices are connectedwith each other by an application of the Minkowskii metric tensor. Both aspects are of little significancefor the present discussion.


1.2. †QUANTUM ELECTRODYNAMICS (QED) 13

Quantising the field theory now again amounts to introducing operators Ai 7! Ai, i 7!i, as well as canonical commutation relations between Ai and j . A natural Ansatz for thecommutation relations would be

[Ai(x, t), j(x0, t)] = [Ai(x, t), j(x0, t)] = iij(x x0). (1.20)

Yet a closer inspection reveals that these identities are in fact in conflict with the Coulomb gauge

!!

A(k)

k12

operator-valued

r · A = 0 (cf. Ryder, pp. 142). The way out is to replace ij by a more general symmetrictensor. However as this complication does not alter the general principle of quantisation wedo not discuss them any further here. The further construction of the theory is conceptuallyanalogous to the phonon model and will be sketched only briefly.

Again one introduces momentum ‘modes’ by Fourier transforming the field:

Ai(x) =

Zd3k

(2)32k0

X=1,2

()(k)ha()(k)eikx + a()†(k)eikx

i, (1.21)

where k2 k20

k2 = 0,11 ()(k) are polarisation vectors (cf. Fig. ??) obeying k · ()(k) =0 (Coulomb gauge!). The specific form of the integration measure follows from the generalcondition of relativistic invariance (cf. Ryder, pp. 143). Substituting this representation intothe Hamilton operator of the field theory one obtains

H =X

Zd3k

(2)32k0

k0

2a()†(k)a()(k). (1.22)

As Eq. (1.19), this is an oscillator type Hamiltonian. The di↵erence is that the operators agenerate oscillator quanta of the quantised electromagnetic field, so-called transverse photons,rather than phonons. Eq. (1.21) represents the decomposition of the free quantised vectorpotential in terms of photons. As with phonons, the oscillator quanta of the electromagneticfield can also be interpreted as particles. In this sense, the decomposition (1.21) represents thebridge between the wave and the particle description of electrodynamics. For discussions of thephysical applications of the theory — in both high energy and condensed matter physics — werefer the reader to the literature, e.g. Ryder (high energy) and Ref. [1] (condensed matter).

11The condition k·k = 0 follows from the Coulomb gauge formulation of Maxwell’s equations, @µ

@µA

=0.


Chapter 2

Second Quantisation

In this section we introduce the method of second quantisation, the basic framework forthe formulation of many-body quantum systems. The first part of the section focuses onmethodology and notation, while the remainder is devoted to physically-motivated appli-cations. Examples of the operator formalism are taken from various fields of quantumcondensed matter.

2.1 Notations and Definitions

Second quantisation provides a basic and ecient language in which to formulate many-particle systems. As such, extensive introductions to the concept can be found throughoutthe literature (see, e.g., Feynman’s text on Statistical Mechanics [10]). The first partof this section will be concerned with the introduction of the basic elements of secondquantisation, while the remainder of this section will be concerned with developing fluencyin the method by addressing a number of physical applications.

Let us begin by defining the (normalised) wavefunctions | i and corresponding eigen-values of the single-particle Hamiltonian H, viz.

H| i = | i.

With this definition, populating states 1 and 2, the symmetrised (normalised) two-particlewavefunction for fermions and bosons is respectively given by

F

B

(x1

, x2

) =1p2

( 1

(x1

) 2

(x2

) 2

(x1

) 1

(x2

)) .

In the Dirac bracket representation, we can write

|1, 2iF

B

1p2

(| 1

i | 2

i | 2

i | 1

i) .

More generally, a symmetrised N -particle wavefunction of fermions ( = 1) or bosons( = +1) is expressed in the form

|1

,2

, . . .Ni 1pN !Q1

=0

n!

XP

P | P1

i | P2

i . . . | PN

i


20 CHAPTER 2. SECOND QUANTISATION

where n is the total number of particles in state (for fermions, Pauli exclusion en-forces the constraint n = 0, 1, i.e. n! = 1) – see Fig. 2.1. The summation runs overall N ! permutations of the set of quantum numbers

1

, . . .N, and P denotes the

Enrico Fermi 1901-1954: 1938Nobel Laureate in Physics for hisdemonstrations of the existence ofnew radioactive elements producedby neutron irradiation, and for hisrelated discovery of nuclear reactionsbrought about by slow neutrons.

parity, defined as the number of trans-positions of two elements which bringsthe permutation (P

1

, P2

, · · · PN) backto the ordered sequence (1, 2, · · · N).Note that the summation over per-mutations is necessitated by quantummechanical indistinguishability: forbosons/fermions the wavefunction hasto be symmetric/anti-symmetric under particle exchange. It is straightforward to confirmthat the prefactor 1p

N !

Q

n

!

normalises the many-body wavefunction. In the fermionic

case, the many-body wavefunction is known as a Slater determinant.

n

43

321

1

1

1

1

1

1

0

0

n

0

5

bosons fermions

Figure 2.1: Schematic showing typical occupation numbers for a generic fermionic and bosonicsystem.

The expression above makes it clear that this ‘first quantised’ representation of themany-body wavefunction is clumsy. We will see that the second quantisation provides themeans to heavily condense the representation. Let us define the vacuum state |i, andintroduce a set of field operators a together with their adjoints a†

, as follows:1

a|i = 0,1pQ n!

a†

N

· · · a†1

|i = |1

,2

, . . .Ni (2.1)

Physically, the operator a† creates a particle in state while the operate a annhilates it.

These definitions are far from innocent and deserve some qualification. Firstly, in ordernot to be at conflict with the symmetry of the wavefunction, the operators a have tofulfill the commutation relations,h

a, a†µ

i

= ,µ,ha, aµ

i

= 0,ha†, a

†µ

i

= 0 (2.2)

1As before, it will be convenient to represent these operators without a circumflex.


2.1. NOTATIONS AND DEFINITIONS 21

where [A, B] ABBA is the commutator = 1 (anticommutator = 1) for bosons(fermions).2 The most straightforward way to understand this condition is to check thatthe definition |, µi = a†

a†µ|i and property |, µi = |µ,i in fact necessitate Eqs. (2.2).

Yet even if (2.2) is understood, the definitions above remain non-trivial. Actually, quite astrong statement has been made: for any N , the N -body wavefunction can be generatedby an application of a set of N-independent operators to a unique vacuum state. In order tocheck that Eqs. (2.1) and (2.2) actually represent a valid definition, including, for instance,the right symmetrisation and normalisation properties of N -body wave functions, variousconsistency checks have to be made.

Based on Eqs. (2.1) and (2.2), a formal definition of the general many-body or Fockspace can now be given as follows. First define FN to be the linear span of all N -particle states |

1

, · · ·Ni = 1pQ

n

a†

N

· · · a†1

|i. The Fock space F is then defined as

the direct sum 1N=0

FN (see Fig. 2.2).3 A general state |i of the Fock space is, therefore,

+ a+a

a

0a

...a

F 1F 2 F 0

Figure 2.2: Visualisation of the generation of the Fock-subspaces FN by repeated action ofcreation operators onto the vacuum space F

0

.

a linear combination of states with any number of particles. To turn these rather abstractdefinitions into a valuable tool for practical computation we need to put them into relationwith standard operations performed in quantum mechanics. In particular we have tospecify how changes from one single-particle basis to another e↵ect the operatoralgebra a, and in what way standard operators of (many-body) quantum mechanicscan be represented in terms of the a s:

.Change of basis: Using the resolution of identity, id =P1

=0

|ih|, the relations |i =P |ih|i, |i a†

|i, and |i a†˜|i immediately give rise to the transformation

law

a†˜

=P

h|ia†, a

˜ =P

h|ia (2.3)

In many applications we are not dealing with a set of discrete quantum numbers (spin,quantised momenta, etc.), but rather with a continuum (a continuous position coordinate,say). In these cases, the quantum numbers are commonly denoted in a bracket notationa ; a(x) =

Phx|ia, and the summations appearing in the transformation formula

above become integrals.

2As a convention, when unspecified by , the notation [·, ·] will be used to denote the commutator and·, · the anticommutator.

3Here, the symbol of the direct sum is used to show that each “submodule”FN

is linearly indepen-dent.



Example: The transformation from the coordinate to the momentum representationin a finite one-dimensional system of length L would read

ak =

Z L

0

dx hk|xia(x), a(x) =Xk

hx|kiak,

where hk|xi hx|ki = 1

L1/2

eikx, cf. Fourier series expansion.

. Representation of operators (one-body): Single particle or one-body operatorsO

1

acting in a N -particle Hilbert space, FN , generally take the form O1

=PN

n=1

on,where on is an ordinary single-particle operator acting on the n-th particle. A typical

David Hilbert 1862-1943: His work ingeometry had the greatest influence inthat area after Euclid. A systematic studyof the axioms of Euclidean geometry ledHilbert to propose 21 such axioms andhe analysed their significance. He con-tributed to many areas of mathematics.

example is the kinetic energy op-erator T =

Pn

p2n

2m , where pn isthe momentum operator acting onthe n-th particle. Other examplesinclude the one-particle potentialoperator V =

Pn V (xn), where

V (x) is a scalar potential, thetotal spin-operator

Pn Sn, etc.

Since we have seen that, by applying field operators to the vacuum space, we can gener-ate the Fock space in general and any N -particle Hilbert space in particular, it must bepossible to represent any operator O

1

in an a-representation.Now, although the representation of n-body operators is after all quite straightforward,

the construction can, at first sight, seem daunting. A convenient way of finding such arepresentation is to express the operator in terms of a basis in which it is diagonal, andonly later transform to an arbitrary basis. For this purpose it is useful to define theoccupation number operator

n = a†a (2.4)

with the property that, for bosons or fermions (exercise), n (a†)

n|i = n (a†)

n|i, i.e.the state (a†

)n|i is an eigenstate of the number operator with eigenvalue n. When acting

upon a state |1

,2

, · · ·Ni, it is a straightforward exercise to confirm that the numberoperator simply counts the number of particles in state ,

n|1,2, · · ·Ni = a†a

1pQ n!

a†

N

· · · a†1

|i =NXi=1

i

|1

,2

, · · ·Ni.

Let us now consider a one-body operator, O1

, which is diagonal in the orthonormalbasis |i, o =

P o|ih|, o = h|o|i. With this definition, one finds

h01

, · · ·0N |O1

|1

, · · ·Ni =

NXi=1

oi

!h0

1

, · · ·0N |1

, · · ·Ni

= h01

, · · ·0N |1X=0

on|1, · · ·Ni.


2.1. NOTATIONS AND DEFINITIONS 23

Since this equality holds for any set of states, we obtain the operator or second quantisedrepresentation O

1

=P1

=0

on =P1

=0

h|o|ia†a. The result is straightforward; a

Wolfgang Pauli 1900-1958:1945 Nobel Laureate in Physicsfor the discovery of the Exclu-sion Principle, also called thePauli Principle.

one-body operator engages a single particleat a time — the others are just spectators.In the diagonal representation, one simplycounts the number of particles in a state and multiplies by the corresponding eigen-value of the one-body operator. Finally, bytransforming from the diagonal representa-tion to a general basis, one obtains the result,

O1

=Xµ

hµ|ioh|ia†µa =

Xµ

hµ|o|ia†µa (2.5)

Formally, the one-body operator, O1

, scatters a particle from a state into a state µ withprobability amplitude hµ|o|i.

Examples: The total spin operator is given by

S =X↵↵0

a†↵S↵↵0a↵0 , S↵↵0 =

1

2↵↵0 (2.6)

where ↵ =", # is the spin quantum number, denotes the set of additional quantumnumbers (e.g. coordinate), and denotes the vector of Pauli spin matrices

x =

0 11 0

, y =

0 ii 0

, z =

1 00 1

, (2.7)

i.e. Sz = 1

2

P(n" n#), and S+ =

P a†

"a#.Second quantised in the position representation, the one-body Hamiltonian is given

as a sum of kinetic and potential energy as (exercise)

H = T + V =

Zdx a†(x)

p2

2m+ V (x)

a(x)

where p = i~@x. (Note that the latter is easily proved by expressing the kinetic energyin the diagonal (i.e. momentum) representation — see problem set.)

Finally, the total occupation number operator is defined as N =R

dx a†(x)a(x).

. Representation of operators (two-body): Two-body operators O2

are needed todescribe pairwise interactions between particles. Although pair-interaction potentials arestraightforwardly included into classical many-body theories, their embedding into con-ventional many-body quantum mechanics is made awkward by particle indistinguishabil-ity. As compared to the conventional description, the formulation of interaction processeswithin the language of second quantisation is considerably more straightforward.

Initially, let us consider particles subject to the symmetric two-body potential V (x, x0) V (x0, x). Acting on two-particle states, the operator is given by

V (2) =1

2

Zdx

Zdx0 |x, x0iV (x, x0)hx, x0|. (2.8)



Our aim is to find an operator V in second quantised form whose action on a many-bodystate gives

V |x1

, x2

, · · · xNi =NX

n<m

V (xn, xm)|x1

, x2

, · · · xNi =1

2

NXn 6=m

V (xn, xm)|x1

, x2

, · · · xNi.

Comparing this expressions with (2.8) one might immediately guess that

V =1

2

Zdx

Zdx0 a†(x)a†(x0)V (x, x0)a(x0)a(x).

That this is the correct answer can be confirmed by applying the operator to a many-bodystate. We first note that

a(x0)a(x)|x1

, x2

, · · · xNi = a(x0)NX

n=1

n1(x xn)|x1

, x2

, · · · (no xn) · · · xNi

=NX

n=1

n1(x xn)NX

m=1,(m 6=n)

mn(x0 xm)|x

1

, x2

, · · · (no xn, xm) · · · xNi

where

mn =

m1 if m < nm if m > n

.

Then, making use of the identity

a†(x)a†(x0)a(x0)a(x)|x1

, x2

, · · · xNi

=NX

m 6=n

n1mn(x xn)(x0 xm)|x, x0, x

1

, x2

, · · · (no xn, xm) · · · xNi

=NX

m 6=n

n1mn(x xn)(x0 xm)|xn, xm, x

1

, x2

, · · · (no xn, xm) · · · xNi

=NX

m 6=n

(x xn)(x0 xm)|x

1

, x2

, · · · xNi,

multiplying by V (x, x0)/2, and integrating over x and x0, one confirms the validity of theexpression. It is left as an exercise to confirm that the expression, 1

2

RdxR

dx0 V (x, x0)n(x)n(x0)although a plausible candidate, does not reproduce the two-body operator.

More generally, turning to a non-diagonal basis, it is easy to confirm that a generaltwo-body operator can be expressed in the form

O2

=X

0µµ0

Oµ,µ0,,0a†µ0a†

µaa0 (2.9)


2.2. APPLICATIONS OF SECOND QUANTISATION 25

where Oµ,µ0,,0 hµ, µ0|O2

|,0i.In principle one may proceed in the same manner and represent general n-body in-

teractions in terms of second quantised operators. However, as n > 2 interactions rarelyappear, we refer to the literature for discussion.

This completes our formal introduction to the method of second quantisation. Tomake these concepts seem less abstract, the remainder of this section is concerned withthe application of this method to a variety of problems.

2.2 Applications of Second Quantisation

Although the second quantisation is a representation and not a solution, its applicationoften leads to a considerable simplification of the analysis of many-particle systems. Toemphasize this fact, and to practice the manipulation of second quantised operators, weturn to several applications. The first example is taken from the physics of correlatedelectron systems, and will engage the manipulation of fermionic creation and annihilationoperators. The second example involves the study of quantum magnetism within theframework of boson creation and annihilation operators. However, before getting to theseapplications, let us first go back and reinterpret our analysis of phonon modes in thequantum chain.

2.2.1 Phonons

Although, at the time, we did not specify in which Hilbert space the field operators ak

act, the answer is that the representation space is again a Fock space; this time a Fockspace of phonons or, more formally, of oscillator states. In contrast to what we’ll find forthe fermion case below, the Fock space in the phonon problem does not have an a prioriinterpretation as a unification of physical N -particle spaces. However, outgoing froma vacuum state, it can be constructively generated by applying the oscillator creationoperators a†

k to a unique vacuum state:

. Info. Define a ground or vacuum state |i by requiring that all operators ak annihilateit. Next define F

0

to be the space generated by |i. We may then introduce a set of states|ki a†k|i, k = 0, 2/L, . . . by applying oscillator creation operators to the vacuum. Physically,the state |ki has the significance of a single harmonic oscillator quantum excited in mode k. Inother words, all oscillator states k0 6= k are in their ground state, whilst mode k is in the firstexited state. The vector space generated by linear combinations of states |ki is called F

1

. Thisprocedure can be iterated in an obvious manner. Simply define the space FN to be generated byall states a†k

1

. . . a†kN

|i |k1

, . . . , kN i. The spaces FN can be defined more concisely by sayingthat they are the eigenspaces of the occupation number operator with eigenvalue N . Finally,the Fock space is just the direct sum of all FN , F 1

N=0

FN . By construction, the applicationof any one a†k or ak to states 2 F does not leave F . A closer analysis actually shows that thecorresponding Fock space F represents a proper representation space for the operators ak. Aparticle interpretation of the phonon states can now be naturally introduced by saying that theFock space sector FN represents a space of bosonic N -particle states. Application of a†k (ak) to



a state 2 FN creates (annihilates) a particle (cf. Fig. 2.2).——————————————–

2.2.2 Interacting Electron Gas

As a second example, we will cast the Hamiltonian of the interacting electron gas insecond quantised form. To emphasize the utility of this approach, in the next section wewill use it to explore the phase diagram of a strongly interacting electron gas. In doingso, we will uncover the limitations of the “nearly free electron theory” of metals.

As we have seen, in second quantised notation, the non-interacting Hamiltonian of aone-dimensional system of electrons subject to a lattice potential is given by

H(0) =

ZdxX

c†(x)

p2

2m+ V (x)

c(x),

where the fermionic electron field operators obey the anticommutation relations [c(x), c†0(x0)]+

= (x x0) 0 . The field operators act on the ‘big’ many-particle Fock space, F =1

N=0

FN . Each N -particle space FN is spanned by states of the form c†N

(xN) · · · c†1

(x1

)|iwhere the ‘no-particle’ state or vacuum |i is annihilated by all operators c(x).

Applying a two-body Coulomb interaction potential, 1

2

Pi 6=j

e2

|xi

xj

| , where xi denotesthe position of the i-th electron, the total many-body Hamiltonian takes the second quan-tised form

H = H(0) +1

2

Zdx

Zdx0X0

c†(x)c†0(x0)e2

|x x0|c0(x0)c(x) (2.10)

. Exercise. Setting V (x) = 0 and switching to the Fourier basis, reexpress the Coulomb

interaction. Show that the latter is non-diagonal, and scatters electrons between di↵erent quasi-momentum states — see Fig. 2.3.

k’,’

k’+q,’ k–q,

k,V(q)

Figure 2.3: Feynman diagram-matic representation of the two-bodyCoulomb interaction.

Having introduced both the field operatorsthemselves and their representation spaces, we arein a position to point out certain conceptual analo-gies between the model theories discussed above.In each case we have described a physical systemin terms of a theory involving a continuum of op-erators, (x) (phonons) and c(x) (electrons). Ofcourse there are also important di↵erences betweenthese examples. Obviously, in the phonon theory,we are dealing with bosons whilst the electron gasis fermionic. However, by far the most importantdi↵erence is that the first example has been a freefield theory. That means that the Hamiltoniancontained field operators at quadratic order but no higher. As a rule, free field theo-ries can be solved (in a sense that will become clear later on) straightforwardly. The



fermionic model, however, represents a typical example of an interacting field theory.There are terms of fourth order in the field operators which arose from the Coulombinteraction term. As compared to free theories, the analysis of interacting theories isinfinitely harder, a fact that will surely become evident later on.

To develop some fluency in the manipulation of second quantised field operators wewill continue by exploring the ‘atomic limit’ of a strongly interacting electron gas. Indoing so, we will derive a model Hamiltonian which has served as a paradigm for thestudy of correlated electron systems.

2.2.3 Tight-binding theory and the Mott transition

According to the conventional band picture of non-interacting electrons, a system witha half-filled band of valence electron states is metallic. However, the strong Coulombinteraction of electrons can induce a phase transition to a (magnetic) insulating electron‘solid’ phase (much as interactions can drive the condensation of a classical liquid into asolid). To explore the nature of this phenomenon, known as the Mott transition afterSir Neville Mott (formerly of the Cavendish Laboratory), it is convenient to reexpress theinteracting Hamiltonian in a tight-binding approximation.

!0

!1

"0

"1

V(x)

x

Es=1

s=0 "0A

E

"0B

"1A

"1B

(n!1)a

E!/ a

E

k0

a

x

(n+1)ana

Figure 2.4: Infinitely separated, each lattice site is associated with a set of states, s = 0, 1, · · ·,bound to the ion core. Bringing together just two atoms, the orbitals weakly overlap and hy-bridise into bonding and anti-bonding combinations. Bringing together a well-separated latticeof atoms, each atomic orbital broadens into a delocalised band of Bloch states indexed by aquasi-momentum k from the Brillouin zone and an orbital or band index s.

To develop an e↵ective Hamiltonian of the strongly interacting electron system webegin by considering a lattice of very widely spaced (almost isolated) atoms — the atomic



limit. In the simplest non-interacting picture, the overlap of the outermost electrons(albeit exponentially weak) leads to a hybridisation of the electronic orbitals and leadsto the ‘delocalisation’ of a narrow band of extended states (see Fig. 2.4). These Bloch

Felix Bloch 1905-1983: 1952 No-bel Laureate in Physics for the de-velopment (with Edward M. Purcell)of new methods for nuclear magneticprecision measurements and discov-eries in connection therewith.

states ks(x) of H(0), which carry aquasi-momentum index k and a bandor orbital index s = 0, 1, · · ·, provide aconvenient basis with which to expandthe interaction. We can, in turn, de-fine a set of local Wannier orbitals

c†ns|iz | | nsi 1p

N

B.Z.Xk2[/a,/a]

eikna

c†ks|iz|| ksi , | ksi 1p

N

NXn=1

eikna| nsi

where N denotes the total number of primitive lattice sites (with periodic boundaryconditions). Here the sum on k runs over the N k-points spanning the Brillouin zone, i.e.k = 2m/aN with integers N/2 < m N/2.

!n0(x)

(n!1)a

x

(n+1)ana

Figure 2.5: Diagram illustrating the weak over-lap of Wannier states in the atomic limit.

If the lattice is very widely spaced, theWannier state ns will di↵er little from thes-th bound state of an isolated atom atx = na (see Fig. 2.5). Restricting attentionto the lowest band s = 0, and restoring thespin degrees of freedom , the field opera-tors associated with the Wannier functionsare defined by

c†n|iz|| ni =

Z L=Na

0

dx

c†(x)|iz||xi hx| ni,

i.e.

c†n Z L

0

dx n(x)c†(x), c†(x) =NX

n=1

n(x)c†n. (2.11)

Physically, c†n can be interpreted as an operator creating an electron with spin at siten in the lowest band. Since the transformation (2.11) is unitary, it is straightforwardto confirm that the operators cn, and c†n obey fermionic anticommutation relations[cn, c

†m0 ]

+

= 0nm.In the presence of a two-body Coulomb interaction, a substitution of the field operators

in Eq. (2.10) by Wannier operators generates the generalised tight-binding Hamiltonian(exercise)

H = Xmn

X

tmnc†mcn +

Xmnrs

X0

Umnrs c†mc†n0cr0cs0



where the “hopping” matrix elements are given by

tmn = h m|H(0)| ni = 1

N

Xk

ei(nm)kak = tnm,

and the interaction parameters are set by (exercise)

Umnrs =1

2

Z L

0

dx

Z L

0

dx0 m(x)

n(x0)

e2

|x x0| r(x0) s(x).

Physically tmn represents the probability amplitude for an electron to transfer (hop) froma site m to a site n.

. Exercise. Show that, in the Fourier basis, ck = 1pN

Pn e

inkacn, the non-interacting

Hamiltonian takes the diagonal form H(0) =P

B.Z.k kc

†kck.

Expressed in the Wannier basis, the representation above is exact (at least for statescontained entirely within the lowest band). However, for a widely spaced lattice, most ofthe matrix elements of the general tight-binding model are small and can be neglected.Focusing on the most relevant:

. The direct terms Umnnm Vmn involve integrals over square moduli of Wannierfunctions and couple density fluctuations at di↵erent sites,X

m 6=n

Vmnnmnn,

where nm =P

c†mcm. Such terms have the capacity to induce charge densityinstabilities. Here we will focus on transitions to a magnetic phase where suchcontributions are inconsequential and can be safely neglected.

. A second important contribution derives from the exchange coupling which in-duces magnetic spin correlations. Setting JF

mn Umnmn, and making use of theidentity ↵ · = 2↵ ↵, one obtains (exercise)

Xm 6=n

X0

Umnmnc†mc

†n0cm0cn = 2

Xm 6=n

JFmn

Sm · Sn +

1

4nmnn

.

Such contributions tend to induce weak ferromagnetic coupling of neighbouringspins (i.e. JF > 0). Physically, the origin of the coupling is easily understood asderiving from a competition between kinetic and potential energies. By aligningwith each other and forming a symmetric spin state, two electrons can reduce theirpotential energy arising from their mutual Coulomb repulsion. To enforce the anti-symmetry of the two-electron state, the orbital wavefunction would have to vanishat x = x0 where the Coulomb potential is largest. This mechanism is familiar fromatomic physics where it is manifest as Hund’s rule.



. However, in the atomic limit where the atoms are well-separated and the overlapbetween neighbouring orbitals is weak, the matrix elements tij and JF

ij are expo-nentially small in the interatomic separation. By contrast the ‘on-site’ Coulomb orHubbard interactionX

m

X

Ummmmc†mc†m0cm0cm = U

Xm

nm"nm#,

where U 2Ummmm, increases as the atomic wavefunctions become more localised.

Therefore, dropping the constant energy o↵-set 0

= tnn, in the atomic limit, a stronglyinteracting many-body system of electrons can be described e↵ectively by the (single-band) Hubbard Hamiltonian

H = tXhmni

X

c†mcn + UXm

nm"nm# (2.12)

where we have introduced the notation hmni to indicate a sum over neighbouring latticesites, and t = tmn (assumed real and usually positive). In hindsight, a model of this struc-ture could have been guessed on phenomenological grounds from the outset. Electronstunnel between atomic orbitals localised on individual lattice sites and experience a localCoulomb interaction with other electrons.

Deceptive in its simplicity, the Hubbard model is acknowledged as a paradigm of strongelectron correlation in condensed matter. Yet, after forty years of intense investigation,the properties of this seemingly simple model system — the character of the ground stateand nature of the quasi-particle excitations — is still the subject of heated controversey(at least in dimensions higher than one — see below). Nevertheless, given the importanceattached to this system, we will close this section with a brief discussion of the remarkablephenomenology that is believed to characterise the Hubbard system.

As well as dimensionality, the phase behaviour of the Hubbard Hamiltonian is char-acterised by three dimensionless parameters; the ratio of the Coulomb interaction scaleto the bandwidth U/t, the particle density or filling fraction n (i.e. the average numberof electrons per site), and the (dimensionless) temperature, T/t. The symmetry of theHamiltonian under particle–hole interchange allows one to limit consideration to densitiesin the range 0 n 1 while densities 1 < n 2 can be inferred by ‘reflection’.

Focussing first on the low temperature system, in the dilute limit n 1, the typicalelectron wavelength is greatly in excess of the particle separation and the dynamics is free.Here the local interaction presents only a weak perturbation and one can expect the prop-erties of the Hubbard system to mirror those of the weakly interacting nearly free electronsystem. While the interaction remains weak one expects a metallic behaviour to persist.By contrast, let us consider the half–filled system where the average site occupancy isunity. Here, if the interaction is weak U/t 1, one may again expect properties remi-niscent of the weakly interacting electron system.4 If, on the other hand, the interactionis very strong U/t 1, site double occupancy is inhibited and electrons in the half–filled

4In fact, one has to exercise some caution since the commensurability of the Fermi wavelength withthe lattice can initiate a transition to an insulating spin density wave state characterised by a smallquasi-particle energy gap — the Slater Instability



Figure 2.6: Conductivity of Cr–dopedV

2

O3

as a function of decreasing pres-sure and temperature. At tempera-tures below the Mott–Hubbard transi-tion point (Pc = 3738bar, Tc = 457.5K)the conductivity reveals hysteretic be-haviour characteristic of a first ordertransition. Reproduced from Limeletteet al., Universality and critical behaviorat the Mott transition, Science 302, 89(2003).

Sir Neville Mott 1905–1996: 1977Nobel Laureate in Physics (withPhilip W. Anderson and John H. vanVleck) for their fundamental theoret-ical investigations of the electronicstructure of magnetic and disorderedsystems.

system become ‘jammed’: migrationof an electron to a neighbouring latticesite necessitates site double occupancyincurring an energy cost U . In thisstrongly correlated phase, the mutualCoulomb interaction between the elec-trons drives the system from a metalto an insulator.

. Info. Despite the ubiquity of the experimental phenomenon (first predicted in a cele-brated work by Mott) the nature of the Mott–Hubbard transtion from the metallic to theinsulating phase in the half–filled system has been the subject of considerable debate. In theoriginal formulation, following a suggestion of Rudolf Peierls, Mott conceived of an insulatorcharacterised by two ‘Hubbard bands’ with a bandwidth t separated by a charge gap U .5

States of the upper band engage site double occupancy while those states that make up thelower band do not. The transition between the metallic and insulating phase was predicted tooccur when the interaction was suciently strong that a charge gap develops between the bands.Later, starting from the weakly interacting Fermi–liquid, Brinkman and Rice6 proposed that thetransition was associated with the localisation of quasi–particles created by an interaction-drivenrenormalisation of the e↵ective mass. Finally, a third school considers the transition to the Mottinsulating phase as inexorably linked to the development of magnetic correlations in the weakcoupling system — the Slater instability.

——————————————–To summarise, we have shown how the method of second quantisation provides a

useful and ecient way of formulating and investigating interacting electron systems. Inthe next section we will employ methods of second quantisation involving bosonic degreesof freedom to explore the collective excitations of quantum magnets.

5N. F. Mott, Proc. Roy. Soc. A 62, 416 (1949) — for a review see, e.g. N. F. Mott, Metal–Insulatortransition, Rev. Mod. Phys. 40, 677 (1968) or N. F. Mott, Metal–Insulator Transitions, 2nd ed. (Taylorand Francis, London, 1990).

6W. Brinkman and T. M. Rice, Application of Gutzwiller’s variational method to the metal–insulatortransition, Phys. Rev. B 2, 4302 (1970).



2.2.4 Quantum Spin Chains

In the previous section, emphasis was placed on charging e↵ects generated by Coulombinteraction. However, as we have seen, Coulomb interaction may also lead to the indirectgeneration of magnetic interactions of both ferromagnetic and antiferromagnetic charac-ter. To address the phenomena brought about by quantum magnetic correlations, it isinstructive to begin by considering systems where the charge degrees of freedom are frozenand only spin excitations remain. Such systems are realized, for example, in Mott insu-lators where magnetic interactions between the local moments of localised electrons aremediated by virtual exchange processes between neighbouring electrons. Here, one candescribe the magnetic correlations through models of localised quantum spins embeddedon lattices. We begin our discussion with the ferromagnetic spin chain.

Quantum Ferromagnet

The quantum ferromagnetic chain is specified by the Heisenberg Hamiltonian

H = JXm

Sm · Sm+1

(2.13)

where J > 0, and Sm represents the quantum mechanical spin operator at lattice site m.

Werner Heisenberg 1901–1976: 1932 Nobel Laureate inPhysics “for the creation ofquantum mechanics, the appli-cation of which has, inter alia,led to the discovery of the al-lotropic forms of hydrogen”.

In section 2.1 (cf. Eq. (2.6)) the quantummechanical spin was represented throughan electron basis. However, one can con-ceive of situations where the spin sitting atsite m is carried by a di↵erent object (e.g.an atom with non–vanishing magnetic mo-ment). At any rate, for the purposes of ourpresent discussion, we need not specify themicroscopic origin of the spin. All we need to know is (i) that the lattice operators Si

m

obey the SU(2) commutator algebra (for clarity, we have set ~ = 1 in this section)

[Sim, Sj

n] = imnijkSk

n (2.14)

characteristic of quantum mechanical spins, and (ii) that the total spin at each lattice siteis S.7

Now, due to the positivity of the coupling constant J , the Hamiltonian favours con-figurations where the spins at neighbouring sites are aligned in the same direction (cf.Fig. 2.7). A ground state of the system is given by |i m|Smi, where |Smi representsa state with maximal spin–z component: Sz

m|Smi = S|Smi. We have written ‘a’ groundstate instead of ‘the’ ground state because the system is highly degenerate: A simulta-neous change of the orientation of all spins does not change the ground state energy, i.e.the system posesses a global spin rotation symmetry.

7Remember that the finite–dimensional representations of the spin operator are of dimension 2S + 1where S may be integer or half integer. While a single electron has spin S = 1/2, the total magneticmoment of electrons bound to an atom may be much larger.



Figure 2.7: Schematic showing the spin con-figuration of an elementary spin-wave excita-tion from the spin polarized ground state.

. Exercise. Compute the energy expectation value of the state |i. Defining global

spin operator as S P

m Sm, consider the state |↵↵↵i exp(i(/2)↵↵↵ · S)|i. Making use of

the Baker-Hausdor↵ identity, eiˆOAei ˆO = A + i[O, A] + (i)2

2!

[O, [O, A]] + · · · or otherwise,verify that the state |↵↵↵i is degenerate with |i. Explicitly compute the state |(1, 0, 0)i. Con-vince yourself that for general ↵↵↵, |↵↵↵i can be interpreted as a state with rotated quantisation axis.

As with our previous examples, we expect that a global continuous symmetry willinvolve the presence of energetically low–lying exciations. Indeed, it is obvious that inthe limit of long wavelength , a weak distortion of a ground state configuration (cf.Fig. 2.7) will cost vanishingly small energy. To quantitatively explore the physics of thesespin–waves, we adopt a ‘semi–classical’ picture, where the spin S 1 is assumed tobe large. In this limit, the rotation of the spins around the ground state configurationbecomes similar to the rotation of a classical magnetic moment.

. Info. To better understand the mechanism behind the semi–classical approximation,consider the Heisenberg uncertainty relation, SiSj |h[Si, Sj ]i| = ijk|hSki|, where Si isthe root mean square of the quantum uncertainty of spin component i. Using the fact that

|hSki| S, we obtain for the relative uncertainty, Si/S, Si

SSj

S SS2

S1! 0, i.e. for S 1,quantum fluctuations of the spin become less important.

——————————————–In the limit of large spin S, and at low excitation energies, it is natural to describe the

ordered phase in terms of small fluctuations of the spins around their expectation values(cf. the description of the ordered phase of a crystal in terms of small fluctuations of theatoms around the ordered lattice sites). These fluctuations are conveniently representedin terms of spin raising and lowering operators: with S±

m Sxm± iSy

m, it is straightforwardto verify that

[Szm, S±

n ] = ±mnS±m, [S+

m, Sn ] = 2mnS

zm. (2.15)

Application of S(+)

m lowers (raises) the z–component of the spin at site m by one. Toactually make use of the fact that deviations around |i are small, a representationknown as the Holstein–Primako↵ transformation8 was introduced in which the spinoperators S±, S are specified in terms of bosonic creation and annihilation operators a†

and a:

Sm = a†

m(2S a†mam)1/2, S+

m = (2S a†mam)1/2am, Sz

m = S a†mam

8T. Holstein and H. Primako↵, Field dependence of the intrinsic domain magnetisation of a ferromag-net, Phys. Rev. 58, 1098 (1940).



Figure 2.8: Measurements ofthe spin-wave dispersion re-lations for the ferromagnetLa

0.7Sr0.3MnO3

.

The utility of this representation is clear: When the spin is large S 1, an expansionin powers of 1/S gives Sz

m = Sa†mam, S

m = (2S)1/2a†m+O(S1/2), and S+

m = (2S)1/2am+O(S1/2). In this approximation, the one-dimensional Hamiltonian takes the form

H = JXm

SzmSz

m+1

+1

2

S+

mSm+1

+ SmS+

m+1

= JNS2 + JS

Xm

na†mam + a†

m+1

am+1

a†mam+1

+ h.c.o

+ O(S0)

= JNS2 + SXm

(a†m+1

a†m)(am+1

am) + O(S0).

Bilinear in Bose operators, the approximate Hamiltonian can be diagonalised by Fouriertransformation. With periodic boundary conditions, Sm+N = Sm, am+N = am, defining

ak =1pN

NXm=1

eikmam, am =1pN

B.Z.Xk

eikmak, [ak, a†k0 ] = kk0 ,

the Hamiltonian for the one dimensional lattice system takes the form (exercise)

H = JNS2 +B.Z.Xk

!ka†kak + O(S0) (2.16)

where !k = 2JS(1 cos k) = 4JS sin2(k/2) represents the dispersion relation of the spinexcitations. In particular, in the limit k ! 0, the energy of the elementary excitationsvanishes, !k ! JSk2 (cf. Fig. 2.8). These massless low–energy excitations, known asmagnons, describe the elementary spin–wave excitations of the ferromagnet. Takinginto account terms at higher order in the parameter 1/S, one finds interactions betweenthe magnons.



Quantum Antiferromagnet

Having explored the low-energy excitation spectrum of the ferromagnet, we turn now tothe spin S Heisenberg antiferromagnetic chain,

H = JXm

Sm · Sm+1

where J > 0. Such local moment antiferromagnetic phases frequently occur in the arenaof strongly correlated electron systems. Although the Hamiltonian di↵ers from its fer-romagnetic relative ‘only’ by a change of sign, the di↵erences in the physics are dras-tic. Firstly, the phenomenology displayed by the antiferromagnetic Hamiltonian H de-pend sensitively on the geometry of the underlying lattice: For a bipartite lattice, i.e.one in which the neighbours of one sublattice A belong to the other sublattice B (cf.Fig. 2.9a), the ground states of the Heisenberg antiferromagnet are close9 to a staggered

Louis Neel 1904–2000: 1970 NobelLaureate in physics for fundamentalwork and discoveries concerning anti-ferromagnetism and ferrimagnetismwhich have led to important appli-cations in solid state physics

spin configuration, known as a Neelstate, where all neighbouring spinsare antiparallel. Again the groundstate is degenerate, i.e. a global ro-tation of all spins by the same amountdoes not change the energy. By con-trast, on non–bipartite lattices suchas the triangular lattice shown inFig. 2.9b, no spin arrangement can be found wherein which each and every bond canrecover the full exchange energy J . Spin models of this kind are said to be frustrated.

: A

: B

a) b)

Figure 2.9: (a) Example of a two–dimensional bipartite lattice and (b) a non–bipartite lattice.Notice that, with the latter, no antiferromagnetic arrangement of the spins can be made thatrecovers the maximum exchange energy from each and every bond.

. Exercise. Using only symmetry arguments, specify one of the possible ground statesof a classical three site triangular lattice antiferromagnet. (Note that the invariance of theHamiltonian under a global rotation of the spins means that the there is manifold of continuous

9It is straightforward to verify that the classical ground state — the Neel state — is now not anexact eigenstate of the quantum Hamiltonian. The true ground state exhibits zero–point fluctuationsreminiscent of the quantum harmonic oscillator or atomic chain. However, when S 1, it serves as auseful reference state from which fluctuations can be examined.



degeneracy in the ground state.) Using this result, construct one of the classical ground statesof the infinite triangular lattice.

Returning to the one–dimensional system, we first note that a chain is trivially bi-partite. As before, our strategy will be to expand the Hamiltonian in terms of bosonicoperators. However, before doing so, it is convenient to apply a canonical transformationto the Hamiltonian in which the spins on one sublattice, say B, are rotated through 180o

about the x–axis, i.e. SxB 7! Sx

B, SyB 7! Sy

B, and SzB 7! Sz

B, i.e. when representedin terms of the new operators, the Neel ground state looks like a ferromagnetic state,with all spins aligned. We expect that a gradual distortion of this state will produce theantiferromagnetic analogue of the spin–waves discussed in the previous section.

Represented in terms of the transformed operators, the Hamiltonian takes the form

H = JXm

SzmSz

m+1

1

2

S+

mS+

m+1

+ SmS

m+1

.

Once again, applying an expansion of the Holstein–Primako↵ representation, Sm ' (2S)1/2a†

m,etc., one obtains the Hamiltonian

H = NJS2 + JSXm

ha†mam + a†

m+1

am+1

+ amam+1

+ a†ma†

m+1

i+ O(S0) .

At first sight the structure of this Hamiltonian, albeit bilinear in the Bose operators, looksakward. However, after Fourier transformation, am = N1/2

Pk eikmak, it assumes the

more accessible form (exercise)

H = NJS(S + 1) + JSXk

( a†k ak )

1 kk 1

ak

a†k

+ O(S0),

where k = cos k.Quadratic in the bosonic operators, the Hamiltonian can be again diagonalised by

N. N. Bogoliubov 1909-1992: Theoreticalphyscists aclaimed for his works in nonlinearmechanics, statistical physics, theory of super-fluidity and superconductivity, quantum fieldtheory, renormalization group theory, proof ofdispersion relations, and elementary particletheory.

canonical transformation, i.e.a transformation of the fieldoperators that preserves thecommutation relations. Inthe present case, this isachieved by a Bogoliubovtransformation .

↵k

↵†k

=

cosh k sinh k

sinh k cosh k

ak

a†k

. (2.17)

. Exercise. Construct the inverse transformation. Considering the commutation relationsof the operators a↵, where a1 = a and a

2

= a†, explain why the Bogoliubov transformation is ofthe form of a Lorentz transformation. If operators a obeyed fermionic commutation relations,what form would the transformation take?



Figure 2.10: Experimentally obtainedspin–wave dispersion of the high–Tc par-ent compound LaCuO

4

— a prominentspin 1/2 antiferromagnet. Figure repro-duced from R. Coldea et al., Phys. Rev.Lett. 86, 5377 (2001).

Applying the Bogoliubov transformation, and setting tanh 2k = k, the Hamiltonianassumes the diagonal form (exercise)

H = NJS2 + 2JSXk

| sin k|↵†k↵k (2.18)

Thus, in contrast to the ferromagnet, the spin–wave excitations of the antiferromagnetexhibit a linear dispersion in the limit k ! 0. Surprisingly, although developed in thelimit of large spin, experiment shows that even for S = 1/2 spin chains, the integrity ofthe linear dispersion is maintained (see Fig. 2.10).

2.2.5 Bogoliubov theory of the weakly interacting Bose gas

Ealier, we explored the influence of interactions on the electron gas. When interactionsare weak, it was noted that the elementary collective excitations are reminiscent of theexcitations of the free electron gas — the Fermi-liquid phase. When the interactions arestrong, we discussed a scenario in which the electron liquid can condense into a solidinsulating phase — the Mott transition. In the following section, we will discuss theproperties of a quantum liquid comprised of Bose particles — the weakly interactingdilute Bose gas.

Let us consider a system of N Bose particles confined to a volume Ld and subject tothe Hamiltonian

H =Xk

k

a†k

ak

+1

2

Zddx ddx0 a†(x)a†(x0)V (x x0)a(x0)a(x)

where (0)k

= ~2k2

2m , and V (x) denotes a weak repulsive pairwise interaction. In the caseof a Bose gas, this assumption is connected with the fact that, even for infinitesimalattractive forces, a Bose gas cannot stay dilute at low temperatures. In Fourier space, thecorresponding two-body interaction can be expressed as (exercise)

HI =1

2Ld

Xk,k0,q

Vq

a†k+q

a†k

0ak

ak

0+q

,



where ak

= 1

Ld/2

Rddx eik·xa(x), and V

q

=R

ddx eiq·xV (x). In the following, we will beinterested in the ground state and low-lying excitations of the dilute system. In this case,we may distill the relevant components of the interaction and considerably simplify themodel.

In the ground state, the particles of an ideal (i.e. non-interacting) Bose gas condenseinto the lowest energy level. In a dilute gas, because of the weakness of the interactions,the ground state will di↵er only slightly from the ground state of the ideal gas, i.e. thenumber of particles N

0

in the condensate will still greatly exceed the number of particlesin other levels, so that N N

0

N . Since the number of particles in the condensate isspecified by the number operator N

0

= a†k=0

ak=0

= O(N) 1, matrix elements of theBose operators scale as a

0

O(p

N0

).10 This means that, from the whole sum in theinteraction, it is sucient to retain only those terms which involve interaction with thecondensate itself. Taking V

q

= V constant, one obtains (exercise)

HI =V

2LdN2

0

+V

LdN

0

Xk 6=0

a†k

ak

+ a†k

ak

+1

2

ak

ak

+ a†k

a†k

+ O(N0

0

).

Terms involving the excited states of the ideal gas have the following physical inter-pretation:

. V a†k

ak

represents the ‘Hartree-Fock energy’ of excited particles interacting with thecondensate;11

. V (ak

ak

+ a†k

a†k

) represents creation or annihilation of excited particles from thecondensate. Note that, in the present approximation, the total number of particlesis not conserved.

Now, using the identity N = N0

+P

k 6=0

a†k

ak

to trade for N0

, the total Hamiltoniantakes the form

H =V nN

2+Xk 6=0

(0)k

+ V n

a†k

ak

+ a†k

ak

+

V n

2

ak

ak

+ a†k

a†k

,

where n = N/Ld represents the total number density. This result may be comparedwith that obtained for the Hamiltonian of the quantum antiferromagnet in the spinwave approximation. Applying the Bogoluibov transformation (2.17): a

k

= cosh k

↵k

sinh

k

↵†k

, etc., with (exercise)

sinh2 k

=1

2

(0)k

+ V n

k

1

!,

where k

= [((0)k

+ V n)2 (V n)2]1/2, one obtains

H =V nN

2 1

2

Xk 6=0

((0)k

+ nV k

) +Xk 6=0

k

↵†k

↵k

.

10Note that the commutator [a0, a†0] = 1 is small as compared to a0 and a†0 allowing the field operators

to be replaced by the ordinary c-numberpN0.

11Note that the contact nature of the interaction disguises the presence of the direct and exchangecontributions.



From this result, we find that the spectrum of low energy excitations scales linearly ask

' ~c|k| where the velocity is given by c = (V n/m)1/2 while, at high energies (whenk k

0

= mc/~), the spectrum becomes free particle-like.12

. Info. Since the number operator ↵†k

↵k

can assume only positive values, one can inferthe ground state wavefunction from the condition ↵

k

|g.s.i = 0. Noting that the Bogoliubov

transformation can be written as ↵k

= Uak

U1, with U = exp[P

k 6=0

k2

(a†k

a†k

ak

ak

)]

(exercise), one can obtain the ground state as |g.s.i = U |i, where |i = (a†k=0

)N |i denotesthe ground state of the ideal Bose gas and |i the vacuum. The proof follows as

0 = ak 6=0

|i = U1

↵kz |

Uak

U1 U |i.

For the contact interaction, the corresponding ground state energy diverges and must be ‘regu-

larised’.13 In doing so, one obtains E0

= V nN2

1

2

Pk 6=0

((0)k

+ nV k

(nV )

2

2(0)k

) which, when

summed over k, translates to the energy density

E0

Ld=

n2V

2

1 +

128

15p(na3)1/2

,

where a ' (m/4~2)V denotes the scattering length of the interaction.Finally, one may estimate the depletion of the condensate due to interaction.

N N0

N=

1

N

Xk 6=0

hg.s.|a†k

ak

|g.s.i = 1

N

Xk 6=0

sinh2 k

=1

n

Zd3k

(2)3sinh2

k

=1

32nk30

,

i.e. ca. one particle per “coherence length” 1/k0

. Recast using the scattering length, oneobtains

N N0

N=

8

3p(na3)1/2.

——————————————–How do these prediction compare with experiment?14 When cooled to temperatures

below 4K, 4He condenses from a gas into a liquid. The 4He atoms obey Bose statisticsand, on cooling still further, the liquid undergoes a transition to a superfluid phase inwhich a fraction of the Helium atoms undergo Bose-Einstein condensation. Within thisphase, neutron scattering can be used to probe the elementary excitations of the system.Fig. 2.11 shows the excitation spectrum below the transition temperature. As predictedby the Bogoluibov theory, the spectrum of low-energy excitations is linear. The data alsoshow the limitations of the weakly interacting theory. In Helium, the steric interactionsare strong. At higher energy scales an important second branch of excitations known asrotons appear. The latter lie outside the simple hydrodynamic scheme described above.

12Physically, the e↵ect of the interaction is to displace particles from the condensate even at T = 0.13For details see, e.g., Ref. [1].14For a review of the history of Bose-Einstein condensation see, e.g. Grin, cond-mat/9901123



Figure 2.11: The dispersion curve for4He for the elementary excitations at1.1K (from Cowley and Woods 1971).

While studies of 4He allow only for the indirect manifestations of Bose-Einstein con-densation (viz. superfluidity and elementary excitations of the condensate), recent investi-gations of dilute atomic gases allow momentum distributions to be explored directly. Fol-lowing a remarkable sequence of technological breakthroughs in the 90s, dilute vapours ofalkali atoms, confined in magnetic traps, were cooled down to extremely low temperatures,of the order of fractions of microkelvins! Here the atoms in the vapour behave as quantum

Steven Chu 1948-, ClaudeCohen-Tannoudji 1933-and William D. Phillips1948-: 1997 Nobel Laureatesin Physics for developmentof methods to cool and trapatoms with laser light.

particles obeying Bose orFermi statistics depend-ing on the atomic num-ber. By abruptly re-moving the trap, time-of-flight measurements al-low the momentum dis-tribution to be inferreddirectly (see Fig. 2.12a). Below a certain critical temperature, these measurements re-vealed the development of a sharp peak at low momenta in Bose gases providing a clearsignature of Bose-Einstein condensation.

In the condensed phase, one may measure the sound wave velocity by e↵ecting adensity fluctuation using the optical dipole force created by a focused, blue-detuned laserbeam. By measuring the speed of propagation of the density fluctuation, the sound wavevelocity can be inferred. The latter, shown in Fig. 2.12b, shows a good agreement withthe theoretical prediction of the Bogoliubov theory.

2.3 Summary

This concludes our discussion of the second quantisation and its applications to problemsin many-body quantum mechanics. Beyond qualitative discussions, the list of applicationsencountered in this chapter involved problems that were either non–interacting from theoutset, or could be reduced to a quadratic operator structure by a number of suitablemanipulations. However, we carefully avoided dealing with interacting problems where


2.3. SUMMARY 41

10

5

0

Sp

ee

d o

f S

ou

nd

(m

m/s

)

86420

Density (1014

cm-3

)

Figure 2.12: Images of the velocity distribution of rubidium atoms by Anderson et al. (1995),taken by means of the expansion method. The left frame corresponds to a gas at a temperaturejust above condensation; the center frame, just after the appearance of the condensate; the rightframe, after further evaporation leaves a sample of nearly pure condensate. The field of view is200µm 200µm, and corresponds to the distance the atoms have moved in about 1/20s. Speedof sound, c vs. condensate peak density N

0

for waves propagating along the axial direction in the

condensate. Data taken from Kurn et al. (1997) compared to theoretical prediction c N1/20

.

no such reductions are possible. Yet it should be clear already at this stage of our dis-cussion that completely or nearly solvable systems represent only a small minority of thesystems encountered in condensed matter physics. What can be done in situations whereinteractions, i.e. operator contributions of fourth or higher order, are present?

Generically, interacting problems of many-body physics are either fundamentally inac-cessible to perturbation theory, or they necessitate perturbative analyses of infinite orderin the interaction contribution. Situations where a satisfactory result can be obtained byfirst or second order perturbation theory are exceptional. Within second quantisation,large order perturbative expansions in interaction operators leads to complex polynomialsof creation and annihilation operators. Quantum expectation values taken over such struc-tures can be computed by a reductive algorithm, known as Wick’s theorem. However,from a modern perspective, the formulation of perturbation theory in this way is not veryecient. More importantly, problems that are principally non-perturbative have emergedas a focus of interest. To understand the language of modern quantum condensed matter,we thus need to develop another layer of theory, known as field integration. However,before discussing quantum field theory, we should understand how the concept works inprinciple, i.e. on the level of point particle quantum mechanics. This will be the subjectof the next chapter.


Chapter 3

Feynman Path Integral

The aim of this chapter is to introduce the concept of the Feynman path integral. As well asdeveloping the general construction scheme, particular emphasis is placed on establishingthe interconnections between the quantum mechanical path integral, classical Hamiltonianmechanics and classical statistical mechanics. The practice of path integration is discussedin the context of several pedagogical applications: As well as the canonical examples of aquantum particle in a single and double potential well, we discuss the generalisation ofthe path integral scheme to tunneling of extended objects (quantum fields), dissipative andthermally assisted quantum tunneling, and the quantum mechanical spin.

In this chapter we will temporarily leave the arena of many–body physics and secondquantisation and, at least superficially, return to single–particle quantum mechanics. Byestablishing the path integral approach for ordinary quantum mechanics, we will set thestage for the introduction of functional field integral methods for many–body theoriesexplored in the next chapter. We will see that the path integral not only represents agateway to higher dimensional functional integral methods but, when viewed from anappropriate perspective, already represents a field theoretical approach in its own right.Exploiting this connection, various techniques and concepts of field theory, viz. stationaryphase analyses of functional integrals, the Euclidean formulation of field theory, instantontechniques, and the role of topological concepts in field theory will be motivated andintroduced in this chapter.

3.1 The Path Integral: General Formalism

Broadly speaking, there are two basic approaches to the formulation of quantum mechan-ics: the ‘operator approach’ based on the canonical quantisation of physical observables

Concepts in Theoretical Physics

64 CHAPTER 3. FEYNMAN PATH INTEGRAL

together with the associated operator algebra, and the Feynman1 path integral.2 Whereascanonical quantisation is usually taught first in elementary courses on quantum mechan-ics, path integrals seem to have acquired the reputation of being a sophisticated conceptthat is better reserved for advanced courses. Yet this treatment is hardly justified! In fact,the path integral formulation has many advantages most of which explicitly support anintuitive understanding of quantum mechanics. Moreover, integrals — even the infinitedimensional ones encountered below — are hardly more abstract than infinite dimensionallinear operators. Further merits of the path integral include the following:

⊲ Whereas the classical limit is not always easy to retrieve within the canonical for-mulation of quantum mechanics, it constantly remains visible in the path integralapproach. In other words, the path integral makes explicit use of classical mechan-ics as a basic ‘platform’ on which to construct a theory of quantum fluctuations.The classical solutions of Hamilton’s equation of motion always remain a centralingredient of the formalism.3

⊲ Path integrals allow for an efficient formulation of non–perturbative approaches tothe solution of quantum mechanical problems. For example, the ‘instanton’ for-mulation of quantum tunnelling discussed below — whose extension to continuumtheories has led to some of the most powerful concepts of quantum field theory —makes extensive use of the classical equations of motion when it is tailored to a pathintegral formulation.

⊲ The Feynman path integral represents a prototype of the higher dimensional func-tional field integrals to be introduced in the next chapter. However,...

⊲ ...even in its ‘zero–dimensional’ form discussed in this chapter, the path integralis of relevance to a wide variety of applications in many–body physics: Very of-ten, one encounters enviroments such as the superconductor, superfluid, or stronglycorrelated few electron devices where a macroscopically large number of degrees of

1

Richard P. Feynman 1918–1988:1965 Nobel Laureate in Physics(with Sin–Itiro Tomonaga, and Ju-lian Schwinger) for fundamentalwork in quantum electrodynamics,with deep–ploughing consequencesfor the physics of elementary parti-cles.

2For a more extensive introduction to the Feynman path integral, one can refer to one of the manystandard texts including Refs. [9, 16, 20] or, indeed, one may turn to the original paper, R. P. Feyn-man, Space–time approach to non–relativistic quantum mechanics, Rev. Mod. Phys. 20, 367 (1948).Historically, Feynman’s development of the path integral was motivated by earlier work by Dirac on theconnection between classical and quantum mechanics, P. A. M. Dirac, On the analogy between classical

and quantum mechanics, Rev. Mod. Phys. 17, 195 (1945).3For this reason, path integration has turned out to be an indispensable tool in fields such as quantum

chaos where the quantum manifestations of classically non–trivial behaviour are investigated — for moredetails, see section 3.2.2 below.


3.1. THE PATH INTEGRAL: GENERAL FORMALISM 65

freedom ‘lock’ to form a single collective variable. (For example, to a first approx-imation, the phase information carried by the order parameter field in moderatlylarge superconducting grains can often be described in terms of a single phase de-gree of freedom, i.e. a ‘quantum particle’ living on the complex unit circle.) Pathintegral techniques have proven ideally suited to the analysis of such systems.

What, then, is the basic idea of the path integral approach? More than any otherformulation of quantum mechanics, the path integral formalism is based on connectionsto classical mechanics. The variational approach employed in chapter ?? relied on the factthat classically allowed trajectories in configuration space extremize an action functional.A principal constraint to be imposed on any such trajectory is energy conservation. Bycontrast, quantum particles have more freedom than their classical counterparts. In par-ticular, by the Uncertainty Principle, energy conservation can be violated by an amount∆E over a time ∼ ~/∆E (here, and throughout this chapter, we will reinstall ~ forclarity). The connection to action principles of classical mechanics becomes particularlyapparent in problems of quantum tunneling: A particle of energy E may tunnel through apotential barrier of height V > E. However, this process is penalized by a damping factor∼ exp(i

∫barrier

dx p/~), where p =√

2m(E − V ), i.e. the exponent of the (imaginary)action associated with the classically forbidden path.

These observations motivate the idea of a new formulation of quantum propagation:Could it be that, as in classical mechanics, the quantum amplitude A for propagationbetween any two points in coordinate space is again controlled by the action functional?— controlled in a relaxed sense where not just a single extremal path xcl(t), but an entiremanifold of neighbouring paths contribute. More specifically, one might speculate thatthe quantum amplitude is obtained as A ∼

∑x(t) exp(iS[x]/~), where

∑x(t) symbolically

stands for a summation over all paths compatible with the initial conditions of the prob-lem, and S denotes the classical action. Although, at this stage, no formal justification forthe path integral has been presented, with this ansatz, some features of quantum mechan-ics would obviously be born out correctly: Specifically, in the classical limit (~ → 0), thequantum mechanical amplitude would become increasingly dominated by the contributionto the sum from the classical path xcl(t). This is because non–extremal configurationswould be weighted by a rapidly oscillating amplitude associated with the large phase S/~and would, therefore, average to zero.4 Secondly, quantum mechanical tunneling would bea natural element of the theory; non–classical paths do contribute to the net–amplitude,but at the cost of a damping factor specified by the imaginary action (as in the traditionalformulation).

Fortunately, no fundamentally novel ‘picture’ of quantum mechanics needs to be de-clared to promote the idea of the path ‘integral’

∑x(t) exp(iS[x]/~) to a working theory.

As we will see in the next section, the new formulation can quantitatively be developedfrom the same principles of canonical quantization.

4More precisely, in the limit of small ~, the path sum can be evaluated by saddle–point methods, asdetailed below.



3.2 Construction of the Path Integral

All information about any autonomous5 quantum mechanical system is contained inthe matrix elements of its time evolution operator. A formal integration of the time–dependent Schrodinger equation i~∂t|Ψ〉 = H|Ψ〉 obtains the time evolution operator

|Ψ(t′)〉 = U(t′, t)|Ψ(t)〉, U(t′, t) = e−i~H(t′−t)Θ(t′ − t) . (3.1)

The operator U(t′, t) describes dynamical evolution under the influence of the Hamiltonianfrom a time t to time t′. Causality implies that t′ > t as indicated by the step or HeavisideΘ–function. In the real space representation we can write

Ψ(q′, t′) = 〈q′|Ψ(t′)〉 = 〈q′|U(t′, t)Ψ(t)〉 =

∫dq U(q′, t′; q, t)Ψ(q, t) ,

where U(q′, t′; q, t) = 〈q′|e− i~H(t′−t)|q〉Θ(t′ − t) defines the (q′, q) component of the time

evolution operator. As the matrix element expresses the probability amplitude for aparticle to propagate between points q and q′ in a time t′ − t, it is sometimes known asthe propagator of the theory.

The basic idea behind Feynman’s path integral approach is easy to formulate. Ratherthan attacking the Schrodinger equation governing the time evolution for general times t,one may first attempt to solve the much simpler problem of describing the time evolutionfor infinitesimally small times ∆t. In order to formulate this idea quantitatively one mustfirst ‘divide’ the time evolution operator into N ≫ 1 discrete ‘time steps’,

e−iHt/~ =[e−iH∆t/~

]N, (3.2)

where ∆t = t/N . Albeit nothing more than a formal rewriting of Eq. (3.1), the repre-

sentation (3.2) has the advantage that the factors e−iH∆t/~ (or, rather, their expectationvalues) are small. (More precisely, if ∆t is much smaller than the (reciprocal of the)eigenvalues of the Hamiltonian in the regime of physical interest, the exponents are smallin comparison with unity and, as such, can be treated perturbatively.) A first simplifica-tion arising from this fact is that the exponentials can be factorised into two pieces eachof which can be readily diagonalised. To achieve this factorisation, we make use of theidentity

e−iH∆t/~ = e−iT∆t/~e−iV∆t/~ +O(∆t2) ,

where the Hamiltonian H = T + V is the sum of a kinetic energy T = p2/2m, and somepotential energy operator V .6 (The following analysis, restricted for simplicity to a one–dimensional Hamiltonian, is easily generalised to arbitrary spatial dimension.) The key

5A system is classified as autonomous if its Hamiltonian does not explicitly depend on time. Actuallythe construction of the path integral can be straightforwardly extended so as to include time–dependentproblems. However, in order to keep the introductory discussion as simple as possible, here we assumetime–independence.

6Although this ansatz already covers a wide class of quantum mechanical problems, many applicationsof practical importance (e.g. Hamiltonians involving spin or magnetic fields) do not fit into this frame-work. For a detailed exposition covering its realm of applicability, we refer to the specialist literaturesuch as, e.g., Schulman’s text [20].


3.2. CONSTRUCTION OF THE PATH INTEGRAL 67

advantage of this factorisation is that the eigenstates of each of the factors e−iT∆t/~ ande−iV∆t/~ are known independently. To exploit this fact we consider the time evolutionoperator factorised as a product,

〈qF |[e−iH∆t/~

]N|qI〉 ≃ 〈qF |∧e

−iT∆t/~e−iV∆t/~

∧ . . .∧e−iT∆t/~e−iV∆t/~|qI〉 (3.3)

and insert at each of the positions indicated by the symbol ‘∧’ the resolution of identity

id. =

∫dqn

∫dpn|qn〉〈qn|pn〉〈pn| . (3.4)

Here |qn〉 and |pn〉 represent a complete set of position and momentum eigenstates respec-tively, and n = 1, . . . , N serves as an index keeping track of the time steps at which theunit operator is inserted. The rational behind the particular choice (3.4) is clear. The unitoperator is arranged in such a way that both T and V act on the corresponding eigenstates.Inserting (3.4) into (3.3), and making use of the identity 〈q|p〉 = 〈p|q〉∗ = eiqp/~/(2π~),one obtains

〈qF |e−iHt/~|qI〉 ≃∫ N−1∏

n=1qN =qF ,q0=qI

dqn

N∏

n=1

dpn2π~

e−i∆t

~

PN−1n=0

“

V (qn)+T (pn+1)−pn+1qn+1−qn

∆t

”

. (3.5)

Thus, the matrix element of the time evolution operator has been expressed as a 2N − 1dimensional integral over eigenvalues. Up to corrections of higher order in V∆t/~ andT∆t/~, the expression (3.5) is exact. At each ‘time step’ tn = n∆t, n = 1, . . . , N we areintegrating over a pair of coordinates xn ≡ (qn, pn) parametrising the classical phase

space. Taken together, the points xn form an N–point discretization of a path in thisspace (see Fig. 3.1).

q I

qF

tn

p

0N N-1

PhaseSpace

t

Figure 3.1: Left: visualisation of a set of phase space points contributing to the discretetime configuration integral (3.5). Right: in the continuum limit, the set of points becomesa smooth curve.

To make further progress, we need to develop some intuition for the behaviour of theintegral (3.5). We first notice that rapid fluctuations of the integration arguments xn as



a function of the index n are strongly inhibited by the structure of the integrand. Whentaken together, contributions for which (qn+1−qn)pn+1 > O(~) (i.e. when the phase of theexponential exceeds 2π) tend to lead to a ‘random phase cancellation’. In the language ofwave mechanics, the ‘incoherent’ superposition of different Feynman paths destructivelyinterferes. The smooth variation of the paths which contribute significantly motivate theapplication of a continuum limit analogous to that employed in chapter ??.

To be specific, sending N → ∞ whilst keeping t = N∆t fixed, the formerly discreteset tn = n∆t, n = 1, . . . , N becomes dense on the time interval [0, t], and the set of phasespace points xn becomes a continuous curve x(t). In the same limit,

∆t

N−1∑

n=0

7→∫ t

0

dt′,qn+1 − qn

∆t7→ ∂t′q

∣∣∣t′=tn

≡ q|t′=tn ,

while [V (qn) + T (pn+1)] 7→ [T (p|t′=tn) + V (q|t′=tn)] ≡ H(x|t′=tn) denotes the classicalHamiltonian. In the limit N → ∞, the fact that kinetic and potential energies areevaluated at neighbouring time slices, n and n+ 1, becomes irrelevant.7 Finally,

limN→∞

∫ N−1∏

n=1qN =qF ,q0=qI

dqn

N∏

n=1

dpn2π~

≡∫

q(t)=qFq(0)=qI

Dx

defines the integration measure of the integral.

⊲ Info. Integrals extending over infinite dimensional integration measures like D(q, p)

are generally called functional integrals (recall our discussion of functionals in chapter ??).

The question of how functional integration can be rigorously defined is far from innocent and

represents a subject of current, and partly controversial mathematical research. In this book —

as in most applications in physics — we take a pragmatic point of view and deal with the infinite

dimensional integration naively unless mathematical problems arise (which actually won’t be the

case!).

——————————————–

Then, applying these conventions to Eq. (3.5), one finally obtains

〈qF |e−iHt/~|qI〉 =

∫

q(t)=qFq(0)=qI

Dx exp

[i

~

∫ t

0

dt′ (pq −H(p, q))

](3.6)

7To see this formally, one may Taylor expand T (pn+1) = T (p(t′+∆t))|t′=n∆t around p(t′). For smoothp(t′), all but the zeroth order contribution T (p(t′)), scale with powers of ∆t, thereby becoming irrelevant.Note, however, that all of these arguments are based on the assertion that the dominant contributions tothe path integral are smooth in the sense qn+1− qn ∼ O(∆t). A closer inspection, however, shows that infact qn+1 − qn ∼ O(

√∆t) [20]. In some cases, the most prominent one being the quantum mechanics of

a particle in a magnetic field, the lowered power of ∆t spoils the naive form of the continuity argumentabove, and more care must be applied in taking the continuum limit. In cases where a ‘new’ pathintegral description of a quantum mechanical problem is developed, it is imperative to delay taking thecontinuum limit until the fluctuation behaviour of the discrete integral across individual time slices hasbeen thoroughly examined.



Eq. (3.6) represents the Hamiltonian formulation of the path integral: The integra-tion extends over all possible paths through the classical phase space of the system whichbegin and end at the same configuration points qI and qF respectively (cf. Fig. 3.1). Thecontribution of each path is weighted by its Hamiltonian action.

Before we turn to the discussion of the path integral (3.6), it is first useful to recastthe integral in an alternative form which will be both convenient in various applicationsand physically instructive. The search for an alternative formulation is motivated by theobservation of the close resemblance of (3.6) with the Hamiltonian formulation of classicalmechanics. Given that, classically, Hamiltonian and Lagrangian mechanics can be equallyemployed to describe dynamical evolution, it is natural to seek a Lagrangian analogue of(3.6). Until now, we have made no assumption about the momentum dependence of thekinetic energy T (p). However, if we focus on Hamiltonians in which the dynamics is free,i.e. the kinetic energy dependence is quadratic in p, the Lagrangian form of the pathintegral can be inferred from (3.6) by Gaussian integration.

To make this point clear, let us rewrite the integral in a way that emphasises itsdependence on the momentum variable p:


∫

q(t)=qFq(0)=qI

Dq e−i~

R t

0dt′V (q)

∫Dp e

− i~

R t

0dt′

„

p2

2m−pq

«

. (3.7)

The exponent of the integral is quadratic in the momentum variable or, equivalently, theintegral is Gaussian in p. Carrying out the integration by means of Eq. (3.13) below, oneobtains


∫

q(t)=qFq(0)=qI

Dq exp

[i

~

∫ t

0

dt′L(q, q)

](3.8)

where Dq = limN→∞( Nmit2π~

)N/2∏N−1

n=1 dqn denotes the functional measure of the remainingq–integration, and L(q, q) = mq2/2 − V (q) represents the classical Lagrangian. Strictlyspeaking, the (finite–dimensional) integral formula (3.13) is not directly applicable tothe infinite dimensional Gaussian integral (3.7). This, however, does not represent asubstantial problem as we can always rediscretise the integral (3.7), apply Eq. (3.13), andreinstate the continuum limit after integration (exercise).

Together Eqs. (3.6) and (3.8) represent the central results of this section. A quantummechanical transition amplitude has been expressed in terms of an infinite dimensionalintegral extending over paths through phase space (3.6) or coordinate space (3.8). Allpaths begin (end) at the initial (final) coordinate of the matrix element. Each pathis weighted by its classical action. Notice in particular that the quantum transitionamplitude has been cast in a form which does not contain quantum mechanical operators.Nonetheless, quantum mechanics is still fully present! The point is that the integrationextends over all paths and not just the subset of solutions of the classical equations ofmotion. (The distinguished role classical paths play in the path integral will be discussedbelow in section 3.2.2.) The two forms of the path integral, (3.6) and (3.8), representthe formal implementation of the ‘alternative picture’ of quantum mechanics proposedheuristically at the beginning of the chapter.



⊲ Info. Gaussian Integration: Apart from a few rare exceptions, all integrals encounteredin this course will be of Gaussian8 form. In most cases the dimension of the integrals will belarge if not infinite. Yet, after a bit of practice, it will become clear that high dimensionalrepresentatives of Gaussian integrals are no more difficult to handle than their one–dimensionalcounterparts. Therefore, considering the important role played by Gaussian integration in fieldtheory, we will here derive the principle formulae once and for all. Our starting point is theone–dimensional integral (both real and complex). The basic ideas underlying the proofs ofthe one–dimensional formulae, will provide the key to the derivation of more complex, multi–dimensional and functional identities which will be used liberally throughout the remainder ofthe text.

One–dimensional Gaussian integral: The basic ancestor of all Gaussian integrals is theidentity

∫ ∞

−∞dx e−

a2x2

=

√2π

a, Re a > 0 (3.9)

In the following we will need various generalisations of Eq. (3.9). Firstly, we have∫∞−∞ dx e−ax

2/2x2 =√2π/a3, a result established either by substituting a→ a+ ǫ in Eq. (3.9) and expanding both

the left and the right side of the equation to leading order in ǫ, or by differentiating Eq. (3.9).Often one encounters integrals where the exponent is not purely quadratic from the outset butrather contains both quadratic and linear pieces. The generalisation of Eq. (3.9) to this casereads ∫ ∞

−∞dx e−

a2x2+bx =

√2π

ae

b2

2a . (3.10)

To prove this identity, one simply eliminates the linear term by means of the change of variablesx → x + b/a which transforms the exponent to ax2/2 + bx → −ax2/2 + b2/2a. The constantfactor scales out and we are left with Eq. (3.9). Note that Eq. (3.10) holds even for complex b.The reason is that by shifting the integration contour into the complex plane no singularitiesare encountered, i.e. the integral remains invariant.

Later, we will be concerned with the generalisation of the Gaussian integral to complexarguments. In this case, the extension of Eq. (3.9) reads

∫d(z, z)e−zwz =

π

w, Re w > 0 ,

where z represents the complex conjugate of z. Here,∫d(z, z) ≡

∫∞−∞ dxdy represents the

independent integration over the real and imaginary parts of z = x + iy. The identity is easyto prove: Owing to the fact that zz = x2 + y2, the integral factorizes into two pieces each ofwhich is equivalent to Eq. (3.9) with a = w. Similarly, it may be checked that the complex

8

Johann Carl Friedrich Gauss 1777-1855: worked ina wide variety of fields in both mathematics and physicsincuding number theory, analysis, differential geometry,geodesy, magnetism, astronomy and optics. Portrait takenfrom the former German 10–Mark note. (Unfortunately,the subsequently introduced Euro notes no longer displayGauss’ portrait.)



generalisation of Eq. (3.10) is given by∫d(z, z)e−zwz+uz+zv =

π

we

uvw , Re w > 0 . (3.11)

More importantly u and v may be independent complex numbers; they need not be related toeach other by complex conjugation (exercise).

Gaussian integration in more than one dimension: All of the integrals above have higherdimensional counterparts. Although the real and complex versions of the N–dimensional integralformulae can be derived in a perfectly analogous manner, it is better to discuss them seperatelyin order not to confuse the notation.

(a) Real Case: The multi–dimensional generalisation of the prototype integral (3.9) reads∫dve−

12vT Av = (2π)N/2detA−1/2 , (3.12)

where A is a positive definite real symmetric N–dimensional matrix and v is an N–componentreal vector. The proof makes use of the fact that A (by virtue of being symmetric) can bediagonalised by orthogonal transformation, A = OTDO, where the matrix O is orthogonal,and all elements of the diagonal matrix D are positive. The matrix O can be absorbed into theintegration vector by means of the variable transformation, v 7→ Ov which has unit Jacobian,detO = 1. As a result, we are left with a Gaussian integral with exponent −vTDv/2. Dueto the diagonality of D, the integral factorizes into N independent Gaussian integrals each ofwhich contributes a factor

√2π/di, where di, i = 1, . . . , N is the ith entry of the matrix D.

Noting that∏Ni=1 di = detD = detA, (3.12) is derived.

The multi–dimensional generalization of (3.10) reads∫dve−

12vT Av+jT ·v = (2π)N/2detA−1/2e

12jT A−1j (3.13)

where j is an arbitrary N–component vector. Eq. (3.13) is proven by analogy with Eq. (3.10),i.e. by shifting the integration vector according to v → v + A−1j, which does not changethe value of the integral but removes the linear term from the exponent, −1

2vTAv + jT · v →

−12v

TAv + 12 jTA−1j. The resulting integral is of the type (3.12), and we arrive at Eq. (3.13).

The integral (3.13) is not only of importance in its own right, but it also serves as a ‘gen-erator’ of other useful integral identities. Applying the differentiation operation ∂2

jmjn |j=0 to

the left and the right hand side of Eq. (3.13), one obtains the identity9∫dve−

12vT Avvmvn =

(2π)N/2detA−1/2A−1mn. This result can be more compactly formulated as

〈vmvn〉 = A−1mn, (3.14)

where we have introduced the shorthand notation

〈. . .〉 ≡ (2π)−N/2detA1/2

∫dve−

12vT Av(. . .) , (3.15)

suggesting an interpretation of the Gaussian weight as a probability distribution.Indeed, the differentiation operation leading to (3.14) can be iterated: Differentiating four

times, one obtains 〈vmvnvqvp〉 = A−1mnA

−1qp + A−1

mqA−1np + A−1

mpA−1nq . One way of memorising the

9Note that the notation A−1mn refers to the mn element of the matrix A−1.



structure of this — important — identity is that the Gaussian ‘expectation’ value 〈vmvnvpvq〉is given by all ‘pairings’ of type (3.14) that can be formed from the four components vm. Thisrule generalises to expectation values of arbitrary order: 2n–fold differentiation of (3.13) yields

〈vi1vi2 . . . vi2n〉 =∑

all possiblepairings ofi1,...i2n

A−1ik1

ik2. . . A−1

ik2n−1ik2n

(3.16)

This result is the mathematical identity underlying Wick’s theorem (for real bosonic fields).

(b) Complex Case: The results above are straightforwardly extended to multi–dimensionalcomplex Gaussian integrals. The complex version of Eq. (3.12) is given by

∫d(v†,v)e−v†Av = πNdetA−1 , (3.17)

where v is a complex N–component vector, d(v†,v) ≡ ∏Ni=1 dRe vi dIm vi, and A is a complex

matrix with positive definite Hermitian part. (Remember that every matrix can be decomposedinto a Hermitian and an anti–Hermitian component, A = 1

2(A+A†)+ 12(A−A†).) For Hermitian

A, the proof of (3.17) is analogous to (3.12), i.e. A is unitarily diagonalisable, A = U†AU; thematrices U can be transformed into v, the resulting integral factorises, etc. For non–HermitianA the proof is more elaborate, if unedifying, and we refer to the literature for details. Thegeneralization of Eq. (3.17) to exponents with linear contributions reads

∫d(v†,v)e−v†Av+w†·v+v†·w′

= πNdetA−1ew†A−1w′

(3.18)

Note that w and w′ may be independent complex vectors. The proof of this identity mirrors thatof (3.13), i.e. by effecting the shift v† → v† + w†, v → v + w′.10 As with Eq. (3.13), Eq. (3.18)may also serve as a generator of related integral identities. Differentiating the integral twiceaccording to ∂2

wm,w′n|w=w′=0 gives

〈vmvn〉 = A−1nm ,

where 〈· · ·〉 ≡ π−NdetA∫d(v†,v)e−v†Av(· · ·). The iteration to more than two derivatives gives

〈vnvmvpvq〉 = A−1pmA

−1qn +A−1

pnA−1qm and, eventually,

〈vi1 vi2 . . . vinvj1vj2 . . . vjn〉 =∑

P

A−1j1iP1

. . . A−1jniPn

where∑

P represents for the sum over all permutations of N integers.

Gaussian Functional Integration: With this preparation, we are in a position to in-vestigate the main practice of quantum and statistical field theory — the method of Gaussianfunctional integration. Turning to Eq. (3.13), let us suppose that the components of the vectorv parameterise the weight of a real scalar field on the sites of a one–dimensional lattice. In thecontinuum limit, the set vi translates to a function v(x), and the matrix Aij is replaced by an

10For an explanation of why v and v† may be shifted independently of each other, cf. the analyticityremarks made in connection with (3.11).



operator kernel or propagator A(x, x′). In this limit, the natural generalisation of Eq. (3.13)is

∫Dv(x) exp

[−1

2

∫dx dx′v(x)A(x, x′)v(x′) +

∫dx j(x)v(x)

]

∝ (detA)−1/2 exp

[1

2

∫dx dx′ j(x)A−1(x, x′)j(x′)

], (3.19)

where the inverse kernel A−1(x, x′) satisfies the equation

∫dx′ A(x, x′)A−1(x′, x′′) = δ(x− x′′) (3.20)

i.e. A−1(x, x′) can be interpreted as the Green function of the operator A(x, x′). The notationDv(x) is used to denote the measure of the functional integral. Although the constant ofproportionality, (2π)N left out of Eq. (3.19) is formally divergent in the thermodynamic limitN → ∞, it does not effect averages that are obtained from derivatives of such integrals. Forexample, for Gaussian distributed functions, Eq. (3.14) has the generalisation

〈v(x)v(x′)〉 = A−1(x, x′)

Accordingly, Eq. (3.16) assumes the form

〈v(x1)v(x2) . . . v(x2n)〉 =∑

all possiblepairings ofx1,...x2n

A−1(xk1 , xk2) . . . A−1(xk2n−1 , xk2n

) (3.21)

The generalization of the other Gaussian averaging formulae discussed above should be obvious.

To make sense of Eq. (3.19) one must interpret the meaning of the determinant, detA.

When the variables entering the Gaussian integral were discrete, the latter simply represented

the determinant of the (real symmetric) matrix. In the present case, one must interpret A as

an Hermitian operator having an infinite set of eigenvalues. The determinant simply represents

the product over this infinite set (see, e.g., section 3.3.1). This completes our discussion of the

method of Gaussian integration. Although, in the following section, we will employ only a few

of the integral identities above, later we will have occasion to draw on the properties of the ‘field

averages’.

——————————————–

Before turning to specific applications of the Feynman path integral, let us stay withthe general structure of the formalism and identify two fundamental connections of thepath integral to classical point mechanics and classical and quantum statistical mechanics.

3.2.1 Path Integral and Statistical Mechanics

The path integral reveals a connection between quantum mechanics and classical (andquantum) statistical mechanics whose importance to all areas of field theory and statis-tical physics can hardly be exaggerated. To reveal this link, let us for a moment forget



about quantum mechanics and consider, by way of an example, a perfectly classical, one–dimensional continuum model describing a ‘flexible string’. We assume that our stringis held under constant tension, and confined to a ‘gutter–like potential’ (as shown inFig. 3.2). For simplicity, we also assume that the mass density of the string is pretty high,so that its fluctuations are ‘asymptotically slow’ (the kinetic contribution to its energyis negligible). Transverse fluctuations of the string are then penalised by its line tension,and by the external potential.

x

u

V(u)

Figure 3.2: A string held under tension and confined to a potential well V .

Assuming that the transverse displacement of the string u(x) is small, the potentialenergy stored in the string separates into two parts. The first arises form the line tensionstored in the string, and the second comes from the external potential. Starting with theformer, a transverse fluctuation of a line segment of length dx by an amount du, leads toa potential energy of magnitude δVtension = σ[(dx2 +du2)1/2−−dx] ≃ σdx(∂xu)

2/2, whereσ denotes the tension. Integrated over the length of the string, one obtains Vtension[∂xu] ≡∫δVtension = 1

2

∫ L0dx σ(∂xu(x))

2. The second contribution arising from the external po-

tential is given by Vexternal[u] ≡∫ L0dx V (u(x)). Adding the two contributions, we find that

the total energy of the string is given by V = Vtension + Vexternal =∫ L0dx[σ

2(∂xu)

2 + V (u)].According to the general prinicples of statistical mechanics, the equilibrium properties

of a system are encoded in the partition function Z = tr[e−βV

], where ‘tr’ denotes a

summation over all possible configurations of the system and V is the total potentialenergy functional. Applied to the present case, tr →

∫Du, where

∫Du stands for the

functional integration over all configurations of the string u(x), x ∈ [0, L]. Thus, thepartition function of the string is given by

Z =

∫Du exp

[−β∫ L

0

dx(σ

2(∂xu)

2 + V (u))]

. (3.22)

A comparison of this result with Eq. (3.8) shows that the partition function of the classicalsystem coincides with the quantum mechanical amplitude

Z =

∫dq 〈q|eiS[q]/~|q〉

∣∣∣t=−iL

evaluated at an imaginary ‘time’ t→ −iτ ≡ −iL, where H = p2/2σ+V (q), and Planck’sconstant is identified with the ‘temperature’, ~ = 1/β. (Here we have assumed that ourstring is subject to periodic boundary conditions.)



To see this explicitly, let us assume that we had reason to consider quantum propa-gation in imaginary time, i.e. e−itH/~ → e−τH/~, or t → −iτ . Assuming convergence (i.e.positivity of the eigenvalues of H), a construction scheme perfectly analogous to the oneoutlined in section 3.1 would have led to a path integral formula of the structure (3.8).Formally, the only differece would be that (a) the Lagrangian would be integrated alongthe imaginary time axis t′ → −iτ ′ ∈ [0,−iτ ] and (b) that there would be a change ofthe sign of the kinetic energy term, viz. (∂t′q)

2 → −(∂τ ′q)2. After a suitable exchange of

variables, τ → L, ~ → 1/β, the coincidence of the resulting expression with the partitionfunction (3.22) is clear.

The connection between quantum mechanics and classical statistical mechanics out-lined above generalises to higher dimensions: There are close analogies between quan-tum field theories in d dimensions and classical statistical mechanics in d + 1. (Theequality of the path integral above with the one–dimensional statistical model is merelythe d = 0 version of this connection.) In fact, this connection turned out to be one of themajor driving forces behind the success of path integral techniques in modern field the-ory/statistical mechanics. It offered, for the first time, a possibility to draw connectionsbetween systems which had seemed unrelated.

However, the concept of imaginary times not only provides a bridge between quantumand classical statistical mechanics, but also plays a role within a purely quantum me-chanical context. Consider the quantum partition function of a single particle quantummechanical system,

Z = tr[e−βH ] =

∫dq 〈q|e−βH |q〉

The partition function can be interpreted as a trace over the transition amplitude 〈q|e−iHt/~|q〉evaluated at an imaginary time t = −i~β. Thus, real time dynamics and quantum sta-tistical mechanics can be treated on the same footing, provided that we allow for theappearance of imaginary times.

Later we will see that the concept of imaginary or even generalized complex timesplays an important role in all of field theory. There is even some nomenclature regardingimaginary times. The transformation t → −iτ is denoted as a Wick rotation (alludingto the fact that a multiplication with the imaginary unit can be interpreted as a π/2–rotation in the complex plane). Imaginary time representations of Lagrangian actions aretermed Euclidean, whereas the real time forms are called Minkowski11 actions.

⊲ Info. The origin of this terminology can be understood by considering the structure of

the action of, say, the phonon model (1.2). Forgetting for a moment about the magnitude of the

11

Hermann Minkowski 1864–1909: A pure mathematiciancredited with the development ofa four–dimensional treatment ofelectrodynamics and, separately,the geometry of numbers.



coupling constants, we see that the action has the bilinear structure ∼ xµgµνxν , where µ = 0, 1,

the vector xµ = ∂µφ and the diagonal matrix g = diag(−1, 1) is the two dimensional version

of a Minkowski metric. (In three spatial dimensions, g would take the form of the standard

Minkowski metric of special relativity.) Wick rotating time, the −1 in the metric changes sign

and g becomes a positive definite Euclidean metric. The nature of this transformation motivates

the notation above.

——————————————–

Once one has grown accustomed to the idea that the interpretation of time as animaginary quantity can be useful, yet more general concepts can be conceived. For exam-ple, one may contemplate quantum propagation along temporal contours that are neitherpurely real nor purely imaginary but rather are generally complex. Indeed, it has turnedout that path integrals with curvelinear integration contours in the complex ‘time plane’find numerous applications in statistical and quantum field theory.

3.2.2 Semiclassics from the Path Integral

In deriving the two path integral representations (3.6) and (3.8) no approximations weremade. Yet the vast majority of quantum mechanical problems cannot be solved in closedform, and it would be hoping for too much to expect that within the path integral ap-proach this situation would be any different. In fact no more than the path integralsof problems with a quadratic Hamiltonian — corresponding to the quantum mechanicalharmonic oscillator and generalisations thereof — can be carried out in closed form. Yetwhat counts more than the (rare) availability of exact solutions is the flexibility withwhich approximation schemes can be developed. As for the path integral formulation, itis particularly strong in cases where semiclassical limits of quantum theories areexplored. Here, by ‘semiclassical’, we mean the limit ~ → 0, i.e. the case where the the-ory is expected to be largely governed by classical structures with quantum fluctuationssuperimposed.

To see more formally how classical structures enter the path integral approach, letus explore Eqs. (3.6) and (3.8) in the limit of small ~. In this case the path integralsare dominated by path configurations with stationary action. (Non–stationary contri-butions to the integral imply massive phase fluctuations which largely average to zero.)Now, since the exponents of the two path integrals (3.6) and (3.8) involve the classicalaction functionals in their Hamiltonian respectively Lagrangian form, the extremal pathconfigurations are simply the solutions of the classical equations of motion, viz.

Hamiltonian : δS[x] = 0 ⇒ dtx = H(x), x ≡ ∂pH∂qx− ∂qH∂px,

Lagrangian : δS[q] = 0 ⇒ (dt∂q − ∂q)L(q, q) = 0 . (3.23)

These equations are to be solved subject to the boundary conditions q(0) = qI andq(t) = qF . (Note that these boundary conditions do not uniquely specify a solution, i.e.in general there are many solutions to the equations (3.23). As an exercise, one may tryto invent examples!)

Now the very fact that the stationary phase configurations are classical does not implythat quantum mechanics has disappeared completely. As with saddle–point approxima-



tions in general, it is not just the saddle–point itself that matters but also the fluctua-tions around it. At least it is necessary to integrate out Gaussian (quadratic) fluctuationsaround the point of stationary phase. In the case of the path integral, fluctuations ofthe action around the stationary phase configurations involve non–classical (in that theydo not solve the classical equations of motion) trajectories through phase or coordinatespace. Before exploring how this mechanism works in detail, let us consider the stationaryphase analysis of functional integrals in general.

⊲ Info. Stationary Phase Approximation: Consider a general functional integral∫Dxe−F [x] where Dx = limN→∞

∏Nn=1 dxn represents a functional measure resulting from

taking the continuum limit of some finite dimensional integration space, and the ‘action’ F [x]may be an arbitrary complex functional of x (leading to convergence of the integral). Thefunction resulting from taking the limit of infinitely many discretisation points, xn is denotedby x : t 7→ x(t) (where t plays the role of the formerly discrete index n). Evaluating the integralabove within a stationary phase approximation amounts to performing the following steps:

1. Firstly, find the ‘points’ of stationary phase, i.e. configurations x qualified by the conditionof vanishing functional derivative,

DxF = 0 ⇔ ∀t :δF [x]

δx(t)

∣∣∣∣x=x

= 0.

Although there may, in principle, be one or many solutions, for clarity, we first discussthe case in which the stationary phase configuration x is unique.

2. Secondly, Taylor expand the functional to second order around x, viz.

F [x] = F [x+ y] = F [x] +1

2

∫dt

∫dt′y(t′)A(t, t′)y(t) + . . . (3.24)

where A(t, t′) = δ2F [x]δx(t)δx(t′)

∣∣∣x=x

denotes the second functional derivative. Due to the sta-

tionarity of x, one may note that no first order contribution can appear.

3. Thirdly, check that the operator A ≡ A(t, t′) is positive definite. If it is not, there is aproblem — the integration over the Gaussian fluctuations y below diverges. (In practice,where the analysis is rooted in a physical context, such eventualities arise only rarely.In situations were problems do occur, the resolution can usually be found in a judiciousrotation of the integration contour.) For positive definite A, however, the functional

integral over y can be performed after which one obtains∫Dx e−F [x] ≃ e−F [x] det( A2π )−1/2,

(cf. the discussion of Gaussian integrals above and, in particular, Eq. (3.19)).

4. Finally, if there are many stationary phase configurations, xi, the individual contributionshave to be added:

∫Dx e−F [x] ≃

∑

i

e−F [xi] det

(Ai2π

)−1/2

. (3.25)

Eq. (3.25) represents the most general form of the stationary phase evaluation of a (real) func-tional integral.



⊲ Exercise. Applied to the Gamma function, Γ(z + 1) =∫∞0 dxxze−x, with z complex,

show that the stationary phase approximation is consistent with Stirling’s approximation, viz.

Γ(s+ 1) =√

2πses(ln s−1).

——————————————–

qq

I

qFα h

1/2

t

q

q(t)

Figure 3.3: Quantum fluctuations around a classical path in coordinate space (here weassume a set of two–dimensional coordinates). Non–classical paths q fluctuating arounda classical solution qcl typically extend a distance O(h1/2). All paths begin and end at qIand qF , respectively.

Applied to the Lagrangian form of the Feynman path integral, this program can beimplemented directly. In this case, the extremal field configuration q(t) is identified asthe classical solution associated with the Lagrangian, i.e. q(t) ≡ qcl(t). Defining r(t) =q(t) − qcl(t) as the deviation of a general path, q(t), from a nearby classical path, qcl(t)(see Fig. 3.3), and assuming for simplicity that there exists only one classical solutionconnecting qI with qF in a time t, a stationary phase analysis obtains

〈qF |e−iHt/~|qI〉 ≃ eiS[qcl]/~

∫

r(0)=r(t)=0

Dr exp

[i

2~

∫ t

0

dt′dt′′r(t′)δ2S[q]

δq(t′)δq(t′′)

∣∣∣∣q=qcl

r(t′′)

](3.26)

as the Gaussian approximation to the path integral (cf. Eq. (3.24)). For free Lagrangiansof the form, L(q, q) = mq2/2−V (q), the second functional derivative of the action can bestraightforwardly computed by means of the rules of functional differentiation formulatedin chapter 1. Alternatively, one can obtain this result by simply expanding the action asa Taylor series in the deviation r(t). As a result, one obtains (exercise)

1

2

∫ t

0

dt

∫dt′r(t)

δ2S[q]

δq(t)δq(t′)

∣∣∣∣q=qcl

r(t′) = −1

2

∫dt r(t)

[m∂2

t + V ′′(qcl(t))]r(t) , (3.27)

where V ′′(qcl(t)) ≡ ∂2qV (q)|q=qcl represents the ordinary (second) derivative of the potential

function evaluated at qcl(t). Thus, the Gaussian integration over r yields the square rootof the determinant of the operator m∂2

t + V ′′(qcl(t)) — interpreted as an operator actingin the space of functions r(t) with boundary conditions r(0) = r(t) = 0. (Note that, aswe are dealing with a differential operator, the issue of boundary conditions is crucial.)



⊲ Info. More generally, Gaussian integration over fluctuations around the stationary phaseconfiguration obtains the formal expression

〈qF |e−iHt/~|qI〉 ≃ det

(i

2π~

∂2S[qcl]

∂qI∂qF

)1/2

ei~S[qcl] , (3.28)

as the final result for the transition amplitude evaluated in the semiclassical approxima-tion. (In cases where there is more than one classical solution, the individual contributions haveto be added.) To derive this expression, one shows that the operator controlling the quadraticaction (3.27) fulfills some differential relations which can be, again, related back to the classicalaction. While a detailed formulation of this calculation (see, for example, Ref. [20], page 94) isbeyond the scope of the present text, the heuristic interpretation of the result is straightforward:

According to the rules of quantum mechanics p(qF , qI , t) = |〈qF |e−iHt/~|qI〉|2 defines theprobability density for a particle injected at coordinate qI to arrive at coordinate qF after a timet. In the semiclassical approximation, the probability density assumes the form p(qF , qI , t) =

|det( 12π~

∂2S[qcl]∂qI∂qF

)|. We can gain some physical insight into this expression from the followingconsideration: For a fixed inital coordinate qI , the final coordinate qF (qI , pI) becomes a functionof the initial momentum pI . The classical probability density p(qI , qF ) can then be related tothe probabilty density p(qI , pI) for a particle to leave from the initial phase space coordinate(qI , pI) according to

p(qI , qF )dqIdqF = p(qI , qF )

∣∣∣∣det

(∂qF∂pI

)∣∣∣∣ dqIdpI = p(qI , pI)dqIdpI .

Now, when we say that our particle actually left at the phase space coordinate (qI , pI), p becomessingular at (qI , pI) while being zero everywhere else. In quantum mechanics, however, all wecan say is that our particle was initially confined to a Planck cell centered around (qI , pI):p(qI , pI) = 1/(2π~)d. We thus conclude that p(qI , qF ) = |det(∂pI/∂qF )|~−d. Finally, noticingthat pI = −∂qIS we arrive at the result of the semiclassical analysis above.

p

q

q q

p

I F I I (q ,p )

I

Figure 3.4: Trajectory in two-dimensional phase space: For fixed initial coordinate qI ,the final coordinate qF (qI , pI) becomes a function of the initial momentum. In quantummechanics, the Planck cell ~d (indicated by the rectangle) limits the accuracy at whichthe initial coordinate can be set.

In deriving (3.28) we have restricted ourselves to the consideration of quadratic fluctuations

around the classical paths — the essence of the semiclassical approximation. Under what condi-

tions is this approximation justified? Unfortunately there is no rigorous and generally applicable



answer to this question: For finite ~, the quality of the approximation depends largely on the

sensitivity of the action to path variations. Whether or not the approximation is legitimate is

a question that has to be judged from case to case. However, the asymptotic stability of the

semiclassical approximation in the limit ~ → 0, can be deduced simply from power counting.

From the structure of Eq. (3.28) it is clear that the typical magnitude of fluctuations r(t) scales

as r ∼ (~/δ2qS)1/2, where δ2qS is a symbolic shorthand for the functional variation of the ac-

tion. (Variations larger than that lead to phase fluctuations > 2π, thereby being negligible.)

Non–Gaussian contributions to the action would have the structure ∼ ~−1rnδnq S, n > 2. For a

typical r, this is of the order ∼ δnq S/(δ2qS)n/2~

n/2−1. Since the S-dependent factors are classical

(~–independent), these contributions scale to zero as ~ → 0.

——————————————–

This concludes the conceptual part of the chapter. Before turning to the discussionof specific applications of the path integral, let us first briefly recapitulate the main stepstaken in its construction:

3.2.3 Construction Recipe of the Path Integral

Consider a general quantum transition amplitude 〈ψ|e−iHt/~|ψ′〉, where t may be real,purely imaginary or generally complex. To construct a functional integral representationof the amplitude:

1. Partition the time interval into N ≫ 1 steps,

e−iHt/~ =[e−iH∆t/~

]N, ∆t = t/N .

2. Regroup the operator content appearing in the expansion of each factor e−iH∆t/~

according to the relation

e−iH∆t/~ = 1 + ∆t∑

mn

cmnAmBn +O(∆t2) ,

where the eigenstates |a〉, |b〉 of A, B are known and the coefficients cmn are c–numbers. (In the quantum mechanical application above A = p, B = q.) This‘normal ordering’ procedure emphasizes that many distinct quantum mechanicalsystems are associated with the same classical action.

3. Insert resolutions of identity according to

e−iH∆t/~ =∑

a,b

|a〉〈a|(1 + ∆t∑

mn

cmnAmBn +O(∆t2))|b〉〈b|

=∑

a,b

|a〉〈a|e−iH(a,b)∆t/~|b〉〈b| +O(∆t2) ,

where H(a, b) is the Hamiltonian evaluated at the eigenvalues of A and B.


3.3. APPLICATIONS OF THE FEYNMAN PATH INTEGRAL 81

4. Regroup terms in the exponent: Due to the ‘mismatch’ of the eigenstates at neigh-bouring time slices n and n + 1, not only the Hamiltonians H(a, b), but also sumsover differences of eigenvalues appear (cf. the last term in the action (3.5)).

5. Take the continuum limit.

3.3 Applications of the Feynman Path Integral

Having introduced the general machinery of path integration we now turn to the discus-sion of specific applications. Our starting point will be an investigation of a low energyquantum particle confined to a single potential well, and the phenomenon of tunnelingin a double well. With the latter, we will become acquainted with instanton techniquesand the role of topology in field theory. The ideas developed in this section will be gener-alised further to the investigation of quantum mechanical decay and quantum dissipation.Finally, we will turn our attention to the development of the path integral for quantummechanical spin and, as a case study, explore the semiclassical trace formulae for quantumchaos.

The simplest example of a quantum mechanical problem is that of a free particle

(H = p2/2m). Yet, within the framework of the path integral, this example, which canbe dealt with straightforwardly by elementary means, is far from trivial: the Gaussianfunctional integral engaged in its construction involves divergences which must be regu-larised by rediscretising the path integral. Nevertheless, its knowledge will be useful as ameans to normalise the path integral in the applications below. Therefore, we leave it asan exercise to show12

Gfree(qF , qI ; t) ≡ 〈qF |e−i~

p2

2mt|qI〉Θ(t) =

( m

2πi~t

)1/2

ei~

m2t

(qF−qI)2Θ(t) (3.29)

where the Heaviside Θ–function reflects causality.13

⊲ Exercise. Derive Eq. (3.29) by the standard methodology of quantum mechanics. Hint:

insert a resolution of identity and perform a Gaussian integral.

⊲ Exercise. Using the path integral, obtain a perturbative expansion for the scattering

amplitude 〈p′|U(t → ∞, t′ → −∞)|p〉 of a free particle from a short–ranged central potential

V (r). In particular, show that the first order term in the expansion recovers the Born scattering

amplitude −i~e−i(t−t′)E(p)/~δ(E(p) − E(p′)) 〈p′|V |p〉.

12Compare this result to the solution of a classical diffusion equation.13Motivated by its interpretation as a Green function, in the following we will refer to the quantum

transition probability amplitude by the symbol G (as opposed to U used above).



V

q

ω

Figure 3.5: Solid: Potential well. Dashed: Quadratic fit approximating the potentialshape close to the minimum.

3.3.1 Quantum Particle in a Well

As a first application of the path integral, let us consider the problem of a quantumparticle in a one–dimensional potential well (see the figure). The discussion of thisexample will illustrate how the semiclassical evaluation scheme discussed above worksin practice. For simplicity we assume the potential to be symmetric, V (q) = V (−q)with V (0) = 0. The quantity we wish to compute is the probability amplitude thata particle injected at q = 0 returns after a time t, i.e. with H = p2/2m + V (q),

G(0, 0; t) ≡ 〈qF = 0|e−iHt/~|qI = 0〉Θ(t). Drawing on our previous discussion, the pathintegral representation of the transition amplitude is given by

G(0, 0; t) =

∫

q(t)=q(0)=0

Dq exp

[i

~

∫ t

0

dt′L(q, q)

],

where L = mq2/2 − V (q) represents the corresponding Lagrangian.

Now, for a generic potential V (q), the path integral can not be evaluated exactly.Instead, we wish to invoke the semiclassical analysis outlined conceptually above. Ac-cordingly, we must first find solutions to the classical equation of motion. Minimisingthe action with respect to variations of q(t), one obtains the Euler–Lagrange equationof motion mq = −V ′(q) where, as usual, we have used the shorthand V ′(q) ≡ ∂qV (q).According to the Feynman path integral, this equation must be solved subject to theboundary conditions q(t) = q(0) = 0. One solution is obvious, viz. qcl(t) = 0. Assumingthat this is in fact the only solution,14 we obtain (cf. Eqs. (3.26) and (3.27))

G(0, 0; t) ≃∫

r(0)=r(t)=0

Dr exp

[− i

~

∫ t

0

dt′r(t′)m

2

(∂2t′ + ω2

)r(t′)

],

14In general, this assumption is wrong: For smooth potentials V (q), a Taylor expansion of V at small qobtains the harmonic oscillator potential, V (q) = V0 +mω2q2/2+ · · ·. For times t that are commensuratewith π/ω, one has periodic solutions, qcl(t) ∝ sin(ωt) that start out from the origin at time t = 0 andrevisit it at just the right time t. In the next section we will see why the restriction to just the trivialsolution was nonetheless legitimate (for arbitrary times t).



where, by definition, mω2 ≡ V ′′(0) is the second derivative of the potential at the ori-gin.15 Note that, in this case, the contribution to the action from the stationary phasefield configuration vanishes S[qcl] = 0. Following the discussion of section 3.2, Gaussianfunctional integration over r then leads to the semiclassical expansion

G(0, 0; t) ≃ Jdet(−m(∂2

t + ω2)/2)−1/2

, (3.30)

where the prefactor J absorbs various constant prefactors.Operator determinants are usually most conveniently obtained by presenting them as

a product over eigenvalues. In the present case, the eigenvalues ǫn are determined by theequation

−m2

(∂2t + ω2

)rn = ǫnrn ,

which is to be solved subject to the boundary condition rn(t) = rn(0) = 0. A completeset of solutions to this equation is given by16 rn(t

′) = sin(nπt′/t), n = 1, 2, . . ., witheigenvalues ǫn = m[(nπ/t)2 − ω2]/2. Applied to the determinant, one therefore finds

det(−m(∂2

t + ω2)/2)−1/2

=∞∏

n=1

[m

2

((nπt

)2

− ω2

)]−1/2

.

To interpret this result, one must first make sense of the infinite product (which evenseems divergent for times commensurate with π/ω!). Moreover the value of the constantJ has yet to be properly determined. To resolve these difficulties, one may exploit thefact that (a) we do know the transition amplitude (3.29) of the free particle system, and(b) the latter coincides with the transition amplitude G in the special case where thepotential V ≡ 0. In other words, had we computed Gfree via the path integral, we wouldhave obtained the same constant J and, more importantly, an infinite product like theone above, but with ω = 0. This allows the transition amplitude to be regularised as

G(0, 0; t) ≡ G(0, 0; t)

Gfree(0, 0; t)Gfree(0, 0; t) =

∞∏

n=1

[1 −

(ωt

nπ

)2]−1/2 ( m

2πi~t

)1/2

Θ(t) .

Then, with the help of the mathematical identity∏∞

n=1[1 − (x/nπ)2]−1 = x/ sin x, onefinally arrives at the result

G(0, 0; t) ≃√

mω

2πi~ sin(ωt)Θ(t) . (3.31)

In the case of the harmonic oscillator, the expansion of the potential necessarily trun-cates at quadratic order and, in this case, the expression above is exact. (For a moreranging discussion of the path integral for the quantum harmonic oscillator system, see

15Those who are uncomfortable with functional differentiation can arrive at the same expression simplyby substituting q(t) = qcl(t) + r(t) into the action and expanding in r.

16To find the solutions of this equation, recall the structure of the Schrodinger equation of a particlein a one–dimensional box of width L = t!



problem 3.5.) For a general potential, the semiclassical approximation effectively involvesthe replacement of V (q) by a quadratic potential with the same curvature. The calcula-tion above also illustrates how coordinate space fluctuations around a completely staticsolution may reinstate the zero–point fluctuations characteristic of quantum mechanicalbound states.

q

V

Figure 3.6: Solid: Double well potential. Dashed: Inverted potential

3.3.2 Double Well Potential: Tunneling and Instantons

As a second application of the path integral let us now consider the motion of a particlein a double well potential (see the figure). Our aim will be to estimate the quantumprobability amplitude for a particle to either stay at the bottom of one of the localminima or to go from one minimum to the other. In doing so, it is understood thatthe energy range accessible to the particle (i.e. ∆E ∼ ~/t) is well below the potentialbarrier height, i.e. quantum mechanical transfer between minima is by tunnelling. Here,in contrast to the single well system, it is far from clear what kind of classical stationaryphase solutions may serve as a basis for a description of quantum tunnelling; there appearto be no classical paths connecting the two minima. Of course one may think of particles‘rolling’ over the potential hill. Yet, these are singular and, by assumption, energeticallyinaccessible.

The key to resolving these difficulties is an observation, already made above, that thetime argument appearing in the path integral should be considered as a general complexquantity that can (according to convenience) be sent to any value in the complex plane.In the present case, a Wick rotation to imaginary times will reveal a stationary pointof the action. At the end of the calculation, the real time amplitudes we seek can bestraightforwardly obtained by analytic continuation.

To be specific, let us consider the imaginary time transition amplitudes

GE(a,±a; τ) ≡ 〈±a| exp[−τ

~H]|a〉 = G(−a,∓a; τ) (3.32)



where the coordinates ±a coincide with the two minima of the potential. From (3.32) thereal time amplitudes G(a,±a; t) can be recovered by the analytic continuation τ → it.According to section 3.2.1, the Euclidean path integral formulation of the transitionamplitudes is given by

G(a,±a; τ) =

∫

q(0)=±a,q(τ)=a

Dq exp

[−1

~

∫ τ

0

dτ ′(m

2q2 + V (q)

)](3.33)

where the function q now depends on imaginary time. From (3.33) we obtain the station-ary phase (or saddle–point) equations

−mq + V ′(q) = 0. (3.34)

From this result, one can infer that, as a consequence of the Wick rotation, there is aneffective inversion of the potential, V → −V (shown dashed in the figure above). Thecrucial point is that, within the inverted potential landscape, the barrier has become asink, i.e. within the new formulation, there are classical solutions connecting the twopoints, ±a. More precisely, there are three different types of classical solutions whichfulfill the condition to be at coordinates ±a at times 0 and/or τ : (a) The solution whereinthe particle rests permanently at a,17 (b) the corresponding solution staying at −a and,most importantly, (c) the solution in which the particle leaves its initial position at ±a,accelerates through the minimum at 0 and eventually reaches the final position ∓a at timeτ . In computing the transition amplitudes, all three types of paths have to be taken intoaccount. As for (a) and (b), by computing quantum fluctuations around these solutions,one can recover the physics of the zero–point motion described in section 3.3.1 for eachwell individually (exercise: convince yourself that this is true!). Now let us see whathappens if the paths connecting the two coordinates are added to this picture.

The Instanton Gas

The classical solution of the Euclidean equation of motion that connects the two potentialmaxima is called an instanton solution while a solution traversing the same path butin the opposite direction (‘−a → a’ ‘a → −a’) is called an anti–instanton. The name‘instanton’ was invented by ’t Hooft18 with the idea that these objects are very similar in

17Note that the potential inversion answers a question that arose above, i.e. whether or not the classicalsolution staying at the bottom of the single well was actually the only one to be considered. As with thedouble well, we could have treated the single well within an imaginary time representation, whereupon thewell would have become a hill. Clearly there is just one classical solution being at two different times atthe top of the hill, viz. the solution that stays there forever. By formulating the semiclassical expansionaround that path, we would have obtained (3.31) with t→ −iτ , which, upon analytic continuation, wouldhave led back to the real time result.

18

Gerardus ’t Hooft1946– : 1999 NobelLaureate in Physics forelucidating the quantumstructure of electroweakinteractions in physics.



their mathematical structure to ‘solitons’, particle–like solutions of classical field theories.However, unlike solitons, they are structures in time (albeit Euclidean time); thus the‘instant–’. As another etymographic remark, note that the syllable ‘–on’ in ‘instanton’hints to an interpretation of these states as a kind of particle. The background is that, asa function of the time coordinate, instantons are almost everywhere constant save for ashort region of variation (see below). Alluding to the interpretation of time as somethingakin to a spatial dimension, these states can be interpreted as a well–localised excitationor, according to standard field theoretical practice, a particle.19

To proceed, we must first compute the classical action associated with a single instan-ton solution. Multiplying (3.34) by qcl, integrating over time (i.e. performing the firstintegral of the equation of motion), and using the fact that at qcl = ±a, qcl = 0 and V = 0,one finds that

m

2q2cl = V (qcl). (3.35)

With this result, one obtains the instanton action

Sinst =

∫ τ

0

dτ ′

mq2cl︷︸︸︷(m

2q2cl + V (qcl)

)=

∫dτ ′

dqcldτ ′

(mqcl) =

∫ a

−a

dq (2mV (q))1/2. (3.36)

Notice that Sinst is determined solely by the functional profile of the potential V (i.e. doesnot depend on the structure of the solution qcl).

Secondly, let us explore the structure of the instanton as a function of time. Definingthe second derivative of the potential at ±a by V ′′(±a) = mω2, Eq. (3.35) implies thatfor large times (where the particle is close to the right maximum), qcl = −ω(qcl−a) whichintegrates to qcl(τ)

τ→∞−→ a − e−τω. Thus the temporal extension of the instanton is setby the oscillator frequencies of the local potential minima (the maxima of the invertedpotential) and, in cases where tunnelling takes place on time scales much larger than that,can be regarded as short (see Fig. 3.7).

The confinement of the instanton configuration to a narrow interval of time has animportant implication — there must exist approximate solutions of the stationary equa-tion involving further anti–instanton/instanton pairs (physically, the particle repeatedlybouncing to and fro in the inverted potential). According to the general philosophy ofthe saddle–point scheme, the path integral is obtained by summing over all solutions ofthe saddle–point equations and hence over all instanton configurations. The summationover multi–instanton configurations — termed the ‘instanton gas’ — is substantiallysimplified by the fact that individual instantons have short temporal support (events ofoverlapping configurations are rare) and that not too many instantons can be accommo-dated in a finite time interval (the instanton gas is dilute). The actual density is dictatedby the competition between the configurational ‘entropy’ (favouring high density), and

19In addition to the original literature, the importance that has been attached to the instanton methodhas inspired a variety of excellent and pedagogical reviews of the field. Of these, the following are highlyrecommended: A. M. Polyakov, Quark confinement and topology of gauge theories, Nucl. Phys. B120,429 (1977); S. Coleman, in Aspects of symmetry — selected Erice lectures, (Cambridge University Press1985) chapter 7.



q

qV

-a

-a aa

t

ω–1

Figure 3.7: Single instanton configuration.

the ‘energetics’, the exponential weight implied by the action (favouring low density) —see the estimate below.

In practice, multi–instanton configurations imply a transition amplitude

G(a,±a; τ) ≃∑

n even / odd

Kn

∫ τ

0

dτ1

∫ τ1

0

dτ2 · · ·∫ τn−1

0

dτnAn(τ1, . . . , τn), (3.37)

where An denotes the amplitude associated with n instantons, and we have taken intoaccount the fact that in order to connect a with ±a, the number of instantons must beeven/odd. The n instanton bounces contributing to each An can take place at arbitrarytimes τi ∈ [0, τ ], i = 1, . . . , n and all these possibilities have to be added (i.e. integrated).Here K denotes a (dimensionful) constant absorbing the temporal dimension [time]n intro-duced by the time integrations, and An(τ1, . . . , τn) is the transition amplitude, evaluatedwithin the semiclassical approximation around a configuration of n instanton bounces attimes 0 ≤ τ1 < τ2 < . . . < τn ≤ τ (see Fig. 3.8). In the following, we will first focus onthe transition amplitude An which controls the exponential dependence of the tunnelingamplitude returning later to consider the prefactor K.

τ 1 ττ5 τ4 τ 3 τ 2

q

–a

a

Figure 3.8: Dilute instanton gas configuration.

According to the general semiclassical principle, each amplitude An = An,cl. × An,qu.

factorises into two parts, a classical contribution An,cl. accounting for the action of theinstanton configuration; and a quantum contribution An,qu. resulting from quadratic fluc-tuations around the classical path. Focusing initially on An,cl. we note that, at intermedi-ate times, τi ≪ τ ′ ≪ τi+1, where the particle rests on top of either of the maxima at ±a,



no action accumulates (cf. the previous section). However, each instanton bounce has afinite action Sinst (see Eq. (3.36)) and these contributions add up to give the full classicalaction,

An,cl.(τ1, . . . , τn) = e−nSinst/~, (3.38)

which is independent of the time coordinates τi. (The individual instantons ‘don’t knowof each other’; their action is independent of their relative position.)

As for the quantum factor An,qu., there are, in principle, two contributions. Whilstthe particle rests on either of the hills (the straight segments in Fig. 3.8), quadraticfluctuations around the classical (i.e. spatially constant) configuration play the samerole as the quantum fluctuations considered in the previous section, the only differencebeing that we are working in a Wick rotated picture. There it was found that quantumfluctuations around a classical configuration which stays for a (real) time t at the bottomof the well, result in a factor

√1/ sin(ωt) (the remaining constants being absorbed into

the prefactor Kn). Rotating to imaginary times, t→ −iτ , one can infer that the quantumfluctuation accumulated during the stationary time τi+1 − τi is given by

√1

sin(−iω(τi+1 − τi))∼ e−ω(τi+1−τi)/2,

where we have used the fact that, for the dilute configuration, the typical separation timesbetween bounces are much larger than the inverse of the characteristic oscillator scales ofeach of the minima. (It takes the particle much longer to tunnel through a high barrierthan to oscillate in either of the wells of the real potential.)

Now, in principle, there are also fluctuations around the ‘bouncing’ segments of thepath. However, due to the fact that a bounce takes a time of O(ω−1) ≪ ∆τ , where ∆τrepresents the typical time between bounces, one can neglect these contributions (whichis to say that they can be absorbed into the prefactor K without explicit calculation).Within this approximation, setting τ0 ≡ 0, τn+1 ≡ τ , the overall quantum fluctuationcorrection is given by

An,qu.(τ1, . . . , τn) =n∏

i=0

e−ω(τi+1−τi)/2 = e−ωτ/2, (3.39)

again independent of the particular spacial configuration τi. Combining (3.38) and(3.39), one finds that

G(a,±a; τ) ≃∞∑

n even / odd

Kne−nSinst/~e−ωτ/2

τn/n!︷︸︸︷∫ τ

0

dτ1

∫ τ1

0

dτ2 · · ·∫ τn−1

0

dτn

= e−ωτ/2∑

n even / odd

1

n!

(τKe−Sinst/~

)n. (3.40)

Finally, performing the summation, one obtains the transition amplitude

G(a,±a; τ) ≃ Ce−ωτ/2

cosh(τKe−Sinst./~

)

sinh(τKe−Sinst./~

) (3.41)



where C is some factor that depends in a non–exponential way on the transition time.Before we turn to a discussion of the physical content of this result, let us check the

self–consistency of our central working hypothesis — the diluteness of the instanton gas.To this end, consider the representation of G in terms of the partial amplitudes (3.40). Todetermine the typical number of instantons contributing to the sum, one may make use ofthe fact that, for a general sum

∑n cn of positive quantities cn > 0, the ‘typical’ value of

the summation index can be estimated as 〈n〉 ≡∑n cnn/∑

n cn. With the abbreviationX ≡ τKe−Sinst/~, the application of this estimate to our current sum yields

〈n〉 ≡∑

n nXn/n!∑

nXn/n!

= X,

where we have used the fact that, as long as 〈n〉 ≫ 1, the even/odd distinction in the sumis irrelevant. Thus, we can infer that the average instanton density, 〈n〉/τ = Ke−Sinst/~ isboth exponentially small in the instanton action Sinst, and independent of τ confirmingthe validity of our diluteness assumptions above.

q

q

V

S

A

ψ

Figure 3.9: Quantum states in the double well: Dashed: Harmonic oscillator states. Solid:Exact eigenstates.

Finally, let us discuss how the form of the transition amplitude (3.41) can be under-stood in physical terms. To this end, let us reconsider the basic structure of the problemwe are dealing with (see Fig. 3.9). While there is no coupling across the barrier, theHamiltonian has two independent, oscillator–like sets of low lying eigenstates sitting inthe two local minima. Allowing for a weak inter–barrier coupling, the oscillator groundstates (like all higher states) split into a doublet of a symmetric and an antisymmetriceigenstate, |S〉 and |A〉 with energies ǫA and ǫS, respectively. Focusing on the low energysector formed by the ground state doublet, we can express the transition amplitudes (3.32)as

G(a,±a; τ) ≃ 〈a|(|S〉e−ǫSτ/~〈S| + |A〉e−ǫAτ/~〈A|

)| ± a〉.



Setting ǫA/S = ~ω/2 ± ∆ǫ/2, where ∆ǫ represents the tunnel-splitting, the symmetryproperties |〈a|S〉|2 = |〈−a|S〉|2 = C/2 and 〈a|A〉〈A| − a〉 = −|〈a|A〉|2 = −C/2 imply that

G(a,±a; τ) ≃ C

2

(e−(~ω−∆ǫ)τ/2~ ± e−(~ω+∆ǫ)τ/2~

)= Ce−ωτ/2

cosh(∆ǫτ/~)sinh(∆ǫτ/~)

.

Comparing this expression with Eq. (3.41) the interpretation of the instanton calculationbecomes clear: At long times, the transition amplitude engages the two lowest states —the symmetric and anti–symmetric combination of the two oscillator ground states. Theenergy splitting ∆ǫ accommodates the energy shift due to the tunneling between the twowells. Remarkably, the effect of tunneling was obtained from a purely classical picture(formulated in imaginary time!). The instanton calculation also produced a prediction forthe tunnel splitting of the energies, viz.

∆ǫ = ~K exp(−Sinst/~),

which, up to the prefactor, agrees with the result of a WKB–type analysis of the tunnelprocess.

Before leaving this section, two general remarks on instantons are in order:

⊲ In hindsight, was the approximation scheme used above consistent? In particular,terms at second order in ~ were neglected, while terms non–perturbative in ~ (theinstanton) were kept. Yet, the former typically give rise to a larger correction to theenergy than the latter. However, the large perturbative shift effects the energies ofthe symmetric and antisymmetric state equally. The instanton contribution givesthe leading correction to the splitting of the levels. It is the latter which is likely tobe of more physical significance.

⊲ Secondly, it may — legitimately — appear as though the development of the ma-chinery above was a bit of an “overkill” for describing a simple tunnelling process.As a matter of fact, the basic result (3.41) could have been obtained in a simplerway by more elementary means (using, for example, the WKB method). Why thendid we discuss instantons at such length? One reason is that, even within a purelyquantum mechanical framework, the instanton formulation of tunnelling is muchstronger than WKB. The latter represents, by and large, an uncontrolled approxi-mation. In general it is hard to tell whether WKB results are accurate or not. Incontrast, the instanton approximation to the path integral is controlled by a numberof well–defined expansion parameters. For example, by going beyond the semiclas-sical approximation and/or softening the diluteness assumption, the calculation ofthe transition amplitudes can, in principle, be driven to arbitrary accuracy.

⊲ A second, and for our purposes, more important motivation is that instanton tech-niques are of crucial importance within higher dimensional field theories (here weregard the path integral formulation of quantum mechanics as a 0 space +1 time= 1–dimensional field theory). The reason is that instantons are intrinsically non–perturbative objects, which is to say that instanton solutions to stationary phaseequations describe a type of physics that cannot be obtained by a perturbative ex-pansion around a non–instanton sector of the theory. (For example, the bouncing



orbits in the example above cannot be incorporated into the analysis by doing a kindof perturbative expansion around a trivial orbit.) This non–perturbative nature ofinstantons can be understood by topological reasoning:

Relatedly, one of the features of the instanton analysis above was that the numberof instantons involved was a stable quantity; ‘stable’ in the sense that by includingperturbative fluctuations around the n instanton sector, say, one does not connectwith the n+ 2 sector. Although no rigorous proof of this statement has been given,it should be heuristically clear: a trajectory involving n bounces between the hills ofthe inverted potential cannot be smoothly connected with one of a different number.Suppose for instance we would forcably attempt to interpolate between two pathswith different bounce numbers: Inevitably, some of the intermediate configurationswould be charged with actions that are far apart from any stationary phase likevalue. Thus, the different instanton sectors are separated by an energetic barrierthat cannot be penetrated by smooth interpolation and, in this sense, they aretopologically distinct.

⊲ Info. Fluctuation determinant: Our analysis above provided a method to extract thetunneling rate between the quantum wells to a level of exponential accuracy. However, in someapplications, it is useful to compute the exponential prefactor K. Although such a computationfollows the general principles outlined above and implemented explicitly for the single well, thereare some idiosyncracies in the tunneling system which warrant discussion.

According to the general principles outlined in section 3.2.2, integrating over Gaussian fluctu-ations around the saddle–point field configurations, the contribution to the transition amplitudefrom the n–instanton section is given by

Gn = Jdet(−m∂2

τ + V ′′(qcl,n))e−nSinst./~

where qcl,n(τ) represents an n–instanton configuration and J the normalisation. Now, in thezero instanton sector, the evaluation of the functional determinant recovers the familiar harmonicoscillator result, G(0, 0; τ) = (mω/π~)1/2 exp[−ωτ/2]. Let us now consider the one instantonsector of the theory. To evaluate the functional determinant, one must consider the spectrumof the operator −m∂2

τ + V ′′(qcl,1). Differentiating the defining equation for qcl,1 (3.34), one mayconfirm that

(−m∂2

τ + V ′′(qcl,1))∂τqcl,1 = 0,

i.e. the function ∂τqcl,1 presents a zero mode of the operator!. Physically, the origin of thezero mode is elucidated by noting that a translation of the instanton along the time axis,qcl,1(τ) → qcl,1(τ + δτ) should leave the action approximately invariant. However, for small δτ ,qcl,1(τ+δτ) ≃ qcl,1(τ)+δτ∂τ qcl,1, i.e. to first order, the addition of the increment function ∂τqcl,1leaves the action invariant, and δτ is a ‘zero mode coordinate’.

With this interpretation, it becomes clear how to repair the formula for the fluctuation deter-minant. While the Gaussian integral over fluctuations is controlled for the non–zero eigenvalues,its execution for the zero mode must be rethought. Indeed, by integrating over the coodinateof the instanton, viz.

∫ τ0 dτ0 = τ , one finds that the contribution to the transition amplitude in

the one instanton sector is given by

Jτ

√Sinst.

2π~det′

[−m∂2

τ + V ′′(qcl,1)]−1/2

e−Sinst./~



where the prime indicates the exclusion of the zero mode from the determinant, and the factor√Sinst./2π~ reflects the Jacobian associated with the change to a new set of integration variables

which contains the zero mode coordinate τ as one of its elements.20 To fix the, as yet, undeter-mined coupling constant J , we normalize by the fluctuation determinant of the (imaginary time)harmonic oscillator, i.e. we use the fact that (cf. section 3.3.1), for the harmonic oscillator, the

return amplitude evaluates to G(0, 0, τ) = J det(m(−∂2τ + ω2)/2)−1/2 =

(mωπ~

)1/2e−ωτ/2, where

the first/second representation is the imaginary time variant of Eq. (3.30)/Eq.(3.31). Using thisresult, and noting that the zero mode analysis above generalizes to the n–instanton sector, wefind that the pre–exponential constant K used in our analysis of the double well problem aboveaffords the explicit representation

K = ω

√Sinst.

2π~

[mω2det′

[−m∂2

τ + V ′′(qcl,1)]

det [−m∂2τ +mω2]

]−1/2

.

Naturally, the instanton determinant depends sensitively on the particular nature of the potentialV (q). For the quartic potential V (q) = mω2(x2 − a2)2/8a2, it may be confirmed that the

mω2det′[−m∂2

τ + V ′′(qcl,1)]

det [−m∂2τ +mω2]

=1

12,

while Sinst =√

2/3mωa2. For further details of the calculation, we refer to, e.g., Zinn-Justin.

——————————————–

Escape From a Metastable Minimum: “Bounces”

The instanton gas approximation for the double well system can be easily adapted toexplore the problem of quantum mechanical tunneling from a metastable state such asthat presented by an unstable nucleus. In particular, suppose one wishes to estimatethe “survival probability” of a particle captured in a metastable minimum of a one–dimensional potential such as that shown in Fig. 3.10.

q q

V

q

q–V

m

qm

τ

Figure 3.10: Effective potential showing a metastable minimum together with the invertedpotential and a sketch of a bounce solution. To obtain the tunnelling rate it is necessaryto sum over a dilute gas of bounce trajectories.

According to the path integral scheme, the survival probability, defined by the proba-bility amplitude to remain at the potential minimum qm, i.e. the propagator G(qm, qm; t),

20For an explicit calculation of this Jacobian see, e.g., J. Zinn-Justin, Quantum Field Theory and

Critical Phenomena (Oxford University Press, 1993).



can be evaluated by making use of the Euclidean time formulation of the Feynman pathintegral. As with the double well, in the Euclidean time formalism, the dominant contri-bution to the transition probability arises from the classical path minimising the actioncorresponding to the inverted potential (see Fig. 3.10). However, in contrast to the doublewell potential, the classical solution takes the form of a ‘bounce’ (i.e. the particle spendsonly a short time away from the potential minimum — there is only one metastableminimum of the potential). As with the double well, one can expect mutliple bouncetrajectories to present a significant contribution. Summing over all bounce trajectories(note that in this case we have an exponential series — no even/odd parity effect), oneobtains the survival probability

G(qm, qm; τ) = Ce−ωτ/2 exp[τKe−Sbounce/~

].

Applying an analytic continuation to real time, one findsG(θm, θm; t) = Ce−iωt/2 exp[−Γ2t],

where the decay rate is given by Γ/2 = |K|e−Sbounce/~. (Note that on physical grounds wecan see that K must be imaginary.21

⊲ Exercise. Consider a heavy nucleus having a finite rate of α-decay. The nuclear forces

can be considered very short-ranged so that the rate of α particle emission is controlled by

tunneling under a Coulomb barrier. Taking the effective potential to be spherically symmetric

with a deep minimum core of radius r0 beyond which it decays as U(r) = 2(Z − 1)e2/r where Z

is the nuclear charge, find the temperature of the nuclei above which α-decay will be thermally

assisted if the energy of the emitted particles is E0. Estimate the mean energy of the α particles

as a function of temperature.

⊲ Exercise. A uniform electric field E is applied perpendicular to the surface of a metal

with work function W . Assuming that the electrons in the metal describe a Fermi gas of density

n, with exponential accuracy, find the tunneling current at zero temperature (“cold emission”).

Show that, effectively, only electrons with energy near the Fermi level are tunneling. With the

same accuracy, find the current at finite temperature (“hot emission”). What is the most prob-

able energy of tunneling electrons as function of temperature?

3.3.3 †Tunneling of Quantum Fields: ‘Fate of the False Vacuum’

⊲ Additional Example: Hitherto we have focussed on applications of the Feynman pathintegral to the quantum mechanics of isolated point–like particles. In this setting, the merit ofthe path integral scheme over, say, standard perturbative methods or the ‘WKB’ approach isperhaps not compelling. Therefore, by way of motivation, let us here present an example whichbuilds upon the structures elucidated above and which illustrates the power of the path integralmethod.

21In fact, a more careful analysis shows that this estimate of the decay rate is too large by a factor of2 (for further details see, e.g., Coleman, Aspects of Symmetry: Selected Erice Lectures, CUP.)



Figure 3.11: Snapshot of a field configuration φ(x, t = const.) in a potential landscapewith two nearly degenerate minima. For further discussion, see the text.

To this end, let us consider a theory involving a continuous classical field which can adopttwo homogeneous equilibrium states with different energy densities: To be concrete, one mayconsider an harmonic chain confined to one or other minimum of an asymmetric quasi one–dimensional ‘gutter–like’ double well potential (see Fig. 3.11). When quantised, the state ofhigher energy density becomes unstable through barrier penetration — it is said to be a “falsevacuum”.22 Specifically, drawing on our discussion of the harmonic chain in chapter 1, let usconsider a quantum system specified by the Hamiltonian density

H =π2

2m+ksa

2

2(∂xφ)2 + V (φ), (3.42)

where [π(x), φ(x′)] = −i~δ(x−x′). Here we have included a potential V (φ) which, in the presentcase, assumes the form of a double well. The inclusion of a weak bias −fφ into V (φ) identifies astable and a metastable potential minimum. Previously, we have seen that, in the absence of theconfining potential, the quantum string exhibits low–energy collective wave–like excitations —phonons. In the confining potential, these harmonic fluctuations are rendered massive. However,drawing on the quantum mechanical principles established in the single–particle system, onemight assume that the string tunnels freely between the two potential minima. To explore thecapacity of the system to tunnel, let us suppose that, at some time t = 0, the system adopts afield configuration in which the string is located in the (metastable) minimum of the potentialat, say, φ = −a. What is the probability that the entire string of length L will tunnel acrossthe barrier into the potential minimum at φ = a in a time t?

⊲ Info. The tunneling of fields between nearly degenerate ground state plays a role innumerous physical contexts. By way of example, consider a superheated liquid. In thiscontext, the ‘false’ vacuum is the liquid state, and the true one the gaseous phase. The role ofthe field is taken by the local density distribution in the liquid. Thermodynamic fluctuationstrigger the continuous appearance of vapor bubbles in the liquid. For bubbles of too small a

22 For a detailed discussion of the history and ramifications of this idea, we refer on the originalinsightful paper by Sidney Coleman, Fate of the false vacuum: semiclassical theory, Phys. Rev. D 15,2929 (1977). Indeed, many of the ideas developed in this work were anticipated in an earlier analysis ofmetastability in the context of classical field theories by J. S. Langer, Theory of the condensation point,Ann. Phys. (N.Y.) 41, 108 (1967).



diameter, the gain in volume energy is outweighed by the surface energy cost — the bubble willcollapse. However, for bubbles beyond a certain critical size, the energy balance is positive. Thebubble will grow and, eventually, swallow the entire mass density of the system; the liquid hasvaporised or, more formally, the density field has tunneled23 from the false ground state into thetrue one.

More speculative (but also potentially more damaging) manifestations of the phenomenonhave been suggested in the context of cosmology:22 What if the big bang released our universenot into its true vacuum configuration but into a state separated by a huge barrier from a morefavourable sector of the energy landscape. In this case, everything would depend on the tunnelingrate: ‘If this time scale is of the order of milliseconds, the universe is still hot when the false

vacuum decays... if this time is on the order of years, the decay will lead to a sort of secondary

big bang with interesting cosmological consequences. If this time is of the order of 10 9 years, we

have occasion for anxiety.’ (S. Coleman, ibid.).——————————————–

Previously, for the point–particle system, we have seen that the transition probability be-tween the minima of the double well is most easily accessed by exploring the classical fieldconfigurations of the Euclidean time action. In the present case, anticipating to some extent ourdiscussion of the quantum field integral in the next chapter, the Euclidean time action associatedwith the Hamiltonian density (3.42) assumes the form24

S[φ] =

∫ T

0dτ

∫ L

0dx

[m

2(∂τφ)2 +

ksa2

2(∂xφ)2 + V (φ)

],

where the time integral runs over the interval [0, T = it]. Here, for simplicity, let us assumethat the string obeys periodic boundary conditions in space, viz. φ(x + L, τ) ≡ φ(x, τ). Toestimate the tunneling amplitude, we will explore the survival probability of the metastablestate imposing the boundary conditions φ(x, τ = 0) = φ(x, τ = T ) = −a on the path integral.Once again, when the potential barrier is high, and the time T is long, one may assume that thepath integral is dominated by the saddle–point field configuration of the Euclidean action. Inthis case, varying the action with respect to the field φ(x, τ), one obtains the classical equationof motion

m∂2τφ+ ksa

2∂2xφ = ∂φV (φ) ,

which must be solved subject to the boundary conditions above.

Now, motivated by our consideration of the point–particle problem, one might seek a solutionin which the string tunnels as a single rigid entity without ‘flexing’. However, it is evident fromthe spatial translational invariance of the system that the instanton action would scale with thesystem size L. In the infinite system L → ∞, such a trajectory would therefore not contributesignificantly to the tunneling amplitude. Instead, one must consider a different type of fieldconfigurataion in which the transfer of the chain is by degree: Elements of the string cross the

23At this point, readers should no longer be confused regarding the mentioning of ‘tunneling’ in thecontext of a classical system: Within the framework of the path integral, the classical partition sum mapsonto the path integral of a fictitious quantum system. It is the tunneling of the latter we have in mind.

24Those readers who wish to develop a more rigorous formulation of the path integral for the stringmay either turn to the discussion of the field integral in the next chapter or, alternatively, may satisfythemselves of the validity of the Euclidean action by (re–)discretisating the harmonic chain, presentingthe transition amplitude as a series of Feynman path integrals for each element of the string and, finally,taking the continuum limit.



Figure 3.12: On the tunneling between two nearly degenerate vacuum states. As timemoves on, a one–dimensional ‘world sheet’ sweeps through a circular structure in Eu-clidean space time. This results in the inflation of a ‘bubble’ of the true vacuum state inreal space.

barrier in a consecutive sequence as two outwardly propagating ‘domain walls’ (see the figurewhere the emergence of such a double–kink configuration is shown as a function of space andtime; notice the spherical shape of the resulting space–time droplet — a consequence of therotational symmetry of the rescaled problem). Such a field configuration can be motivatedfrom symmetry considerations by noting that, after rescaling x 7→ vsx (where vs =

√ksa2/m

denotes the classical sound wave velocity), the saddle–point equation assumes the isotropic formm∂2φ = ∂φV (φ), where ∂2 = ∂2

τ + ∂2x. Then, setting r =

√x2 + (τ − T/2)2, and sending

(T,L) → ∞, the space–time rotational symmetry suggests a solution of the form φ = φ(r)where φ(r) obeys the radial diffusion equation

∂2rφ+

1

r∂rφ = ∂φV ,

with the boundary condition limr→∞ φ(r) = −a. This equation describes the one–dimensionalmotion of a particle in a potential −V and subject to a strange “friction force” −r−1∂rφ whosestrength is inversely proportional to ‘time’ r.

To understand the profile of the non–trivial bounce solution of the problem, suppose that attime r = 0 the particle has been released at rest at a position slightly to the left of the (inverted)potential maximum at a. After rolling through the potential minimum it will climb the potentialhill at −a. Now, the initial position may be fine tuned such that the viscous damping of theparticle compensates for the excess potential energy (which would otherwise make the particleovershoot and disappear to infinity): there exists a solution where the particle starts close toφ = a and eventually winds up at φ = −a, in accord with the imposed boundary conditions. Ingeneral, the analytical solution for the bounce depends sensitively on the form of the confiningpotential. However, while we assume that the well asymmetry imposed by external potential−fφ is small, the radial equation may be considerably simplified. In this limit, one may invoke a



“thin-wall” approximation in which one assumes that the bounce configuration is described by adomain wall of thickness ∆r, at a radius r0 ≫ ∆r separating an inner region where φ(r < r0) = afrom the outer region where φ(r > r0) = −a. In this case, and to lowest order in an expansionin f , the action of the friction force is immaterial, i.e. we may set m∂2

rφ = ∂φV — the veryinstanton equation formulated earlier for the point–particle system!

Then, when substituted back into S, one finds that the bounce (or kink–like) solution ischaracterised by the Euclidean action

S = vs[2πr0Sinst. − πr20 2af

]

where Sinst. denotes the action of the instanton associated with the point–particle system (3.36),and the last term accommodates the effect of the potential bias on the field configuration. Cru-cially, one may note that the instanton contribution to the action scales with the circumferenceof the domain wall in the space–time, while that of the potential bias scales with the area of thedomain. From this scaling dependence, it is evident that, however small is the external forcef , at large enough r0, the contribution of the second term will always outweigh the first andthe string will tunnel from the metastable minimum to the global minimum of the potential.More precisely, the optimal size of domain is found by minimising the action with respect to r0.In doing so, one finds that r0 = Sinst./2af . Then, when substituted back into the action, oneobtains the tunneling rate

Γ ∼ exp

[−1

~

πvsS2inst.

2af

].

From this result, one can conclude that, in the absence of an external force f , the tunneling ofthe string across the barrier is completely quenched ! In the zero temperature unbiased system,the symmetry of the quantum Hamiltonian is broken: The ground state exhibits a two–folddegeneracy in which the string is confined to one potential minimum or another.

The ramifications of the tunneling amplitude suppression can be traced to the statisticalmechanics of the corresponding classical system: As emphasized in section 3.2.1, any Euclideantime path integral of a d–dimensional system can be identified with the statistical mechanics ofa classical system (d+ 1)–dimensional problem. In the double well system, the Euclidean timeaction of the point–particle quantum system is isomorphic to the one–dimensional realisation ofthe classical Ising ferromagnet, viz.

βHIsing =

∫ L

0ddx

[t

2m2 + um4 +

K

2(∇m)2

](3.43)

Translated into this context, the saddle–point (or mean–field) analysis suggests that the system

will exhibit a spontaneous symmetry breaking to an ordered phase (m 6= 0) when the parameter t

(the reduced temperature) becomes negative. However, drawing on our analysis of the quantum

point–particle system, in the thermodynamic limit, we see that fluctuations (non–perturbative

in temperature) associated with instanton field configurations of the Hamiltonian m(x) may

restore the symmetry of the system and destroy long–range order at any finite temperature 1/β.

Whether this happens or not depends on the competition between the energy cost of instanton

creation and the entropy gained by integrating over the instanton zero mode coordinates. It

turns out that in d = 1, the latter wins, i.e. the system is ‘disordered’ at any finite temperature.

In contrast, for d ≥ 2, the creation of instantons is too costly, i.e. the system will remain in its

energetically preferred ground state.



3.3.4 †Tunneling in a Dissipative Environment

⊲ Additional Example: In the condensed matter context it is, of course, infeasible tocompletely divorce a system from its environment. Indeed, in addition to the dephasing effectof thermal fluctuations, the realization of quantum mechanical phenomena depends sensitivelyon the strength and nature of the coupling to the external degrees of freedom. For example,the tunneling of an atom from one interstitial site in a crystal to another is likely to be heavilyinfluenced by its coupling to the phonon degrees of freedom that characterise the crystal lattice.By exchanging energy with the phonons, which act in the system as an external bath, a quantumparticle can loose its phase coherence and with it, its quantum mechanical character. Begin-ning with the seminal work of Caldeira and Leggett,25 there have been numerous theoreticalinvestigations of the effect of an enviroment on the quantum mechanical properties of a system.Such effects are particularly acute in systems where the quantum mechanical degree of freedomis macroscopic such as the magnetic flux trapped in a superconducting quantum interferencedevice (SQUID). In the following, we will show that the Feynman path integral provides a nat-ural (and almost unique) setting in which the effects of the environment on a microscopic ormacroscopic quantum mechanical degree of freedom can be explored.

Before we begin, let us note that the phenomenon of macroscopic quantum tunneling repre-sents an extensive and still active area of research recently reinvigorated by the burgeoning fieldof quantum computation. By contrast, our discussion here will be necessarily limited in scope,targeting a particular illustrative application, and highlighting only the guiding principles. Fora more thorough and detailed discussion, we refer the reader to one of the many comprehensivereviews.26

Caldeira–Leggett Model

Previously, we have discussed the ability of the Feynman path integral to describe quantum me-chanical tunneling of a particle q across a potential barrier V (q). In the following, we will invokethe path integral to explore the capacity for quantum mechanical tunneling when the particle iscoupled to degrees of freedom of an external environment. Following Calderia and Leggett’s orig-inal formulation, let us represent the environment by a bath of N quantum harmonic oscillatorscharacterised by a set of frequencies ωα,

Hbath[qα] =N∑

α

[p2α

2mα+mα

2ω2αq

2α

].

For simplicity, let us suppose that in the leading approximation, the coupling of the particleto the degrees of freedom of the bath is linear such that Hc[q, qα] = −∑N

α fα[q]qα, where fα[q]represents some function of the particle coordinate q. Expressed as a Feynman path integral,the survival probability of a particle confined to a metastable minimum at a position q = a, andcoupled to an external environment, can then be expressed as (~ = 1)

〈a|e−iHt/~|a〉 =

∫

q(0)=q(t)=aDq eiSpart.[q]

∫Dqα e

iSbath[qα]+iSc[q,qα] ,

25A. O. Calderia and A. J. Leggett, Influence of Dissipation on Quantum Tunneling in Macroscopic

Systems, Phys. Rev. Lett. 46, 211 (1981).26See, e.g., A. J. Leggett et al., Dynamics of the dissipative two–state system, Rev. Mod. Phys 48, 357

(1976), and the text by Weiss [?].



where H = Hpart. + Hbath + Hc denotes the total Hamiltonian of the system,

Spart.[q] =

∫ t

0dt[m

2q2 − V (q)

], Sbath[qα] =

∫ t

0dt∑

α

mα

2

[q2α − ω2

αq2α

],

denote, respectively, the action of the particle and bath, while

Scoupling[q, qα] = −∫ t

0dt∑

α

fα[q]qα −∫dt∑

a

fα[q]2

2maω2a

,

represents their coupling.27 Here we assume that the functional integral over qα(t) is taken overall field configurations of the bath while, as before, the path integral on q(t) is subject to theboundary conditions q(0) = q(t) = a.

To reveal the effect of the bath on the capacity for tunneling of the particle, it is instructiveto integrate out fluctuations qα and thereby obtain an effective action for q. Fortunately, beingGaussian in the coordinates qα, the integration can performed straightforwardly. Although notcrucial, since we are dealing with quantum mechanical tunneling, it is useful to transfer to theEuclidean time representation. Taking the boundary conditions on the fields qα(τ) to be periodicon the interval [0, T−1 ≡ β], it may be confirmed that the Gaussian functional integral over qαinduces a time non–local interaction of the particle (exercise) 〈a|e−iHt/~|a〉 =

∫Dq e−Seff [q] where

a constant of integration has been absorbed into the measure and

Seff [q] = Spart.[q] +1

2T

∑

ωn,α

ω2nfα[q(ωn)]fα[q(−ωn)]mαω2

α(ω2α + ω2

n).

Here, the sum∑

ωnruns over the discrete set of Fourier frequencies ωn = 2πn/β with n inte-

ger.28 By integrating out the bath degrees of freedom, the particle action acquires an inducedcontribution. To explore its effect on dissipation and tunneling, it is necessary to specialise ourdiscussion to a particular form of coupling.

In the particular case that the coupling to the bath is linear, viz. fα[q(τ)] = cαq(τ), theeffective action assumes the form (exercise)

Seff [q] = Spart.[q] − T

∫ β

0dτ dτ ′K(τ − τ ′)q(τ)q(τ ′)

where K(τ) =∫∞0

dωπ J(ω)Dω(τ), J(ω) = π

2

∑α

c2αmαωα

δ(ω − ωα), and

Dω(τ) = −∑

ωn

2ω2n

ω(ω2 + ω2n)eiωnτ ,

resembles the Green function of a boson with energy ~ω. Physically, the non–locality of theaction is easily understood: By exchanging fluctuations with the external bath, a particle can

27The second term in the coupling action has been added to keep the effect of the environment min-imally invasive (purely dissipative). If it would not be present, the coupling to the oscillator degrees offreedom would effectively shift the extremum of the particle potential, i.e. change its potential landscape.Exercise: substitute the solutions of the Euler–Lagrange equations δqα

S[q, qα] = 0 — computed for afixed realization of q — into the action to obtain the said shift.

28More preceisely, anticipating our discussion of the Matsubara frequency representation, we havedefined the Fourier decomposition on the Euclidean time interval T , viz. q(τ) =

∑m qme

iωmτ , qm =

T∫ β

0dτq(τ)e−iωmτ , where ωm = 2πm/β with m integer.



affect a self–interaction, retarded in time. Taken as a whole, the particle and the bath maintainquantum phase coherence. However, when projected onto the particle degree of freedom, thetotal energy of the system appears to fluctuate and the phase coherence of the particle transportis diminished. To explore the properties of the dissipative action, it is helpful to separate thenon–local interaction according to the identity q(τ)q(τ ′) = [q2(τ) + q2(τ ′)]/2− [q(τ)− q(τ ′)]2/2.The former squared contribution presents an innocuous renormalisation of the potential V (q)and, applying equally to the classically allowed motion as well as quantum tunneling, presentsan unobservable perturbation. Therefore, we will suppose that its effect has been absorbed intoa redefinition of the particle potential V (q). By contrast, the remaining contribution is alwayspositive.

The particular form of the “spectral function” J(ω) may be obtained either from an a priori

knowledge of the microscopic interactions of the bath, or phenomenologically, it can be inferredfrom the structure of the classical damped equations of motion. For example, for a systemsubject to an “ohmic” dissipation (where, in real time, the classical equations of motion obtaina dissipative term −ηq with a “friction coefficient” η), one has J(ω) = η|ω| for all frequenciessmaller than some characteristic cut–off (at the scale of the inverse Drude relaxation time of theenvironment). By contrast, for a defect in a three–dimensional crystal, interaction with acousticphonons present a frequency dependence of ω3 or ω5 depending on whether ω is below or abovethe Debye frequency.

⊲ Info. Consider, for example, the coupling of a particle to a continuum of bosonic modeswhose spectral density J(ω) = η

8ω grows linearly with frequency. In this case,

K(ωn) = −ηω2n

8π

∫ ∞

0dω

1

ω2 + ω2n

= −η4|ωn|.

describes Ohmic dissipation of the particle. Fourier transforming this expression we obtain

K(τ) = −πTη4

1

sin2(πTτ)

τ≪T−1

≃ − η

4πT

1

τ2, (3.44)

i.e. a strongly time non–local ‘self–interaction’ of the particle.——————————————–

Disssipative Quantum Tunneling

Returning to the particular problem at hand, previously, we have seen that the tunneling rateof a particle from a metastable potential minimum can be inferred from the extremal fieldconfigurations of the Euclidean action: the bounce trajectory. To explore the effect of thedissipative coupling, it is necessary to understand how it revises the structure of the bouncesolution. Now, in general, the non–local character of the interaction inhibits access to an exactsolution of the classical equation of motion. In such cases, the effect of the dissipative couplingcan be explored perturbatively or with the assistance of the renormalisation group (see thediscussion in section ??). However, by tailoring our choice potential V (φ), we can gain someintuition about the more general situation.

To this end, let us consider a particle of mass m confined in a metastable minimum by a(semi–infinite) harmonic potential trap (see figure),

V (q) =

mω2

cq2/2 0 < q ≤ a ,

−∞ q > a .



V(q)

qa

Further, let us assume that the environment imparts an ohmic dissipation with a damping orviscosity η. To keep our discussion general, let us consider the combined impact of dissipationand temperature on the rate of tunneling from the potential trap. To do so, following Langer29 itis natural to investigate the “quasi–equilibrium” quantum partition function Z of the combinedsystem. In this case, the tunneling rate appears as an imaginary contribution to the free energyF = −T lnZ, viz. Γ = −2ImF .

Drawing on the path integral, the quantum partition function of the system can be presentedas a functional integral Z =

∫q(β)=q(0)Dqe

−Seff where, as we have seen above, for ohmic coupling,the Euclidean action assumes the form

Seff [q] =

∫ β

0dτ(m

2q2 + V (q)

)+

η

4π

∫ β

0dτ dτ ′

(q(τ) − q(τ ′)

τ − τ ′

)2

.

Once again, to estimate the tunneling rate, we will suppose that the barrier is high and thetemperature is low so that the path integral is dominated by stationary configurations of theaction. In this case, one may identify three distinct solutions: In the first place, the particlemay remain at q = 0 poised precariously on the maximum of the inverted harmonic potential.Contributions from this solution and the associated harmonic fluctuations reproduce terms inthe quantum partition function associated with states of the closed harmonic potential trap.Secondly, there exists a singular solution in which the particle remains at the minimum ofthe inverted potential, i.e. perched on the potential barrier. The latter presents a negligiblecontribution to the quantum partition function and can be neglected. Finally, there exists abounce solution in which the particle injected at a position q inside the well accelerates downthe inverted potential gradient, is reflected from the potential barrier, and returns to the initialposition q in a time β. While, in the limit β → ∞, the path integral singles out the boundarycondition q(0) = q(β) → 0, at finite β, the boundary condition will depart from 0 in a mannerthat depends non–trivially on the temperature. It is this general bounce solution which governsthe decay rate.

Since, in the inverted potential, the classical bounce trajectory stays within the interval overwhich the potential is quadratic, a variation of the Euclidean action with respect to q(τ) obtainsthe classical equation of motion

−mq +mω2cq +

η

π

∫ β

0dτ ′

q(τ) − q(τ ′)

(τ − τ ′)2= Aδ(τ − β/2) ,

where the term on the right hand side of the equation imparts an impulse which changes dis-continuously the velocity of the particle, while the coefficient A is chosen to ensure symmetry ofthe bounce solution on the Euclidean time interval. Turning to the Fourier representation, thesolution of the saddle–point equation then assumes the form

qn = ATe−iωnβ/2g(ωn), g(ωn) ≡ [m(ω2n + ω2

c ) + η|ωm|]−1. (3.45)

29J. S. Langer, Ben:...



Imposing the condition that q(τ = β/2) = a, one finds that A = a/f where f ≡ T∑

n g(ωn).Finally, the action of the bounce is given by

Sbounce =1

2T

∑

n

(m(ω2n + ω2

c ) + η|ωm|)|qn|2 =a2

2f. (3.46)

(a) To make sense of these expressions, as a point of reference, let us first determine the zerotemperature tunneling rate in the absence of dissipation, viz. η → 0 and β → ∞.In this case, the (Matsubara) frequency summation translates to the continuous integral,f =

∫∞−∞

dω2π g(ω) = (2mωc)

−1. Using this result, the bounce action (3.46) takes the form

Sbounce = mωca2. As one would expect, the tunnelling rate Γ ∼ e−Sbounce is controlled

by the ratio of the potential barrier height mω2ca

2/2 to the attempt frequency ωc. Alsonotice that the bounce trajectory is given by

q(τ) =a

f

∫ ∞

−∞

dω

2πeiω(τ−β/2)g(ω) = a e−ωc|τ−β/2| ,

i.e. as expected from our discussion in section 3.3.2, the particle spends only a time 1/ωcin the under barrier region.

(b) Now, restricting attention to the zero temperature limit, let us consider the influenceof dissipation on the nature of the bounce solution and the capacity for tunneling.Focussing on the limit in which the dynamics of the particle is overdamped, η ≫ mωc,

f =∫∞−∞ g(ω) ≃ 2

πη ln (η/mωc), which implies Sbounce = πηa2

4 ln[η/(mωc)]. In particular, this

result shows that, in the limit η → ∞, the coupling of the particle to the ohmic bathleads to an exponential suppression of the tunneling rate while only a weak dependenceon the jump frequency persists. Physically, this result is easy to rationalise: Under–barriertunneling is a feature of the quantum mechanical system. By transferring energy to andfrom the external bath, the phase coherence of the particle is lost. At zero temperature,the tunneling rate becomes suppressed and the particle confined.

(c) Let us now consider the influence of temperature on the tunneling rate when thedissipative coupling is inactive η → 0. In this case, the discrete frequency summationtakes the form30 f = T

∑n g(ωn) = coth(βωc/2)

2ωcm. Using this result, one obtains the action

Sbounce = mωca2 tanh(βωc/2). In the low temperature limit β → ∞, Sbounce = mωca

2

as discussed above. At high temperatures β → 0, as expected, one recovers a classicalactivated dependence of the escape rate, viz. S ≃ βmω2

ca2/2.

(d) Finally, let us briefly remark on the interplay of thermal activation with ohmic dis-

sipation. Applying the the Euler-Maclaurin formula∑∞

m=0 f(m) =∫∞0 dx f(x) + f(0)

2 −f ′(0)12 + . . . to relate discrete sums over Matsubara frequencies to their zero temperature

integral limits, one finds that Sbounce(T )−Sbounce(T = 0) ∝ ηT 2. This shows that, in thedissipative regime, an increase in temperature diminishes the tunneling rate with a scaleproportion to the damping.

This concludes our cursory discussion of the application of the Feynman path integral to

dissipative quantum tunneling. As mentioned above, our brief survey was able only to touch

upon the broad field of research. Those interested in learning more about the field of macroscopic

30For details on how to implement the discrete frequency summation, see the info block on p 137 below.



quantum tunneling are referred to the wider literature. To close this chapter, we turn now to

our penultimate application of the path integral — quantum mechanical spin.

3.3.5 †Path Integral for Spin

⊲ Additional Example: The quantum mechanics of a spin 1/2–particle is a standardexample in introductory courses. Indeed, there is hardly any other system whose quantummechanics is as easy to formulate. Given that, it is perhaps surprising that for a long time thespin problem defied all attempts to cast it in path integral form: Feynman, the architect of thepath integral, did not succeed in incorporating spin into the new formalism. It took severaldecades to fill this gap (for a review of the early history up to 1980, see Schulman’s text [20]),and a fully satisfactory formulation of the subject was obtained no earlier than 1988. (Thepresent exposition follows closely the lines of the review by Michael Stone, Supersymmetry and

the Quantum Mechanics of Spin, Nucl. Phys. B 314, 557 (1989).)

Why then is it so difficult to find a path integral of spin? In hindsight it turns out that thespin path integral is in fact no more complex than any other path integral, it merely appearsto be a bit unfamiliar. The reason is that, on the one hand, the integrand of the path integralis essentially the exponentiated classical action whilst, on the other, the classical mechanicsof spin is a subject that is not standard in introductory or even advanced courses. In otherwords, the path integral approach must, by necessity, lead to an unusual object. The fact thatthe classical mechanics of spin is hardly ever mentioned is not only related to the common viewthat spin is something ‘fundamentally quantum’ but also to the fact that the mechanics of aclassical spin (see below) cannot be expressed within the standard formulation of Hamiltonianmechanics, i.e. there is no formulation in terms of a set of globally defined coordinates andequally many global momenta. It is therefore inevitable that one must resort to the (less widelyapplied) symplectic formulation of Hamiltonian mechanics.31 However, as we will see below, theclassical mechanics of spin can nevertheless be quite easily understood physically.

Besides attempting to elucidate the connections beween quantum and classical mechanics ofspin, there is yet an other motivation for discussing the spin path integral. Pretending that wehave forgotten essential quantum mechanics, we will formulate the path integral ignoring thefact that spin quantum numbers are half integer or integer. The quantization of spin will thenbe derived in hindsight, by way of a geometric consideration. In other words, the path integralformulation demonstrates how quantum mechanical results can be obtained by geometric ratherthan standard algebraic reasoning. Finally, the path integral of spin will serve as a basic platformon which our analysis of higher dimensional spin systems below will be based.

A reminder of finite–dimensional SU(2)–representation theory

In order to formulate the spin path integral, it is neccessary to recapitulate some facts regardingthe role of SU(2) in quantum mechanics. The special unitary group in two dimensions, SU(2),is defined as SU(2) = g ∈ Mat(2× 2,C)|g†g = 12, det g = 1, where 12 is the two–dimensionalunit matrix. Counting independent components one finds that the group has three free real

31Within this formulation, the phase space is regarded as a differential manifold with a symplecticstructure (cf. Arnold’s text on classical mechanics [?]). (In the case of spin, this manifold is the two—sphere S2.)



parameters or, equivalently, that its Lie algebra, su(2), is three dimensional. As we have seen,the basis vectors of the algebra — the group generators — Si, i = x, y, z satisfy the closurerelation [Si, Sj ] = iǫijkS

k, where ǫijk is the familiar fully antisymmetric tensor. An alternative,and often more useful basis representation of su(2) is given by the spin raising and loweringoperators, S± = (Sx ± iSy)/2. Again, as we have seen earlier, the algebra S+, S−, Sz isdefined by the commutation relations [S+, S−] = 2Sz, [Sz, S±] = ±2S±.

Each group element can be uniquely parametrized in terms of the exponentiated algebra.For example, in the Euler angle representation,32 the group is represented as

SU(2) =g(φ, θ, ψ) = e−iφS3e−iθS2e−iψS3

∣∣∣φ,ψ ∈ [0, 2π], θ ∈ [0, π].

The Hilbert space HS of a quantum spin represents an irreducible representation space of SU(2).Within the spaces HS , SU(2) acts in terms of representation matrices (which will be denotedby g) and the matrix representations of its generators Si. The index S is the so–called weightof the representation (physically: the total spin).33 Within each HS , there is a distinguishedstate, a state of highest weight | ↑〉, which is defined as the (normalized) eigenstate of Sz withmaximum eigenvalue, S (physically: a spin state polarised in the 3–direction, often denoted as|S, Sz = S〉, where m is the azimuthal quantum number). Owing to the irreducibility of therepresentation, each (normalized) state of the Hilbert space HS can be obtained by applyingthe Euler–angle–parameterized elements of the representation to the maximum weight state.

Being a compact group, SU(2) can be integrated over; i.e. it makes sense to define objectslike

∫SU(2) dgf(g), where f is some function of g and dg is a realization of a group measure.34

Among the variety of measures that can be defined in principle, the (unique) Haar measureplays a distinguished role. It has the convenient property that it is invariant under left and rightmultiplication of g by fixed group elements; i.e.

∀h ∈ SU(2) :

∫dgf(gh) =

∫dgf(hg) =

∫dgf(g),

where, for notational simplicity, we have omitted the subscript in∫SU(2).

Construction of the path integral

With this background, we are now in a position to formulate the Feynman path integral forquantum mechanical spin. To be specific, let us consider a particle of spin S subject to theHamiltonian

H = B · S,32

Leonhard Euler 1707–1783: Swiss mathematician and physicist, one ofthe founders of pure mathematics. He not only made decisive and forma-tive contributions to the subjects of geometry, calculus, mechanics, andnumber theory but also developed methods for solving problems in obser-vational astronomy and demonstrated useful applications of mathematicsin technology and public affairs.

33The index S is defined in terms of the eigenvalues of the Casimir operator (physically: the total

angular momentum operator) S2 ≡∑i S2i according to the relation ∀|s〉 ∈ HS : S2|s〉 = S(S + 1)|s〉.

34To define group measures in a mathematically clean way, one makes use of the fact that (as a Liegroup) SU(2) is a 3–dimensional differentiable manifold. Group measures can then be defined in termsof the associated volume form (see the primer in differential geometry on page ?? below).



where B is a magnetic field and S ≡ (S1, S2, S3) is a vector of spin operators in the spin–S representation. Our aim is to calculate the imaginary time path integral representation of

the quantum partition function Z ≡ tr e−βH . In constructing the path integral we will followthe general strategy outlined at the end of section 3.2.3, i.e. the first step is to represent

Z as Z = tr (e−ǫH)N , where ǫ = β/N . Next, we have — the most important step in theconstruction — to insert a suitably chosen resolution of identity between each of the factors

e−ǫH . A representation that will lead us directly to the final form of the path integral is specifiedby

id. = C

∫dg |g〉〈g| (3.47)

where ‘id.’ represents the unit operator in HS,∫dg is a group integral over the Haar measure,

C is some constant and |g〉 ≡ g| ↑〉 is the state obtained by letting the representation matrixg act on the maximum weight state | ↑〉 (cf. the summary of the SU(2) representation theoryabove).

Of course it remains to be verified that the integral (3.47) is indeed proportional to the unitoperator. That this is so follows from Schur’s lemma which states that if, and only if, anoperator A commutes with all representation matrices of an irreducible group representation (inour case the g s acting in the Hilbert space HS), A is either zero or proportional to the unitmatrix. That the group above integral fulfils the global commutativity criterion follows fromthe properties of the Haar measure: ∀h ∈ SU(2),

h

∫dg|g〉〈g| =

∫dg|hg〉〈g| Haar

=

∫dg|hh−1g〉〈h−1g| =

∫dg|g〉〈g|h.

Thus,∫dg|g〉〈g| is, indeed, proportional to the unit operator. The proportionality constant

appearing in (3.47) will not be of any concern to us — apart from the fact that it is non–zero.35

Substituting the resolution of identity into the time–sliced partition function and makinguse of the fact that

〈gi+1|e−ǫB·S|gi〉 ≃ 〈gi+1|gi〉 − ǫ〈gi+1|B · S|gi〉〈gi|gi〉=1

= 1 − 〈gi|gi〉 + 〈gi+1|gi〉 − ǫ〈gi+1|B · S|gi〉≃ exp

(〈gi+1|gi〉 − 〈gi|gi〉 − ǫ〈gi+1|B · S|gi〉

),

one obtains

Z = limN→∞

∫

gN=g0

N∏

i=0

dgi exp

[−ǫ

N−1∑

i=0

(−〈gi+1|gi〉 − 〈gi|gi〉

ǫ+ 〈gi+1|B · S|gi〉

)].

Taking the limit N → ∞, the latter can be cast in path integral form,

Z =

∫Dg exp

[−∫ β

0dτ(−〈∂τg|g〉 + 〈g|B · S|g〉

)](3.48)

where the HS–valued function |g(τ)〉 is the continuum limit of |gi〉. Eq. (3.48) is our final, albeitsomewhat over–compact, representation of the path integral. In order to give this expressionsome physical interpretation, we need to examine more thoroughly the meaning of the states|g〉.

35Actually, the constant C can be straightforwardly computed by taking the trace of (3.47) which leadsto C =(dimension of the representation space)/(volume of the group).



In the literature, the states |g〉 expressed in the Euler–angle representation

|g(φ, θ, ψ)〉 ≡ e−iφS3e−iθS2e−iψS3 | ↑〉

are referred to as spin coherent states. Before discussing the origin of this terminology,it is useful to explore the algebraic structure of these states. First, note that the maximumweight state | ↑〉 is, by definition, an eigenstate of S3 with maximum eigenvalue S. Thus,

|g(φ, θ, ψ)〉 ≡ e−iφS3e−iθS2 | ↑〉e−iψS and the angle ψ enters the coherent state merely as a phaseor gauge factor. By contrast, the two remaining angles θ and φ act through true rotations.Now, the angular variables φ ∈ [0, 2π[ and θ ∈ [0, π[ define a standard representation of thetwo–sphere. In view of the fact that (up to normalization factors) the states |g(φ, θ, ψ)〉 coverthe entire Hilbert space HS , we are led to suspect that the latter bears structural similarity witha sphere.36 To substantiate this view, let us compute the expectation values

ni ≡ 〈g(φ, θ, ψ)|Si|g(φ, θ, ψ)〉, i = 1, 2, 3. (3.49)

To this end, we first derive an auxiliary identity which will spare us much of the trouble thatwill arise in expanding the exponentials appearing in the definition of |g〉. Making use of thegeneral identity (i 6= j)

e−iφSi SjeiφSi = e−iφ[Si, ]Sj = Sj cosφ+ ǫijkSk sinφ, (3.50)

where the last equality follows from the fact that cos x (sinx) contain x in even (odd) orders and[Sj , ]2Si = Si, it is a straightforward matter to obtain (exercise) n = S(sin θ cosφ, sin θ sinφ, cos θ),i.e. n is the product of S and a unit vector parameterized in terms of spherical coordinates.This is the key to understanding the terminology ‘spin coherent states’: The states |g(φ, θ, ψ)〉represent the closest approximation of a classical angular momentum one can form out of spinoperators (see the figure).

θ

ψ

Let us now see what happens if we employ the Euler angle representation in formulating thepath integral. A first and important observation is that the path integral is gauge invariant —in the sense that it does not depend on the U(1)–phase, ψ. As for the B–dependent part of theaction, the gauge invariance is manifest: Eq. (3.49) implies that

SB[φ, θ] ≡∫ β

0dτ〈g|B · S|g〉 =

∫ β

0dτ〈g|B · S|g〉 = S

∫ β

0dτ n · B = SB

∫ β

0dτ cos θ.

36There is a group theoretical identity behind this observation, viz. the isomorphism SU(2) ≃ S2×U(1),where U(1) is the ‘gauge’ subgroup contained in SU(2).



Here, we have introduced the gauge–independent part |g〉 of the state vector by setting |g〉 ≡|g〉 exp(−iSψ) or,equivalently, |g(φ, θ)〉 ≡ e−iφS3e−iθS2 | ↑〉. Substituting this representation intothe first term of the action of (3.48), one obtains

Stop[φ, θ] ≡ −∫ β

0dτ〈∂τ g|g〉 = −

∫ β

0dτ〈∂τe−iSψg|ge−iSψ〉

= −∫ β

0dτ (〈∂τg|g〉 − iS ∂τψ〈g|g〉) = −

∫ β

0dτ〈∂τg|g〉, (3.51)

where the last equality holds because 〈g|g〉 = 1 is constant and ψ is periodic in β. As animportant intermediate result we have found that the path integral is overall gauge invariant or,equivalently, that the path integral is one over paths living on the two–sphere (rather than theentire group manifold SU(2)). This finding is reasuring in the sense that a degree of freedomliving on a sphere comes close to what one might intuitively expect to be the classical counterpartof a quantum particle with conserved angular momentum.

Let us now proceed by exploring the action of the path integral. Using the auxiliary identity(3.50) it is a straightforward matter to show that

Stop[φ, θ] = −∫ β

0dτ〈∂τg|g〉 = −iS

∫ β

0dτ ∂τφ cos θ = iS

∫ β

0dτ ∂τφ(1 − cos θ). (3.52)

Combining this with the B–dependent term discussed above, one obtains

S[θ, φ] = SB [φ, θ] + Stop[φ, θ] = S

∫ β

0dτ [B cos θ + i(1 − cos θ)∂τφ] (3.53)

for the action of the path integral for spin.

⊲ Exercise. Derive the Euler–Lagrange equations asasociated with this action. Show thatthey are equivalent to the Bloch equations i∂τn = B × n of a spin with expectation value〈S〉 = Sn subject to a magnetic field. Here, n(φ, θ) ∈ S2 is the unit vector defined by the twoangles φ, θ.

Analysis of the action

To formulate the second term in the action (3.53) in a more suggestive way, we note that thevelocity of the point n moving on the unit sphere is given by n = θeθ + φ sin θ eφ, where(er, eθ, eφ) form a spherical orthonormal system. We can thus rewrite Eq. (3.52) as

Stop[φ, θ] = iS

∫ β

0dτ n · A = iS

∮

γdn ·A, (3.54)

where

A =1 − cos θ

sin θeφ. (3.55)

Notice that, in spite of its compact appearance, Eq. (3.54) does not represent a coordinateinvariant formulation of the action Stop. (The field A(φ, θ) explcitly depends on the coordinates(φ, θ).) In fact, the action Stop cannot be expressed in a coordinate invariant manner, for reasonsdeeply rooted in the topology of the two–sphere.



A second observation is that (3.54) can be read as the (Euclidean time) action of a particleof charge S moving under the influence of a vector potential A (cf., for example, Ref. [?].) Usingstandard formulae of vector calculus (cf. Ref. [?]) one finds Bm ≡ ∇×A = Ser, i.e. our particlemoves in a radial magnetic field of constant strength S. Put differently, the particle moves inthe field of a magnetic ‘charge’ of strength 4π centered on the origin of the sphere.

⊲ Info. If you find this statement difficult to reconcile with the Maxwell equation ∇ · B =0 ↔

∫S B · dS for any closed surface S, notice that ∇ · B = ∇ · (∇ × A) = 0 holds only if

A is non–singular. However, the vector potential (3.55) is manifestly singular along the line(r, θ = π) through the south pole of the sphere. The physical picture behind this singularityis as follows: Imagine an infinitely thin solenoid running from r = ∞ through the south poleof the sphere to its center. Assuming that the solenoid contains a magnetic flux 4π, the centerof the sphere becomes a source of magnetic flux, the so–called Dirac monopole. This pictureis consistent with the presence of a field B = er. It also explains the singularity of A alongthe string. (Of course, the solenoidal construction does not lead to the prediction of a genuinemonopole potential: Somewhere, at r = ∞, our auxiliary magnetic coil has to end, and this iswhere the flux lines emanating from the point r = 0 terminate.) The postulate of a flux lineat the singularity of A merely helps to reconcile the presence of a radial magnetic field withthe principles of electrodynamics. However, as far as our present discussion goes, this extrastructure is not essential, i.e. we may simply interpret r = 0 as the position of a magnetic‘charge’.

——————————————–

To explore the consequences of this phenomenon, we apply Stokes’ theorem37

Stop[n] = iS

∮

γn ·A = iS

∮

Aγ,n

dS · (∇× A) = iS

∮

Aγ,n

dS · er = iSAγ,n. (3.56)

Here, Aγ,n is the domain on the two–sphere which (a) has the curve γ as its boundary, and (b)contains the north pole (see the figure). The integral produces the area of this surface which weagain denote by Aγ,n. Curiously, the action Stop is but a measure of the area bounded by thecurve γ : τ 7→ n(τ). However, simple as it is, this result should raise some suspicion: By assigninga designated role to the northern hemisphere of the sphere some symmetry breaking, not presentin the original problem, has been introduced. Indeed, we might have defined our action byStop[φ, θ] = iS

∮γ dn · A′ where A′ = −1+cos θ

sin θ eφ = A − 2∇φ differs from A only by a gauge

transformation.38 The newly defined vector potential is non–singular in the southern hemisphere,so that application of Stokes’ theorem leads to the conclusion Stop[n] = −iS

∫Aγ,s

dS · Bm =

37

George Gabriel Stokes 1819–1903: As Lucasian Professor ofMathematics at Cambridge Stokes etablished the science of hy-drodynamics with his law of viscosity (1851), describing the ve-locity of a small sphere through a viscous fluid. Furthermore, heinvestigated the wave theory of light, named and explained thephenomenon of fluorescence, and theorised an explanation of theFraunhofer lines in the solar spectrum.

38You may, with some justification, feel uneasy about the fact that φ is not a true ‘function’ on thesphere (or, alternatively, about the fact that

∫dn · ∇φ = φ(β) − φ(0) may be a non–vanishing multiple

of 2π). We will return to the discussion of this ambiguity shortly. (Notice that a similarly hazardousmanipulation is performed in the last equality of Eq. (3.52).)



−iSAγ,s. Here, Aγ,s is the area of a surface bounded by γ but covering the south pole of thesphere. The absolute minus sign is due to the outward orientation of the surface Aγ,s.

γ

γ,nA

One has to concede that the result obtained for the action Stop depends on the chosen gaugeof the monopole vector potential! The difference between the northern and the southern variantof our analysis is given by

iS

∫

Aγ,n

dS ·Bm + iS

∫

Aγ,s

dS ·Bm = iS

∫

S2

dS · er = 4πiS,

where we have made use of the fact that Aγ,n∪Aγ,s = S2 is the full sphere. At first sight, it looksas if our analysis has led us to a gauge dependent, and therefore pathological result. Let’s recall,however, that physical quantities are determined by the exponentiated action exp(iS[n]) and notby the action itself. Now, S is either integer or half integer which implies the factor exp(4πiS) = 1is irrelevant. In the operator representation of the theory, spin quantization follows from therepresentation theory of the algebra su(2). It is a ‘non–local’ feature, in the sense that the actionof the spin operators on all eigenstates has to be considered to fix the dimensionality 2S + 1 ofHS . In hindsight, it is thus not too surprising that the same information is encapsulated in a‘global’ condition (gauge invariance) imposed on the action of the path integral.

Summarizing, we have found that the classical dynamics of a spin is that of a massless pointparticle on a sphere coupled to a monopole field Bm. We have seen that the vector potential ofthe latter cannot be globally continuous on the full sphere. More generally, the phase space S2

cannot be represented in terms of a global system of ‘coordinates and momenta’ which placesit outside the scope of traditional treatments of classical mechanics. This probably explains thefailure of early attempts to describe the spin in terms of a path integral or, equivalently, in termsof a Hamiltonian action.

In chapter ?? we will use the path action (3.53) as a building block for our construction ofthe field theory of higher dimensional spin systems. However, before concluding this section, letus make some more remarks on the curious properties of the monopole action Stop: Contraryto all other Euclidean actions encountered thus far, the action (3.54) is imaginary. In fact,it will stay imaginary upon Wick rotation τ → it back to real times. More generally, Stop isinvariant under the rescaling τ → cτ , and invariant even under arbitrary reparameterizationsτ → g(τ) ≡ τ ′. This invariance is a hallmark of a topological term. Loosely speaking (seechapter ?? for a deeper discussion), a topological term is a contribution to the action of afield theory that depends on the global geometry of a field configuration rather than on itslocal structure. In contrast, ‘conventional’ operators in field theoretical actions measure theenergy cost of dynamical or spatial field fluctuations. In doing to they must relate to a specificspatio–temporal reference frame, i.e. they cannot be invariant under reparamaterisation.

Summarizing our results, we have found that:



1. The classical action of a spin is one of a massless particle (there is no standard kineticenergy term in (3.48)) moving on a unit sphere. The particle carries a magnetic moment ofmagnitude S. It is coupled to (a) a conventional magnetic field via its magnetic moment,and (b) to a monopole field via its orbital motion. Note that we have come, finally, to aposition which hints at the difficulties plaguing attempts to formulate a classical mechanicsof spin. The vector potential of a monopole, A, cannot be globally defined on the entiresphere. The underlying physical reason is that, by the very nature of the monopole (fluxgoing radially outwards everywhere), the associated vector potential must be singular atone point of the surface.39 As a consequence, the classical phase space of the system, thesphere, cannot be covered by a global choice of coordinate system. (Unlike most standardproblems of classical mechanics there is no system of globally defined ‘p’s and ‘q’s.) Thisfact largely spoils a description within the standard — coordinate oriented — formulationof Hamiltonian mechanics (cf. the discussion in the article by Stone).

2. Terms akin to the monopole contribution to the spin action appear quite frequentlywithin path integral formulations of systems with non–trivial topology (like the two–sphereabove). Depending on the particular context under consideration, one distinguishes be-tween Wess–Zumino–Witten (WZW) terms,40 θ–terms, Chern–Simons termsand a few other terms of topological origin. What makes these contributions generallyimportant is that the value taken by these terms depends only on the topology of a fieldconfiguration but not on structural details.

As a final application of the path integral, we turn now to the consideration of problems in

which the dynamics of the classical system is, itself, non–trivial.

3.3.6 †Trace Formulae and Quantum Chaos

⊲ Additional Example: Introductory courses on classical mechanics usually convey theimpression that dynamical systems behave in a regular and, at least in principle, mathemati-cally predictable way. However, experience shows that the majority of dynamical processes innature do not conform with this picture: Partly, or even fully chaotic motion (i.e. motion that

39To better understand this point, consider the integral of A along an infinitesimal closed curve γ onthe sphere. If A were globally continuous, we would have two choices to transform the integral intoa surface integral over B; an integral over the ‘large’ or the ‘small’ surface area bounded by γ. Themonopole nature of B would demand that both integrals are proportional to the respective area of theintegration domain which, by assumption, are different contradiction. The resolution of this paradoxis that A must be discontinuous at one point on the sphere, i.e. we cannot globally set B = ∇× A andthe choice of the integration area is prescribed by the condition that it must not encompass the singularpoint.

40

Edward Witten 1951–: 1990 Fields Medal for his workin superstring theory. He made significant contributionsto Morse theory, supersymmetry, and knot theory. Ad-ditionally, he explored the relationship between quantumfield theory and the differential topology of manifolds oftwo and three dimensions.



depends in a singular and, thereby, in an essentially unpredictable way on initial conditions)is the rule rather than the exception. In view of the drastic differences in the observable be-haviour of classically integrable and chaotic systems, an obvious question arises: In what waydoes the quantum phenomenology of chaotic systems differ from that associated with integrabledynamics? This question defines the field of quantum chaos.

Understanding signatures of classically chaotic motion in quantum mechanics is an issue notonly of conceptual, but also of great practical relevance impinging on areas such as quantumelectron transport in condensed matter systems: The inevitable presence of impurities andimperfections in any macroscopic solid renders the long–time dynamics of electronic chargecarriers chaotic. Relying on a loose interpretation of the Heisenberg principle, ∆t ∼ ~/∆E,i.e. the relation between long–time dynamical behaviour and small scale structures in energy,one would expect that signatures of chaotic quantum dynamics are especially important in thelow–energy response in which one is usually interested. This expectation has been confirmed forinnumerable observables related to low temperature electronic transport in solid state systems.

Disordered conducting media represent but one example of a wide class of dynamical systemswith long–time chaotic dynamics. Indeed, recent experimental advances have made it possibleto realize a plethora of effectively non–disordered chaotic dynamical systems in condensed mat-ter devices. For example, employing modern semiconductor device technology, it has becomepossible to manufacture small two–dimensional conducting systems, of a size O(< 1µm) and ofalmost any geometric shape. Here, the number of imperfections can be reduced to a negligibleminimum, i.e. electrons propagate ballistically along straight trajectories, as in a billiard. Thesmallness of the devices further implies that the ratio between Fermi wavelength and system sizeis of O(10−1 −10−3), i.e. while semiclassical concepts will surely be applicable, the wave aspectsof quantum propagation remain visible. In recent years, the experimental and theoretical studyof electron transport in such quantum billiards has emerged as a field in its own right.

How then can signatures of chaotic dynamics in quantum systems be sought? The most fun-damental characteristic of a quantum system is its spectrum. Although not a direct observable,it determines the majority of properties accessible to measurement. On the other hand, it is clearthat the manifestations of chaos we are looking for must relate back to the classical dynamicalproperties of the system. The question then is, how can a link between classical mechanics and

quantum spectra be drawn? This problem is tailor made for analysis by path integral techniques.

Semiclassical Approximation to the Density of States

The close connection between the path integral and classical mechanics should be evident fromthe previous sections. However, to address the problem raised above, we still need to understandhow the path integral can be employed to analyse the spectrum of a quantum system. The latterare described by the (single–particle) density of states

ρ(ǫ) = tr δ(ǫ− H) =∑

a

δ(ǫ− ǫa) , (3.57)

where ǫa represents the complete set of energy levels. To compute the sum, one commonlyemploys a trick based on the Dirac identity,

limδց0

1

x+ iδ= −iπδ(x) + P 1

x, (3.58)

where P(1/x) denotes for the principal part of 1/x. Taking the imaginary part of (3.58),Eq. (3.57) can be represented as ρ(ǫ) = − 1

π Im∑

a1

ǫ+−ǫa= − 1

π Im tr ( 1ǫ+−H

), where ǫ+ ≡ ǫ+ iδ



and the limit limδց0 is implicit. Using the identity 1/x+ = −i∫ t0 dt e

ix+t, and representing the

trace tr A =∫dq〈q|A|q〉 as a real space integral,

ρ(ǫ) =1

π~

∫ ∞

0dtRe tr(ei(ǫ

+−H)t/~) =1

πRe

∫ ∞

0dt eiǫ

+t/~

∫dq〈q|e−iHt/~|q〉 , (3.59)

we have made the connection between the density of states and the quantum propagation am-plitude explicit.

Without going into full mathematical detail (see, for example, Ref. [?] for a modern discourse)we now outline how this integral is evaluated by path integral techniques within the semiclassicalapproximation. Although, for brevity, some of the more tricky steps of the calculation are sweptunter the carpet, the sketch will be accurate enough to make manifest some aesthetic connectionsbetween the spectral theory of chaotic quantum systems and classically chaotic dynamics. (Fora more formal and thorough discussion, we refer to Gutzwiller and Haake.)

Making use of the semiclassical approximation (3.28) established earlier, when substituted

into Eq. (3.59), one obtains ρ(ǫ) ≃ 1π Re

∫∞0 dt eiǫ

+t/~∫dq A[qcl]e

i~S[qcl], where, following our

discussion in section 3.2.2, we have defined A[qcl] ≡ det(

i2π~

∂2S[qcl]∂q(0)∂q(t)

)1/2and qcl represents a

closed classical path that begins at q at time zero and ends at the same coordinate at timet. Again relying on the semiclassical condition S[qcl] ≫ ~, the integrals over q and t can beperformed in a stationary phase approximation. Beginning with the time integral, and noticingthat ∂tS[qcl] = −ǫqcl is the (conserved) energy of the path qcl, we obtain the saddle point

condition ǫ!= ǫqcl and

ρ(ǫ) ≃ 1

πRe

∫dq A[qcl,ǫ]e

i~S[qcl,ǫ] ,

where the symbol qcl,ǫ indicates that only paths q → q of energy ǫ are taken into account, and thecontribution coming from the quadratic integration around the saddle point has been absorbedinto a redefinition of A[qcl,ǫ].

q

p α

Turning to the q–integration, making use of the fact that ∂qiS[qcl] = −pi, ∂qfS[qcl] = pf ,where qi,f are the initial and final coordinate of a path qcl, and pi,f are the initial and final mo-

mentum, the stationary phase condition assumes the form 0!= dqS[qcl,ǫ] =

(∂qi + ∂qf

)S[qcl,ǫ]|qi=qf=q =

pf−pi, i.e. the stationarity of the integrand under the q–integration requires the initial and finalmomentum of the path qcl,ǫ be identical. We thus find that the paths contributing to the inte-grated transition amplitude are not only periodic in coordinate space but even in phase space.Such paths are called periodic orbits —‘periodic’ because the path comes back to its initial



phase space coordinate after a certain revolution time. As such, the orbit will be traversedrepeatedly as time goes by (see the figure, where a periodic orbit α with initial coordinatesx = (p, q) is shown).

According to our analysis above, each coordinate point q lying on a periodic orbit is astationary phase point of the q–integral. The stationary phase approximation of the integralcan thus be formulated as

ρ(ǫ) ≃ 1

πRe

∫dq A[qcl,ǫ]e

i~S[qcl,ǫ] ≃

∞∑

n=1

∑

α

∫

αdq Aαe

i~nSα ,

where∑

α stands for a sum over all periodic orbits (of energy ǫ) and Sα is the action corre-sponding to one traversal of the orbit (all at fixed energy ǫ). The index n accounts for the factthat, due to its periodicty, the orbit can be traversed repeatedly, with total action nSα. Fur-thermore,

∫α dq is an integral over all coordinates lying on the orbit and we have again absorbed

a contribution coming from the quadratic integration around the stationary phase points in thepre–exponential amplitude Aα.

Finally, noting that∫α dq ∝ Tα, where Tα is the period of one traversal of the orbit α (at

energy ǫ), we arrive at the result

ρ(ǫ) ≃ 1

πRe

∞∑

n=1

∑

α

TαAαei~nSα (3.60)

This is (a simplified41) representation of the famous Gutzwiller trace formula. The result is

actually quite remarkable: The density of states, an observable of quantum mechanical signifi-

41Had we carefully kept track of all determinants arising from the stationary phase integrals, theprefactor Aα would have read

Aα =1

~

ei π

2να

|detM rα − 1| 12

,

where να is known as the Maslov index (an integer valued factor associated with the singular pointson the orbit, i.e. the classical turning points). The meaning of this object can be understood, e.g., byapplying the path integral to the problem of a quantum particle in a box. To correctly reproduce thespectrum, the contribution of each path must be weighted by (−)n = exp(iπn), where n is the number ofits turning points in the box potential), and Mα represents the Monodromy matrix. To understandthe meaning of this object, notice that a phase space point x on a periodic orbit can be interpreted asa fixed point of the classical time evolution operator U(Tα): U(Tα, x) = x, which is just to say that theorbit is periodic. As with any other smooth mapping, U can be linearized in the vicinity of its fixedpoints, U(Tα, x+ y) = x+Mαy, where the linear operator Mα is the monodromy matrix. Evidently, Mα

determines the stability of the orbit under small distortions, which makes it plausible that it appears asa controlling prefactor of the stationary phase approximation to the density of states.

⊲ Exercise. Making use of the Feynman path integral, show that the propagator for a particleof mass m confined by a square well potential of infinite strength is given by

G(qF , qI ; t) =

√m

2πi~t

∞∑

n=−∞

exp

[im(qF − qI + 2na)2

2~t

]− exp

[im(qF + qI + 2na)2

2~t

].



cance, has been expressed entirely in terms of classical quantities.

3.4 Summary

In this chapter we have introduced the path integral formulation of quantum mechanics, anapproach independent of, yet (modulo certain mathematical imponderabilities related tocontinuum functional integration) equivalent to the standard route of canonical operatorquantization. While a few precious exactly solvable quantum problems (e.g. the evolutionof a free particle, the harmonic oscillator, and, perhaps intriguingly, quantum mechanicalspin) are more efficiently formulated by the standard approach, a spectrum of uniquefeatures make the path integral an indispensible tool of modern quantum mechanics: Thepath integral approach is highly intuitive, powerful in the treatment of non-perturbativeproblems, and tailor–made to formulation of semiclassical limits. Perhaps most impor-tantly, we have seen that it provides a unifying link whereby quantum problems can berelated to classical statistical mechanics. Indeed, we have found that the path integral ofa quantum point particle is, in many respects, equivalent to the partition function of aclassical one–dimensional continuum system. We have hinted at a generalization of thisprinicple, i.e. an equivalence priniciple relating d–dimensional quantum field theory tod + 1–dimensional statistical mechanics. However, before exploring this bridge further,we first need to generalize the concept of path integration to problems involving quantumfields. This will be the subject of the next chapter.


Chapter 4

Functional Field Integral

In this chapter, the concept of path integration is generalized to integration over quan-

tum fields. Specifically we will develop an approach to quantum field theory that takes as

its starting point an integration over all configurations of a given field, weighted by an

appropriate action. To emphasize the importance of the formulation which, methodologic-

cally, represents the backbone of the remainder of the text, we have pruned the discussion

to focus only on the essential elements. This being so, conceptual aspects stand in the

foreground and the discussion of applications is postponed to the following chapters.

In this chapter, the concept of path integration will be extended from quantum me-chanics to quantum field theory. Our starting point will be from a situation very muchanalogous to that outlined at the beginning of the previous chapter. Just as there aretwo different approaches to quantum mechanics, quantum field theory can also be formu-lated in two different ways; the formalism of canonically quantised field operators, andfunctional integration. As for the former, although much of the technology needed to effi-ciently implement this framework — essentially Feynman diagrams — originated in highenergy physics, it was with the development of condensed matter physics through the 50s,60s and 70s that this approach was driven to unprecedented sophistication. The reasonis that, almost as a rule, problems in condensed matter investigated at that time neces-sitated perturbative summations to infinite order in the non–trivial content of the theory(typically interactions). This requirement led to the development of advanced techniquesto sum (subsets of) the perturbation series in many–body interaction operators to infiniteorder.

In the 70s, however, essentially non–perturbative problems began to attract more andmore attention — a still prevailing trend — and it turned out that the formalism ofcanonically quantised operators was not tailored to this type of physics. By contrast, thealternative approach to many–body problems, functional integration, is ideally suited!The situation is similar to the one described in the last chapter where we saw thatthe Feynman path integral provided an entire spectrum of novel routes to approach-ing quantum mechanical problems (controlled semi–classical limits, analogies to classicalmechanics, statistical mechanics, concepts of topology and geometry, etc.). Similarly, theintroduction of functional field integration into many–body physics spawned plenty ofnew theoretical developments, many of which were manifestly non–perturbative. More-over, the advantages of the path integral approach in many–body physics is even more


124 CHAPTER 4. FUNCTIONAL FIELD INTEGRAL

pronounced than in single particle quantum mechanics. Higher dimensionality introducesfields of a more complex internal structure allowing for non–trivial topology while, at thesame time, the connections to classical statistical mechanics play a much more importantrole than in single particle quantum mechanics.

q

q

t

t

Φ

x

Φ

x

QM

degrees of freedom path integral

QFT

Figure 4.1: The concept of field integration. Upper panels: path integral of quantummechanics — integration over all time-dependent configurations of a point particle degreeof freedom leads to integrals over curves. Lower panels: field integral — integrationover time dependent configurations of d–dimensional continuum mappings (fields) leadsto integrals over generalized (d+ 1)–dimensional surfaces.

All of these concepts will begin to play a role in subsequent chapters when applicationsof the field integral are discussed. Before embarking on the quantitative construction —the subject of the following sections — let us first anticipate the kind of structures that oneshould expect. In quantum mechanics, we were starting from a single point particle degreeof freedom, characterized by some coordinate q (or some other quantum numbers for thatmatter). Path integration then meant integration over all time–dependent configurationsq(t), i.e. a set of curves t 7→ q(t) (see Fig. 4.1 upper panel). By contrast, the degrees offreedom of field theory are continuous objects Φ(x) by themselves, where x parameterizessome d–dimensional base manifold and Φ takes values in some target manifold (Fig. 4.1,lower panel). The natural generalization of a ‘path’ integral then implies integration overa single copy of these objects at each instant of time, i.e. we shall have to integrateover generalized surfaces, mappings from (d + 1)–dimensional space–time into the fieldmanifold, (x, t) 7→ Φ(x, t). While this notion may sound worrying, it is important torealize that, conceptually, nothing much changes in comparison with the path integral:instead of a one–dimensional manifold — a curve — our object of integration will be a(d+ 1)–dimensional manifold.

We now proceed to formulate these ideas in quantitative terms.

⊲ Exercise. If necessary, recapitulate the general construction scheme of path integrals

(section 3.2.3) and the connection between quantum fields and second quantized operators.


4.1. CONSTRUCTION OF THE MANY–BODY PATH INTEGRAL 125

4.1 Construction of the Many–body Path Integral

The construction of a path integral for field operators follows the general scheme outlinedat the end of section 3.2.3. The basic idea is to segment the time evolution of a quan-tum (many–body) Hamiltonian into infinitesimal time slices and to absorb as much as ispossible of the quantum dynamical phase accumulated during the short time propagationinto a set of suitably chosen eigenstates. But how should these eigenstates be chosen? Inthe context of single particle quantum mechanics, the basic structure of the Hamiltoniansuggested the choice of a representation in terms of coordinate and momentum eigen-states. Now, given that many particle Hamiltonians are conveniently expressed in termsof creation/annihilation operators, an obvious idea would be to search for eigenstates ofthese operators. Such states indeed exist and are called coherent states.

4.1.1 Coherent States (Bosons)

Our goal is, therefore, to find eigenstates of the Fock space (non–Hermitian) operatorsa† and a. Although the general form of these states will turn out to be the same forbosons and fermions, there are major differences regarding their algebraic structure. Thepoint is that the anticommutation relations of fermions require that the eigenvalues ofan annihilation operator themselves anticommute, i.e. they cannot be ordinary numbers.Postponing the introduction of the unfamiliar concept of anticommuting ‘numbers’ to thenext section, we first concentrate on the bosonic case where problems of this kind do notarise.

So what form do the eigenstates |φ〉 of the bosonic Fock space operators a, and a†

take? Being a state of the Fock space, an eigenstate |φ〉 can be expanded as

|φ〉 =∑

n1,n2,···

Cn1,n2,···|n1, n2 · · ·〉, |n1, n2 · · ·〉 =(a†1)

n1

√n1

(a†2)n2

√n2

· · · |0〉 ,

where a†i creates a boson in state i, Cn1,n2,··· represents a set of expansion coefficients,and |0〉 represents the vacuum. Here, for reasons of clarity, it is convenient to adopt thisconvention for the vacuum as opposed to the notation |Ω〉 used previously. Furthermore,the many–body state |n1, n2 · · ·〉 is indexed by a set of occupation numbers: n1 in state|1〉, n2 in state |2〉, and so on. Importantly, the state |φ〉 can, in principle (and will inpractice) contain a superposition of basis states which have different numbers of particles.Now, if the minimum number of particles in state |φ〉 is n0, the minimum of a†i |φ〉 mustbe n0 + 1: Clearly the creation operators a†i themselves cannot possess eigenstates.

However, with annihilation operators this problem does not arise. Indeed, the annihi-lation operators do possess eigenstates known as bosonic coherent states

|φ〉 ≡ exp

[

∑

i

φia†i

]

|0〉 (4.1)

where the elements of φ = φi represent a set of complex numbers. The states |φ〉 are



eigenstates in the sense that, for all i,

ai|φ〉 = φi|φ〉 (4.2)

i.e. they simultaneously diagonalise all annihilation operators. Noting that ai and a†j , with

j 6= i, commute, Eq. (4.2) can be verified by showing that a exp(φa†)|0〉 = φ exp(φa†)|0〉.1Although not crucial to the practice of functional field integration, in the construction ofthe many–body path integral, it will be useful to assimilate some further properties ofcoherent states.

⊲ By taking the Hermitian conjugate of Eq. (4.2), we find that the ‘bra’ associatedwith the ‘ket’ |φ〉 is a left eigenstate of the set of creation operators, i.e. for all i,

〈φ|a†i = 〈φ|φi (4.3)

where φi is the complex conjugate of φi, and 〈φ| = 〈0| exp[∑

i φiai].

⊲ It is a straightforward matter — e.g. by a Taylor expansion of Eq. (4.1) — to showthat the action of a creation operator on a coherent state yields the identity

a†i |φ〉 = ∂φi |φ〉. (4.4)

Reassuringly, it may be confirmed that Eqs. (4.4) and (4.2) are consistent with thecommutation relations [ai, a

†j ] = δij : [ai, a

†j]|φ〉 = (∂φjφi − φi∂φj )|φ〉 = δij|φ〉.

⊲ Making use of the relation 〈θ|φ〉 = 〈0|eP

i θiai |φ〉 = eP

i θiφi〈0|φ〉 one finds that theoverlap between two coherent states is given by

〈θ|φ〉 = exp

[

∑

i

θiφi

]

(4.5)

⊲ From this result, one can infer that the norm of a coherent state is given by

〈φ|φ〉 = exp

[

∑

i

φiφi

]

(4.6)

⊲ Most importantly, the coherent states form a complete — in fact an overcomplete— set of states in Fock space:

∫

∏

i

dφidφiπ

e−P

i φiφi|φ〉〈φ| = 1F , (4.7)

where dφidφi = dReφidImφi, and 1F represents the unit operator or identity in theFock space.

1Using the result [a, (a†)n] = n(a†)n−1 (cf. Eq. ??) a Taylor expansion shows a exp(φa†)|0〉 =

[a, exp(φa†)]|0〉 =∑∞n=0

φn

n! [a, (a†)n]|0〉 =

∑∞n=1

nφn

n! (a†)n−1|0〉 = φ∑∞

n=1φn−1

(n−1)! (a†)n−1|0〉 =

φ exp(φa†)|0〉.



⊲ Info. The proof of Eq. (4.7) proceeds by straightforward application of Schur’s lemma(cf. our discussion of the completeness of the spin coherent states in the previous chapter):

The operator family ai, a†i acts irreducibly in Fock space. According to Schur’s lemma, the

proportionality of the left hand side of Eq. (4.7) to the unit operator is, therefore, equivalent toits commutativity with all creation and annihilation operators. Indeed, this property is easilyconfirmed:

ai

∫

d(φ, φ)e−P

i φiφi |φ〉〈φ| =

∫

d(φ, φ)e−P

i φiφiφi|φ〉〈φ| = −∫

d(φ, φ)(

∂φie−

P

i φiφi)

|φ〉〈φ|

by parts=

∫

d(φ, φ)e−P

i φiφi |φ〉(

∂φi〈φ|)

=

∫

d(φ, φ)e−P

i φiφi |φ〉〈φ|ai, (4.8)

where, for brevity, we have set d(φ, φ) ≡ ∏

i dφidφi/π. Taking the adjoint of Eq. (4.8), one mayfurther check that the left hand side of (4.7) commutes with the set of creation operators, i.e.it must be proportional to the unit operator. To fix the constant of proportionality, one maysimply take the overlap with the vacuum:

∫

d(φ, φ)e−P

i φiφi〈0|φ〉〈φ|0〉 =

∫

d(φ, φ)e−P

i φiφi = 1, (4.9)

where the last equality follows from Eq. (3.11). Taken together, Eqs. (4.8) and (4.9) prove

(4.7). Note that the coherent states are overcomplete in the sense that they are not mutually

orthogonal (see Eq. (4.5)). The exponential weight e−P

i φiφi appearing in the resolution of the

identity compensates for the overcounting achieved by integrating over the whole set of coherent

states.

——————————————–With these definitions we have all that we need to construct the many–body path inte-

gral for the bosonic system. However, before doing so, we will first introduce the fermionicversion of the coherent state. This will allow us to construct the path integrals for bosonsand fermions simultaneously, thereby emphasising the similarity of their structure.

4.1.2 Coherent States (Fermions)

Surprisingly, much of the formalism above generalises to the fermionic case: As before, it isevident that creation operators cannot possess eigenstates. Following the bosonic system,let us suppose that the annihilation operators are characterised by a set of coherent statessuch that, for all i,

ai|η〉 = ηi|η〉 (4.10)

where ηi is the eigenvalue. Although the structure of this equation appears to be equivalentto its bosonic counterpart (4.2) it has one frustrating feature: Anticommutativity of thefermionic operators, [ai, aj]+ = 0, where i 6= j, implies that the eigenvalues ηi also haveto anticommute,

ηiηj = −ηjηi (4.11)

Clearly, these objects cannot be ordinary numbers. In order to define a fermionic versionof coherent states, we now have two choices: We may (a) accept Eq.(4.11) as a working



definition and pragmatically explore its consequences, or (b), first try to remove anymystery from the definitions (4.10) and (4.11). This latter task is tackled in the infoblock below where objects ηi with the desired properties are defined in a mathematicallyconsistent manner. Readers wishing to proceed in a maximally streamlined manner mayskip this exposition and directly turn to the more praxis–oriented discussion below.

⊲ Info. There is a mathematical structure ideally suited to generalize the concept ofordinary number(fields), namely algebras. An algebra A is a vector space endowed with amultiplication rule A×A → A. So, let us construct an algebra A by starting out from a set ofelements, or generators, ηi ∈ A, i = 1, . . . N , and imposing the rules:

(i) The elements ηi can be added and multiplied by complex numbers, viz.

c0 + ciηi + cjηj ,∈ A c0, ci, cj ∈ C , (4.12)

i.e. A is a complex vectorspace.

(ii) The product, A × A → A, (ηi, ηj) 7→ ηiηj , is associative and anticommutative, i.e. itobeys the anti–commutation relation (4.11). Because of the associativity of this operation,there is no ambiguity when it comes to forming products of higher order, i.e. (ηiηj)ηk =ηi(ηjηk) ≡ ηiηjηk. The definition requires that products of odd order in the numberof generators anti–commute, while (even,even) and (even,odd) combinations commute(exercise).

By virtue of (i) and (ii), the set A of all linear combinations c0+∑∞

n=1

∑Ni1,···in=1 ci1,...,inηi1 . . . ηin ,

c0, ci1,...,in ∈ C spans a finite–dimensional associative algebra A,2 known as the Grassmann

algebra3 (and sometimes also the exterior algebra).

For completeness we mention that Grassmann algebras find a number of realizations in math-ematics, the most basic being exterior multiplication in linear algabra: Given an N–dimensionalvector space V , let V ∗ be the dual space, i.e. the space of all linear mappings, or ‘forms’Λ : V → C, v 7→ Λ(v), where v ∈ V . (Like V , V ∗ is a vector space of dimension N .) Next, defineexterior multiplication through, (Λ,Λ′) → Λ ∧ Λ′, where Λ ∧ Λ′ is the mapping

Λ ∧ Λ′ : V × V → C

(v, v′) 7→ Λ(v)Λ′(v′) − Λ(v′)Λ′(v).

This operation is manifestly anti–commutative: Λ∧Λ′ = −Λ∧Λ′. Identifying the N linear basis

forms Λi ↔ ηi with generators and ∧ with the product, we see that the space of exterior forms

of a vector space forms a Grassmann algebra.

——————————————–

2...whose dimension can be shown to be 2N (exercise)

3

Hermann Gunter Grassmann

1809–1877: credited for inventingwhat is now called Exterior Alge-bra.



Apart from their anomalous commutation properties, the generators ηi, and theirproduct generalizations ηiηj , ηiηjηk, . . . resemble ordinary, albeit anti-commutative num-bers. (In practice, the algebraic structure underlying their definition can safely be ignored.All we will need to work with these objects is the basic rule (4.11) and the property (4.12).)We emphasize that A not only contains anticommuting but also commuting elements, i.e.linear combinations of an even number of Grassmann numbers ηi are overall commutative.(This mimics the behaviour of the Fock space algebra: products of an even number ofannihilation operators aiaj . . . commute with all other linear combinations of operatorsai. In spite of this similarity, the ‘numbers’ ηi must not be confused with the Fock spaceoperators; there is nothing on which they act.)

To make practical use of the new concept, we need to go beyond the level of purearithmetic. Specifically, we need to introduce functions of anti–commuting numbers, andelements of calculus. Remarkably, most of the concepts of calculus not only naturallygeneralize to anti–commuting number fields, but contrary to what one might expect, theanti–commutative generalization of differentiation, integration, etc. turns out to be muchsimpler than in ordinary calculus.

⊲ Functions of Grassmann numbers are defined via their Taylor expansion:

ξ1, . . . , ξk ∈ A : f(ξ1, . . . , ξk) =

∞∑

n=0

k∑

i1,···in=1

1

n!

∂nf

∂ξi1 . . . ∂ξin

∣

∣

∣

ξ=0ξin . . . ξi1, (4.13)

where f is an analytic function. Note that the anticommutation properties of thealgebra implies that the series terminates after a finite number of terms. For exam-ple, in the simple case where η is first order in the generators of the algebra, N = 1,and f(η) = f(0) + f ′(0)η (since η2 = 0).

⊲ Differentiation with respect to Grassmann numbers is defined by

∂ηiηj = δij (4.14)

Note that in order to be consistent with the commutation relations, the differential

operator ∂ηi must itself be anti–commutative. In particular, ∂ηiηjηii6=j= −ηj .

⊲ Integration over Grassmann variables is defined by

∫

dηi = 0,

∫

dηiηi = 1 (4.15)

Note that the definitions (4.13), (4.14) and (4.15) imply that the action of Grass-

mann differentiation and integration are effectively identical, viz.

∫

dηf(η) =

∫

dη(f(0) + f ′(0)η) = f ′(0) = ∂ηf(η) .



With this background, let us now proceed to apply the Grassmann algebra to theconstruction of fermion coherent states. To this end we need to enlarge the algebra evenfurther so as to allow for a multiplication of Grassmann numbers by fermion operators.In order to be consistent with the anticommutation relations, we need to require thatfermion operators and Grassmann generators anticommute,

[ηi, aj]+ = 0 . (4.16)

It then becomes a straightforward matter to demonstrate that fermionic coherentstates are defined by

|η〉 = exp

[

−∑

i

ηia†i

]

|0〉 (4.17)

i.e. by a structure perfectly analogous to the bosonic states (4.1).4 It is a straightforwardmatter — and also a good exercise — to demonstrate that the properties (4.3), (4.4),(4.5), (4.6) and, most importantly, (4.7) carry over to the fermionic case. One merely hasto identify ai with a fermionic operator and replace the complex variables φi by ηi ∈ A.Apart from a few sign changes and the A–valued arguments, the fermionic coherent statesdiffer only in two respects from their bosonic counterpart: firstly, the Grassmann variablesηi appearing in the adjoint of a fermion coherent state,

〈η| = 〈0| exp

[

−∑

i

aiηi

]

= 〈0| exp

[

∑

i

ηiai

]

,

are not related to the ηis of the state |η〉 via some kind of complex conjugation. Rather ηiand ηi are strictly independent variables.5 Secondly, the Grassmann version of a Gaussianintegral (exercise),

∫

dηdη e−ηη = 1 does not contain the factors of π characteristic ofstandard Gaussian integrals. Thus, the measure of the fermionic analogue of Eq. (4.7)does not contain a π in the denominator.

For the sake of future reference, the most important properties of Fock space coherentstates are summarised in table 4.1.

4To prove that the states (4.17) indeed fulfil the defining relation (4.10), we note that

ai exp(−ηia†i )|0〉(4.13)= ai(1 − ηia

†i )|0〉

(4.16)= ηiaia

†i |0〉 = ηi|0〉 = ηi(1 − ηia

†i )|0〉 = ηi exp(−ηia†i )|0〉. This,

in combination with the fact that ai and ηja†j (i 6= j) commute proves (4.10). Note that the proof has

actually been simpler than in the bosonic case. The fermionic Taylor series terminates after the firstcontribution. This observation is representative of a general rule: Grassmann calculus is simpler thanstandard calculus; all series are finite, integrals always converge, etc.

5In the literature, complex conjugation of Grassmann variables is sometimes defined. Although ap-pealing from an aesthetic point of view — symmetry between bosons and fermions — this concept isproblematic. The difficulties become apparent when supersymmetric theories are considered, i.e.theories where operator algebras contain both bosons and fermions (the so–called super–algebras). It isnot possible to introduce a complex conjugation that leads to compatibility with the commutation rela-tions of a super–algebra. It therefore seems to be better to abandon the concept of Grassmann complexconjugation altogether. Note that although, in the bosonic case, complex conjugation is inevitable (inorder to define convergent Gaussian integrals, say), no such need arises in the fermionic case.



Definition |ψ〉 = exp

[

ζ∑

i

ψia†i

]

|0〉

Action of ai ai|ψ〉 = ψi|ψ〉, 〈ψ|ai = ∂ψi〈ψ|

Action of a†i a†i |ψ〉 = ζ∂ψi|ψ〉, 〈ψ|a†i = 〈ψ|ψi

Overlap 〈ψ′|ψ〉 = exp

[

∑

i

ψ′iψi

]

Completeness∫

d(ψ, ψ) e−P

i ψiψi |ψ〉〈ψ| = 1F

Table 4.1: Basic properties of coherent states for bosons (ζ = 1, ψi ∈ C) and fermions(ζ = −1, ψi ∈ A). In the last line, the integration measure is defined as d(ψ, ψ) ≡∏

idψidψiπ(1+ζ)/2 .

⊲ Info. Grassmann Gaussian Integration: Finally, before turning to the developmentof the functional field integral, it is useful to digress on the generalization of higher dimen-sional Gaussian integrals for Grassmann variables. The prototype of all Grassmann Gaussianintegration formulae is the identity

∫

dηdη e−ηaη = a (4.18)

where a ∈ C takes arbitrary values. Eq. (4.18) is derived by a first order Taylor expansion ofthe exponential and application of Eq. (4.15). The multi–dimensional generalization of (4.18) isgiven by

∫

d(φ, φ)e−φTAφ = detA , (4.19)

where φ and φ are N–component vectors of Grassmann variables, the measure d(φ, φ) ≡∏Ni=1 dφidφi, and A may be an arbitrary complex matrix. For matrices that are unitarily diag-

onalisable, A = U†DU, with U unitary, and D diagonal, Eq. (4.19) is proven in the same wayas its complex counterpart (3.17): One changes variables φ→ U†φ, φ→ UT φ. Since detU = 1,the transform leaves the measure invariant (see below) and leaves us with N decoupled integralsof the type (4.18). The resulting product of N eigenvalues is just the determinant of A (cf.the later discussion of the partition function of the non–interacting gas). For general (non–diagonalisable) A, the identity is established by a straightforward expansion of the exponent.The expansion terminates at Nth order and, by commuting through integration variables, it may



be shown that the resulting Nth order polynomial of matrix elements of A is the determinant.6

Keeping the analogy with ordinary commuting variables, the Grassmann version of Eq. (3.18)reads

∫

d(φ, φ)e−φTAφ+νT ·φ+φT ·ν = eν

TA

−1ν detA (4.22)

To prove the latter, we note that∫

dηf(η) =∫

dηf(η+ν), i.e. in Grassmann integration one canshift variables as in the ordinary case. The proof of the Gaussian relation above thus proceedsin complete analogy to the complex case. As with Eq. (3.18), Eq. (4.22) can also be employed

to generate further integration formulae. Defining 〈· · ·〉 ≡ detA−1∫

d(φ, φ)e−φTAφ(· · ·), and

expanding both the left and the right hand side of (4.22) to leading order in the ‘monomial’νjνi, one obtains 〈ηj ηi〉 = A−1

ji . Finally, the N–fold iteration of this procedure gives

〈ηj1ηj2 . . . ηjn ηi1 ηi2 . . . ηin〉 =∑

P

( sgn P )A−1j1iP1

. . . A−1jniPn

where the signum of the permutation accounts for the sign changes accompanying the interchange

of Grassmann variables. Finally, as with Gaussian integration over commuting variables, by

taking N → ∞, the Grassmann integration can be translated to a Gaussian functional integral.

——————————————–

4.2 Field Integral for the Quantum Partition Func-

tion

Having introduced the coherent states, we will see that the construction of path inte-grals for many–body systems no longer presents substantial difficulties. However, be-

6As with ordinary integrals, Grassmann integrals can also be subjected to variable transforms.Suppose we are given an integral

∫

d(φ, φ)f(φ, φ) and wish to change variables according to

ν = Mφ, ν = M′φ , (4.20)

where, for simplicity, M and M′ are complex matrices (i.e. we here restrict ourselves to linear transforms).One can show that

ν1 . . . νN = (detM)φ1 . . . φN , ν1 . . . νN = (detM′)φ1 . . . φN . (4.21)

(There are different ways to prove this identity. The most straightforward is by explicitly expanding (4.20)in components and commuting all Grassmann variables to the right. A more elegant way is to argue thatthe coefficient relating the right and the left hand sides of (4.21) must be an Nth order polynomial ofmatrix elements of M. In order to be consistent with the anti–commutation behaviour of Grassmannvariables, the polynomial must obey commutation relations which uniquely characterise a determinant.Excercise: Check the relation for N = 2.) On the other hand, the integral of the new variables must obey

the defining relation,∫

dνν1 . . . νN =∫

dνν1 . . . νN = (−)N+1, where dν =∏Ni=1 dνi and the sign on the

right hand side is attributed to ordering of the integrand, viz.∫

dν1dν2ν1ν2 = −∫

dν1ν1∫

dν2ν2 = −1.Together Eqs. (4.21) and (4.20) enforce the identities dν = (detM)−1dφ, dν = (detM′)−1dφ, whichcombine to give

∫

d(φ, φ)f(φ, φ) = det(MM′)

∫

d(ν, ν′)f(φ(ν), φ(ν)) .


4.2. FIELD INTEGRAL FOR THE QUANTUM PARTITION FUNCTION 133

fore proceeding, we should address the question; what does the phrase ‘path–integral formany–body systems’ actually mean? In the next chapter we will see that much of theinformation about a quantum many–particle systems is encoded in expectation values ofproducts of creation and annihilation operators, i.e. expressions of the structure 〈a†a . . .〉.By an analogy to be explained then, objects of this type are generally called correlationfunctions. More important for our present discussion, at any finite temperature, theaverage 〈. . .〉 entering the definition of the correlation function runs over the quantum

Gibbs7 distribution ρ ≡ e−β(H−µN)/Z, where, as usual,

Z = tr e−β(H−µN) =∑

n

〈n|e−β(H−µN)|n〉 , (4.23)

is the quantum partition function, β ≡ 1/T , µ denotes the chemical potential, and thesum extends over a complete set of Fock space states |n〉. (For the time being we neitherspecify the statistics of the system — bosonic or fermionic — nor the structure of theHamiltonian.)

Ultimately, we will want to construct and evaluate the path integral representations ofmany–body correlation functions. Later we will see that all of these representations can bederived by a few straightforward manipulations form a prototypical path integral, namelythat for Z. Further, the (path integral of the) partition function is of importance in itsown right: It contains much of the information needed to characterise the thermodynamicproperties of a many–body quantum system.8 We thus begin our journey into many–bodyfield theory with a construction of the path integral for Z.

To prepare the representation of the partition function (4.23) in terms of coherentstates, one must insert the resolution of identity

Z =

∫

d(ψ, ψ) e−P

i ψiψi∑

n

〈n|ψ〉〈ψ|e−β(H−µN)|n〉 . (4.24)

We now wish to get rid of the — now redundant — Fock space summation over |n〉(another resolution of identity). To bring the summation to the form

∑

n |n〉〈n| = 1F ,

7

Josiah Willard Gibbs 1839–1903: cred-ited with the development of chemicalthermodynamics, he introduced conceptsof free energy and chemical potential.

8In fact, the statement above is not entirely correct. Strictly speaking thermodynamic propertiesinvolve the thermodynamic potential Ω = −T lnZ rather than the partition function itself. At firstsight it seems that the difference between the two is artificial — one might first calculate Z and thentake the logarithm. However, typically, one is unable to determine Z in closed form, but rather onehas to perform a perturbative expansion, i.e. the result of a calculation of Z will take the form of aseries in some small parameter ǫ. Now a problem arises when the logarithm of the series is taken. Inparticular, the Taylor series expansion of Z to a given order in ǫ does not automatically determine theexpansion of Ω to the same order. Fortunately, the situation is not all that bad. It turns out that thelogarithm essentially rearranges the combinatorial structure of the perturbation series in an order knownas a cumulant expansion.



one must commute the factor 〈n|ψ〉 to the right hand side. However, in performingthis seemingly innocuous operation, we must be careful not to miss a sign change whosepresence will have important consequences for the structure of the fermionic path integral:Indeed, it may be checked that, whilst for bosons, 〈n|ψ〉〈ψ|n〉 = 〈ψ|n〉〈n|ψ〉, the fermioniccoherent states change sign upon permutation, 〈n|ψ〉〈ψ|n〉 = 〈−ψ|n〉〈n|ψ〉 (i.e. 〈−ψ| ≡exp (−

∑

i ψiai)). The presence of the sign is a direct consequence of the anti–commutationrelations between Grassmann variables and Fock space operators (exercise). Note that,as both H and N contain elements even in the creation/annihilation operators, the signis insensitive to the presence of the Boltzmann factor in (4.24). Making use of the signfactor ζ , the result of the interchange can be formulated as the general expression

Z =

∫

d(ψ, ψ) e−P

i ψiψi∑

n

〈ζψ|e−β(H−µN)|n〉〈n|ψ〉

=

∫

d(ψ, ψ) e−P

i ψiψi〈ζψ|e−β(H−µN)|ψ〉 , (4.25)

where the equality is based on the identity∑

n |n〉〈n| = 1F . Eq. (4.25) can now be directlysubjected to the general construction scheme of the path integral.

To be concrete, let us assume that the Hamiltonian is limited to a maximum of two–body interactions (cf. Eqs. (2.5) and (2.9)),

H(a†, a) =∑

ij

hija†iaj +

∑

ijkl

Vijkl a†ia

†jakal . (4.26)

Note that, to facilitate the construction of the field integral, it is helpful to arrange for allof the annihilation operators to stand to the right of the creation operators. Fock spaceoperators of this structure are said to be normal ordered.9 The reason for emphasisingnormal ordering is that such an operator can be readily diagonalised by means of coher-ent states: Dividing the ‘time interval’ β into N segments and inserting coherent stateresolutions of identity (steps 1, 2 and 3 of the general scheme), Eq. (4.25) assumes theform

Z =

∫

ψ0=ζψN

ψ0=ζψN

N∏

n=0

d(ψn, ψn) e−δPN−1n=0 [δ−1(ψn−ψn+1)·ψn+H(ψn+1,ψn)−µN(ψn+1,ψn)] , (4.27)

where δ = β/N and 〈ψ|H(a†,a)|ψ′〉〈ψ|ψ′〉

=∑

ij hijψiψ′j +

∑

ijkl Vijkl ψiψjψ′kψ

′l ≡ H(ψ, ψ′), (simi-

larly N(ψ, ψ′)). Here, in writing Eq. (4.27), we have adopted the shorthand ψn = ψni ,etc. Finally, sending N → ∞ and taking limits analogous to those leading from (3.5) to

9More generally, an operator is defined to be ‘normal ordered’ with respect to a given vacuum state|0〉, if and only if, it annihilates |0〉. Note that the vacuum need not necessarily be defined as a zeroparticle state. If the vacuum contains particles, normal ordering need not lead to a representation whereall annihilators stand to the right. If, for whatever reason, one is given a Hamiltonian whose structurediffers from (4.26), one can always affect a normal ordered form at the expense of introducing commutatorterms. For example, normal ordering the quartic term leads to the appearance of a quadratic contributionwhich can be absorbed into hαβ .



(3.6) we obtain the continuum version of the path integral,10

Z =

∫

D(ψ, ψ)e−S[ψ,ψ], S[ψ, ψ] =

∫ β

0

dτ[

ψ∂τψ +H(ψ, ψ) − µN(ψ, ψ)]

(4.28)

where D(ψ, ψ) = limN→∞

∏Nn=1 d(ψ

n, ψn), and the fields satisfy the boundary condition

ψ(0) = ζψ(β), ψ(0) = ζψ(β) . (4.29)

Written in a more explicit form, the action associated with the general pair–interactionHamiltonian (4.26) can be cast in the form

S =

∫ β

0

dτ

[

∑

ij

ψi(τ) [(∂τ − µ)δij + hij ]ψj(τ) +∑

ijkl

Vijklψi(τ)ψj(τ)ψk(τ)ψl(τ)

]

. (4.30)

Notice that the structure of the action fits nicely into the general scheme discusedin the previous chapter. By analogy, one would expect that the exponent of the many–body path integral carries the significance of the Hamiltonian action, S ∼

∫

(pq − H),where (q, p) symbolically stands for a set of generalized coordinates and momenta. Inthe present case, the natural pair of canonically conjugate operators is (a, a†). One wouldthen interpret the eigenvalues (ψ, ψ) as ‘coordinates’ (much as (q, p) are the eigenvalues ofthe operators (q, p)). Adopting this interpretation, we see that the exponent of the pathintegral indeed has the canonical form of a Hamiltonian action and, therefore, is easy tomemorize.

Eqs. (4.28) and (4.30) define the functional integral in the time representation(in the sense that the fields are functions of a time variable). In practice we shall mostlyfind it useful to represent the action in an alternative, Fourier conjugate representation.To this end, note that, due to the boundary conditions (4.29), the functions ψ(τ) can beinterpreted as functions on the entire Euclidean time axis which are periodic/antiperiodicon the interval [0, β]. As such they can be represented in terms of a Fourier series,

ψ(τ) =1√β

∑

ωn

ψne−iωnτ , ψn =

1√β

∫ β

0

dτψ(τ)eiωnτ ,

where

ωn =

2nπT, bosons,(2n+ 1)πT, fermions

, n ∈ Z (4.31)

are known as Matsubara frequencies. Substituting this representation into (4.28)and (4.30), we obtain Z =

∫

D(ψ, ψ)e−S[ψ,ψ], where D(ψ, ψ) =∏

n d(ψn, ψn) defines the

10Whereas the bosonic continuum limit is indeed perfectly equivalent to the one taken in constructingthe quantum mechanical path integral (limδ→0 δ

−1(ψn+1 − ψn) = ∂τ |τ=nδ gives an ordinary derivativeetc.), a novelty arises in the fermionic case. The notion of replacing differences by derivatives is purelysymbolic for Grassmann variables. There is no sense in which ψn+1− ψn is small. The symbol ∂τ ψ ratherdenotes the formal (and well defined expression) limδ→0 δ

−1(ψn+1 − ψn).



measure (for each Matsubara index n we have an integration over a coherent state basis|ψn〉),11 and the action takes the form

S[ψ, ψ] =∑

ij,ωn

ψin [(−iωn − µ) δij + hij]ψjn +1

β

∑

ijkl,ωni

Vijkl ψin1ψjn2ψkn3ψln4δn1+n2,n3+n4 .(4.32)

Here we have used the identity∫ τ

0dτe−iωnτ = βδωn,0. Eq. (4.32) defines the frequency

representation of the action.12

⊲ Info. In performing calculations in the Matsubara representation, one sometimes runs

into convergence problems (which will manifest themselves in the form of ill–convergent Matsub-

ara frequency summations): In such cases it will be important to remember that Eq. (4.32) does

not actually represent the precise form of the action. What is missing is a convergence generat-

ing factor whose presence follows from the way in which the integral was constructed, and which

will save us in cases of non–convergent sums (except, of course, in cases where divergences have

a physical origin). More precisely, since the fields ψ are evaluated infinitesimally later than the

operators ψ (cf. Eq. (4.27)), the h and µ–dependent contributions to the action acquire a factor

exp(−iωnδ), δ infinitesimal. Similarly, the V contribution acquires a factor exp(−i(ωn1 +ωn2)δ).

In cases where the convergence is not critical, we will omit these contributions. However, once

in a while it is necessary to remember their presence.

——————————————–

4.2.1 Partition Function of Non–Interacting Gas

As a first exercise, let us consider the quantum partition function of the non–interactinggas. (Later, this object will prove useful as a ‘reference’ in the development of weaklyinteracting theories.) In some sense, the field integral formulation of the non–interactingpartition function has a status similar to that of the path integral for the quantum har-monic osciallator: The direct quantum mechanical solution of the problem is straightfor-ward and application of the full artillery of the field integral seems somewhat ludicrous.From a pedagogical point of view, however, the free partition function is a good problem;it provides us with the welcome opportunity to introduce a number of practical conceptsof field integration within a comparatively simple setting. Moreover, the field integralrepresentation of the free partition function will be an important operational buildingblock for our subsequent analysis of interacting problems.

Consider, then the partition function (4.28) with H0(ψ, ψ) =∑

ij ψiH0,ij ψj . Diag-

onalising H0 by a unitary transformation U , H0 = UDU † and transforming integrationvariables U †ψ ≡ φ, the action assumes the form, S =

∑

a

∑

ωnφan(−iωn + ξa)φan, where

ξa ≡ ǫa − µ and ǫa are the single particle eigenvalues. Remembering that the fields φa(τ)

11Notice, however, that the fields ψn carry dimension [energy]−1/2, i.e. the frequency coherent stateintegral is normalized as

∫

d(ψn, ψn) e−ψnǫψn = (βǫ)−ζ .12As for the signs of the Matsubara indices appearing in Eq. (4.32), note that the Fourier representation

of ψ is defined as ψ(τ) = 1√β

∑

n ψne+iωnτ , ψn = 1√

β

∫ β

0 dτ ψ(τ)e−iωnτ . In the bosonic case, this sign

convention is motivated by ψ being the complex conjugate of ψ. For reasons of notational symmetry, thisconvention is also adopted in the fermionic case.



are independent integration variables (exercise: why does the transformation ψ → φ havea Jacobian of unity?), we find that the partition function decouples, Z =

∏

aZa, where

Za =

∫

D(φa, φa) e−

P

n φan(−iωn+ξa)φan =∏

n

[β(−iωn + ξa)]−ζ , (4.33)

and the last equality follows from the fact that the integrals over φan are one–dimensionalcomplex or Grassmann Gaussian integrals. Here, let us recall our convention definingζ = 1(−1) for bosonic (fermionic) fields. At this stage, we have left all aspects of fieldintegration behind us and reduced the problem to one of computing an infinite productover factors iωn − ξa. Since products are usually more difficult to get under control thansums, we take the logarithm of Z to obtain the free energy

F = −T lnZ = −Tζ∑

an

ln[β(−iωn + ξa)] . (4.34)

⊲ Info. Before proceeding with this expression, let us take a second look at the interme-diate identity (4.33). Our calcuation showed the partition function to be the product over alleigenvalues of the operator −iω + H − µN defining the action of the non-interacting system(here, ω = ωnδnn′). As such, it can be written compactly as:

Z = det[

β(−iω + H − µN)]−ζ

(4.35)

This result was derived by first converting to an eigenvalue integration and then performing theone–dimensional integrals over ‘eigencomponents’ φan. While technically straightforward, that— explicitly representation-dependent — procedure is not well suited to generalization to morecomplex problems. (Keep in mind that later on we will want to embed the free action of thenon–interacting problem into the more general framework of an interacting theory.)

Indeed, it is not necessary to refer to an eigenbasis at all: In the bosonic case, Eq. (3.17)

tells us that Gaussian integration over a bilinear ∼ φXφ generates the inverse determinant of

X . Similarly, as we have seen, Gaussian integration extends to the Grassmann case with the

determinants appearing in the numerator rather than in the denominator (as exemplified by

(4.35)). (As a matter of fact, (4.33) is already a proof of this relation.)

——————————————–

We now have to face up to a technical problem: How do we compute Matsubara sumsof the form

∑

n ln(iωn − x)? Indeed, it takes little imagination to foresee that sums ofthe type

∑

n1,n2,...X(ωn1, ωn2, . . .), where X symbolically stands for some function, will

be a recurrent structure in the analysis of functional integrals. A good ansatz wouldbe to argue that, for sufficiently low temperatures (i.e. temperatures smaller than anyother characteristic energy scale in the problem), the sum can be traded for an integral,viz. T

∑

n →∫

dω/(2π). However, this approximation is too crude to capture much ofthe characteristic temperature dependence in which one is usually interested. Yet thereexists an alternative, and much more accurate way of computing sums over Matsubarafrequencies:



⊲ Info. Consider a single Matsubara frequency summation,

S ≡∑

n

h(ωn) , (4.36)

where h is some function and ωn may be either bosonic or fermionic (cf. Eq. (4.31)). Thebasic idea behind the standard scheme of evaluating sums of this type is to introduce a complexauxiliary function g(z) that has simple poles at z = iωn. The sum S then emerges as the sumof residues obtained by integrating the product gh along a suitably chosen path in the complexplane. Typical choices of g include

g(z) =

β

exp(βz) − 1, bosons

β

exp(βz) + 1, fermions

and g(z) =

β

2coth(βz/2), bosons

β

2tanh(βz/2), fermions

(4.37)

where, in much of this section, we will employ the functions of the first column. (Notice thesimilarity between these functions and the familiar Fermi and Bose distribution functions.) Inpractice, the choice of the counting function is mostly a matter of taste, save for some caseswhere one of the two options is dictated by convergence criteria.

w1 w2

Figure 4.2: Left: the integration contour employed in calculating the sum (4.36). Right:the deformed integration contour.

Integration over the path shown in the left part of Fig. 4.2 then produces

ζ

2πi

∮

dzg(z)h(−iz) = ζ∑

n

Res (g(z)h(−iz))|z=iωn =∑

n

h(ωn) = S ,

where, in the third identity, we have used the fact that the ‘counting functions’ g are chosen soas to have residue ζ and it is assumed that the integration contour closes at z → ±i∞. (Thedifference between using the first and the second column of (4.37) lies in the value of the residue.In the latter case, it is equal to unity rather than ζ.) Now, the integral along a contour in theimmediate vicinity of the poles of g is usually intractable. However, as long as we are careful notto cross any singularities of g or the function h(−iz) (the latter symbolically indicated by isolatedcrosses in the figure13) we are free to distort the integration path, ideally to a contour along

13Remember that a function that is bounded and analytic in the entire complex plane is constant, i.e.every ‘interesting’ function will have singularities.



which the integral can be performed. Finding a suitable contour is not always straightforward.If the product hg decays sufficiently fast at |z| → ∞ (i.e. faster than z−1), one will ususally tryto ‘inflate’ the original contour to an infinitely large circle (Fig. 4.2, right).14 The integral alongthe outer perimeter of the contour then vanishes and one is left with the integral around thesingularities of the function h. In the simple case where h(−iz) posesses a number of isolatedsingularities at zk (i.e. the situation indicated in the figure) we thus obtain15

S = − ζ

2πi

∮

h(−iz)g(z) = −ζ∑

k

Resh(−iz)g(z)|z=zk , (4.38)

i.e. the task of computing the infinite sum S has been reduced to that of evaluating a finitenumber of residues — a task that is always possible!

To illustrate the procedure on a simple example, let us consider the function

h(ωn) = − ζT

iωne−iωnδ − ξ,

where δ is a positive infinitesimal.16 To evaluate the sum S =∑

n h(ωn), we first observe thatthe product h(−iz)g(z) has benign convergence properties. Further, the function h(−iz) has asimple pole that, in the limit δ → 0, lies on the real axis at z = ξ. This leads to the result

∑

n

h(ωn) = ζ Res g(z)h(−iz)|z=ξ = − 1

eβξ − ζ.

We have thus arrived at the important identity

ζT∑

n

1

iωn − ξa=

nB(ǫa), bosons,nF(ǫa), fermions

(4.39)

14Notice that the condition lim|z|→∞ |hg| < z−1 is not as restrictive as it may seem. The reason isthat the function h will be mostly related to physical observables that approach some limit (or vanish)for large excitation energies. This implies vanishing in at least portions of the complex plane. Theconvergence properties of g depend on the concrete choice of the counting function. (Exercise: explorethe convergence properties of the functions shown in Eq. (4.37).)

15If you are confused about signs, note that the contour encircles the singularities of h in a clockwisedirection.

16Indeed, this choice of h is actually not as artificial as it may seem. The expectation value of thenumber of particles in the grand canonical ensemble is defined through the identity N ≡ −∂F/∂µwhere F is the free energy. In the non–interacting case, the latter is given by Eq. (4.34) and, rememberingthat ξa = ǫa − µ, one obtains N ≈ −ζT ∑

an1

iωn−ξa

. Now, why did we write ‘≈’ instead of ‘=’? The

reason is that the right hand side, obtained by naive differentiation of (4.34), is ill–convergent. (Thesum

∑∞n=−∞

1n+x , x arbitrary, does not exist!) At this point we have to remember the remark made in

the on page 136, i.e., had we carefully treated the discretisation of the field integral, both the logarithmof the free energy and ∂µF would acquire infinitesimal phases exp(−iωnδ). As an exercise, try to keeptrack of the discretisation of the field integral from its definition to Eq. (4.34) to show that the accurateexpression for N reads

N = −ζT∑

an

1

iωne−iωnδ − ξa=

∑

a

∑

n

h(ωn)∣

∣

ξ=ξa

,

where h is the function introduced above. (Note that the necessity to keep track of the lifebuoy e−iωnδ

does not arise too often. Most Matsubara sums of physical interest relate to functions f that decay fasterthan z−1.)



where

nF(ǫ) =1

eβ(ǫ−µ) + 1, nB(ǫ) =

1

eβ(ǫ−µ) − 1(4.40)

are the Fermi/Bose distribution functions. As a corollary we note that the expectation valuefor the number of particles in a non–interacting quantum gas assumes the familiar form N =∑

a nF/B(ǫa).

Before returning to our discussion of the partition function, let us note that life is not always

as simple as the example above. More often than not, the function h not only contains isolated

singularities but also cuts or worse singularities. Under such circumstances, finding a good

choice of the integration contour can be far from straightforward!

——————————————–

xi

w1

Figure 4.3:

Returning to the problem of computing the sum (4.34), consider for a moment a fixedeigenvalue ξa ≡ a. In this case, we need to evaluate the sum S ≡

∑

n h(ωn), whereh(ωn) ≡ ζT ln[β(−iωn + ξ)] = ζT ln[β(iωn − ξ)] + C and C is an inessential constant. Asdiscussed before, the sum can be represented as S = − ζ

2πi

∮

g(z)h(−iz), where g(z) =β(eβz − ζ)−1 is (β times) the distribution function and the contour encircles the poles ofg as in Fig. 4.2, left. Now, there is an essential difference with the example discussedpreviously, viz. the function h(−iz) = ζT ln(z − ξ) + C has a branch cut along the realaxis, z ∈ (−∞, ξ) (see the figure). To avoid contact with this singularity one must distortthe integration contour as shown in the figure. Noticing that the (suitably regularized, cf.our previous discussion of the particle number N) integral along the perimeter vanishes,we conclude that

S =T

2πi

∫ ∞

−∞

dǫ g(ǫ)(

ln(ǫ+ − ξ) − ln(ǫ− − ξ))

,

where ǫ± = ǫ±iη, η is a positive infinitesimal, and we have used the fact that g(ǫ±) ≃ g(ǫ)is continuous across the cut. (Also, without changing the value of the integral (exercise:why?), we have enlarged the integration interval from (−∞, ξ] to (−∞,∞)). To evaluate


4.3. SUMMARY AND OUTLOOK 141

the integral, we observe that g(ǫ) = −ζ∂ǫ ln(

1 − ζe−βǫ)

and integrate by parts:

S = − ζT

2πi

∫

dǫ ln(

1 − ζe−βǫ)

(

1

ǫ+ − ξ− 1

ǫ− − ξ

)

(3.58)= ζT ln

(

1 − ζe−βξ)

.

Insertion of this result into Eq. (4.34) finally obtains the familiar expression

F = ζT∑

a

ln(

1 − ζe−β(ǫa−µ))

(4.41)

for the free energy of the non–interacting Fermi/Bose gas. While this result could havebeen obtained much more straightforwardly by the methods of quantum statistical me-chanics, we will shortly see how powerful a tool the methods discussed in this section arewhen it comes to the analysis of less elementary problems!

4.3 Summary and Outlook

This concludes our preliminary introduction to the field integral. We have learned how torepresent the partition function of a quantum many–body system in terms of a generalizedpath integral. The field integral representation of the partition function will be the basicplatform on which all our further developments will be based. In fact, we are now in aposition to face up to the main problem addressed in this text: practically none of the‘non–trivial’ field integrals in which one might be interested can be executed in closed form.This reflects the fact that, save for a few exceptions, interacting many–body problems donot admit closed solutions. In the following chapter, we will introduce approximationstratgies for addressing interacting theories by exploring some physical applications ofthe field integral.


Chapter 5

Broken Symmetry and CollectivePhenomena

Previously, we have seen how the field integral method can be deployed in many-particle

theories. In the following chapter, we will learn how elements of perturbation theory can be

formulated efficiently by staying firmly within the framework of the field integral. In doing

so, we will see how the field integral provides a method for identifying and exploring non-

trivial reference ground states — ‘mean–fields’. A fusion of perturbative and mean–field

methods will provide us with analytical machinery powerful enough to address a spectrum

of rich applications.

Historically, the effects of interaction on many-body systems are typically dealt withwithin the framework of “diagrammatic” perturbation theory, a series expansion in theinteraction strength. However, in the following, we will search for a different methodology.Our motivations are two-fold: Firstly, the structures that typically appear from pertur-bative expansions can be assimilated more straightforwardly. But, more importantly, thedevelopment of phase instabilities typically reflect the appearance of non-perturbativestructures. Thus, what we would like to develop is a theoretical framework that is ca-pable of (a) detecting the ‘right’ reference states or ‘mean-fields’ of a system, (b) thatenables us to efficiently apply perturbative methods around these states and, finally, (c)to do this in a language that draws upon the ‘physical’ rather than the plain microscopicdegrees of freedom as the fundamental units.

To this end, in the following sections we will develop a functional integral based ap-proach that meets these criteria. In contrast to the previous chapters, the discussion herewill be decidedly biased towards concrete application to physically motivated problems.After the formulation of the general strategy of field integral based mean-field methods,the next section will address the problem of the weakly interacting electron gas. Theexemplification of the new concepts on “familiar territory” will enable us to better un-derstand the intimate connection between the mean-field approach and straightforwardperturbation theory. In subsequent sections we will then turn to the discussion of problemswhich lie firmly outside the range of direct perturbative summation, viz. the phenomenaof superfluidity and superconductivity.


146CHAPTER 5. BROKEN SYMMETRY AND COLLECTIVE PHENOMENA

5.1 Mean-Field Theory

Roughly speaking, the functional approach to problems with a large parameter proceedsaccording to the following programme:

1. In the first place, one must identify the relevant structural units of the theory. (Thispart of the programme can be efficiently carried out by the straightforward methodsdiscussed earlier.)

2. Secondly, it is necessary to introduce a new field — let us call it φ for concreteness— that encapsulates the relevant degrees of freedom of the low energy theory.

3. With this in hand, one can then trade integration over the ‘microscopic fields’ foran integration over φ, a step often effected by an operation known as the Hubbard–Stratonovich transformation.

4. The low–energy content of the theory can often be explored by subjecting the re-sulting action S[φ] to a stationary phase analysis. (The justification for applyingstationary phase methods is provided by the existence of a large parameter N ≫ 1.)Often, at this stage, instabilities in the theory show up — an indication of a physi-cally interesting problem!

5. Finally, the nature of the elementary (collective) excitations above the ground statecan by explored by expanding the functional integral around the solution of thestationary phase equations — the ‘mean-field’. From this low-energy effective action,one can compute physical observables.

In the next section, we will illustrate how such a programme can be implemented on aspecific example which can also be studied by perturbative means:

5.2 Plasma Theory of the Interacting Electron Gas

In first quantised form, the weakly interacting electron gas is described by the many-bodyHamiltonian,

H =N∑

i=1

p2i

2m+∑

i<j

e2

|ri − rj|(5.1)

For simplicity, we have chosen to neglect to presence of any underlying lattice potential.Formally, the corresponding quantum partition function is obtained as Z = tre−β(H−µN)

where the trace runs over a complete basis of many-body states.

⊲ Info. Before plunging into the computation of the quantum partition function from thecoherent state path integral we should first try to understand in what limit Coulomb interactionsmay be thought of as weak. In fact, the control parameter is the average density of the electrongas: n ≡ 1/r30 where r0 denotes the average interelectron spacing. Coulomb interaction is weak


5.2. PLASMA THEORY OF THE INTERACTING ELECTRON GAS 147

when the average potential energy (measured in units of e2/r0) is small as compared to thetypical kinetic energy (measured in units of ~

2/mr20). The ratio of energy scales defines thedimensionless density parameter

e2

r0

mr20~

=r0a0

≡ rs,

where a0 = ~/e2m denotes the Bohr radius. Physically, rs is the radius of the spherical volumecontaining one electron on average; the denser the electron gas, the smaller rs.

Below, we will be concerned with the limit of high density rs ≪ 1, in which the effects ofCoulomb interaction can be treated perturbatively. In the opposite limit, rs ≫ 1, propertiesbecome increasingly dominated by the Coulomb interaction. Ultimately, for sufficiently large rs(or low density) it is believed that the electron gas undergoes a (first order) phase transitionto a condensed or ‘solid’ phase known as a Wigner crystal. (Indeed, this phenomenon is thecontinuum counterpart of the Mott-Hubbard transition described in section 2.2.3.) AlthoughWigner1 crystals have never been unambiguously observed, several experiments performed onlow density electron gases are consistent with a Wigner crystal ground state. Quantum Monte-Carlo simulation suggests that Wigner crystallisation may occur for densities rs > 37. (Notethat this scenario relies crucially on being at low temperatures, and the long-range nature of theCoulomb interaction. In particular, if the Coulomb interaction is screened V (r) ∼ e−r/λ, rs ∼(r0/a0)e

−r0/λ and the influence of Coulomb interaction at high densities becomes diminished.)For rs ∼ O(1), the potential and kinetic energies are comparable. This regime of intermediate

coupling is notoriously difficult to describe quantitatively. Yet most metals lie in a regime ofintermediate coupling 2 < rs < 6. Fortunately, there is overwhelming evidence to suggest that aweak coupling description holds even well outside the regime over which microscopic theory canbe justified. The phenomenology of the intermediate coupling regime is the realm of Landau’s

Fermi Liquid Theory.

Figure 5.1: Sketch showing the adiabatic continuity of the eigenstates in a one-dimensionalpotential well.

The fundamental principle underlying the Fermi liquid theory is one of “adiabatic conti-

nuity” [5]: In the absence of an electronic phase transition (such as Wigner crystallisation or

1

Eugene P. Wigner 1902-1995;1963 Nobel Laureate in Physicsfor his contributions to the the-ory of the atomic nucleus and theelementary particles, particularlythrough the discovery and appli-cation of fundamental symmetryprinciples.



the Mott transition), a non-interacting ground state evolves smoothly or adiabatically into the

interacting ground state as the strength of interaction is increased.2 An elementary excitation

of the non-interacting system represents an “approximate excitation” of the interacting system

(i.e. the ‘lifetime’ of an elementary excitation is long). Excitations are quasi-particles (and

quasi-holes) above a sharply defined Fermi surface. The remarkable success (as well as the few

notorious failures) of Landau Fermi liquid theory3 make the subject an important area of modern

condensed matter physics but one which we will not have time to explore. Instead, motivated

in part by the success of Fermi liquid theory, we will proceed to explore the quantum partition

function of the weakly interacting electron gas, rs ≪ 1.

——————————————–

To prepare for a discussion of the field integral, we must first recast the Hamiltonianin second quantised form as

H =

∫

d3rc†σp2

2mcσ +

1

2

∫

d3r d3r′c†σ(r)c†σ′(r

′)e2

|r − r′|cσ′(r′)cσ(r),

where the sum over repeated spin indicies σ is here implied. When cast as functional fieldintegral, the corresponding quantum partition function takes the form Z ≡ tr(e−β(H−µN)) =∫D(ψ, ψ)e−S, where

S[ψ, ψ] =

∫ β

0

dτ

∫

d3rψσ

(

∂τ +p2

2m− µ

)

ψσ

+1

2

∫ β

0

dτ

∫

d3r d3r′ ψσ(r)ψσ′(r′)

e2

|r − r′|ψσ′(r′)ψσ(r),

Employing the “four-vector” shorthand x = (τ, r) and q = (ωm,q), where ωm denotesa bosonic Matsubara frequency (exercise: think why), we may define the “density field”

ρq = 1√β

∫dxeiq·x ψσ(x)ψσ(x) = 1√

β

∑

p ψpσψp+qσ (with∫dx ≡

∫ β

0dτ∫d3r and q · x ≡

ωmτ − q · r). Then, expressed in the Fourier basis, the action takes the form (exercise)

S[ψ, ψ] =∑

p

ψpσ

(

−iωn +p2

2m− µ

)

ψpσ +1

2L3

∑

q

′ρqV (q)ρ−q,

2As a simple non-interacting example, consider the adiabatic evolution of the bound states of a quan-tum particle as the confining potential is changed from a box to a harmonic potential well (see Fig. 5.1).While the wavefunctions and energies evolve, the topological characteristics of the wavefunctions, i.e.the number of the nodes, and therefore the assignment of the corresponding quantum numbers remainsunchanged.

3L. D. Landau, Sov. Phys. JETP 3, 920 (1956); ibid. 5, 101 (1957).

Lev D. Landau 1908-1968, 1962Nobel Laureate in Physics for hispioneering theories for condensedmatter, especially liquid helium.



where V (q) =∫d3re−iq·rV (r) = 4πe2/q2.4 Here, the prime on the q summation de-

notes the exclusion of the q = 0 contribution and reflects the presence of a neutralisingbackground charge.

Now, being quartic in the fields ψσ, the Coulomb interaction prevents an explicitcomputation of the functional ψ–integral. However, it is actually a straightforward matterto reduce, or ‘decouple’ the interaction operator bringing it to a form quadratic in thefields ψ. Let us multiply the functional integral by the ‘fat unity’

1 ≡ N∫

Dφ exp

[

−e2

2

∑

q

φqV−1(q)φ−q

]

,

where φ represents a complex bosonic field variable, and a normalization constant hasbeen absorbed in the definition of the functional measuere Dφ. Employing the variableshift eφq 7→ eφq + i

L3/2V (q)ρq, one obtains

1 =

∫

Dφ exp

[∑

q

(

−e2

2φqV

−1(q)φ−q −i

L3/2eρqφ−q +

1

2L3ρqV (q)ρ−q

)]

.

The rational behind this exercise can be seen in the last contribution to the exponent: Thisterm is equivalent to the quartic interaction contribution to the fermionic path integral,albeit with opposite sign. Therefore, multiplication of Z by our unity leads to the fieldintegral Z =

∫Dφ

∫D(ψσ, ψσ)e

−S, where

S =1

8π

∑

q

φqq2φ−q +

∑

pp′

ψpσ

[(

−iωn +p2

2m− µ

)

δpp′ +i

√

βL3eφ(p′ − p)

]

ψp′σ , (5.2)

denotes the action, i.e. an expression that is free of quartic field interactions of ψσ. Beforeproceeding, to acquire some intuition for the nature of the action, it is helpful to rewrite Sin a real space/time representation. With φq = 1√

βL3

∫dx e−iq·xφ(x) (four-vector notation

as above), one may confirm that,

S[φ, ψ, ψ] =

∫ β

0

dτ

∫

d3r

1

8π(∂φ)2 + ψσ

[

∂τ −∂2

2m− µ+ ieφ

]

ψσ

.

Physically, φ couples to the electron degrees of freedom as a space/time dependent (imag-inary) scaler potential, while the first term reflects the Lagrangian energy density asso-ciated with the electric component of the electromagnetic (alias the photon) field. Saiddifferently, the field φ represents the gauge particle — in this case, the photon — thatmediates the Coulomb interaction between electrons. Before proceeding, let us now stepback and discuss the general philosophy of the manipulations that led from the originalpartition function to the two-field representation (5.2).

⊲ Info. The sequence of manipulations developed above, i.e. the ‘decoupling’ of a quarticinteraction through an auxiliary field, is known more generally as a Hubbard–Stratonovich

4By representing Poisson’s equation for a point charge in the Fourier space, one may confirm that1

L3

∑

qeiq·rV (q) = e2

|r| .



transformation. The essence of the transformation is a straightforward manipulation of aGaussian integral. To make this point more transparent, let us reformulate the Hubbard-Stratonovich transformation in a notation that is not burdened by the presence of model–specificconstants. Consider an interaction operator of the form Sint = Vαβγδ ψαψβψγψδ (summation con-vention implied), where ψ and ψ may be either bosonic or fermionic field variables, the indicesα, β, . . . refer to an unspecified set of quantum numbers, Matsubara frequencies, etc., and Vαβγδis an interaction matrix element. Now, let us introduce composite operators ραβ ≡ ψαψβ torewrite the interaction as Sint = Vαβγδ ραβ ργδ. The notation can be compactified still furtherby introducing composite indices m ≡ (αβ), n ≡ (γδ), whereupon the action Sint = ρmVmnρnacquires the structure of a generalized bilinear form. To reduce the action to a form quadraticin the ψs one may simply multiply the exponentiated action by unity, viz.

e−ρmVmnρn =

∫

Dφe−14φmV

−1mnφn

︸︷︷︸

1

e−ρmVmnρn ,

where φ is bosonic. (Notice that here V −1mn represents the matrix elements of the inverse and

not the inverse (Vmn)−1 of individual matrix elements.) Finally, applying the variable change

φm → φm + 2i(V ρ)m where the notation (V ρ) is shorthand for Vmnρn, one obtains

exp [−ρmVmnρn] =

∫

Dφ exp

[

−1

4φmV

−1mnφn − iφmρn

]

I.e. the term quadratic in ρ is cancelled.5 This completes the formulation of the Hubbard-Stratonovich transformation. The interaction operator has been traded for (a) an integrationover an auxiliary field coupled (b) to a ψ-bilinear (the operator φmρm).

⊲ In essence, the Hubbard-Stratonovich transformation is tantamount to Gaussian integralidentity (3.13) but read in reverse. An exponentiated square is removed in exchange for alinear coupling. (In (3.13) we showed how terms linear in the integration variable can beremoved.)

⊲ To make the skeleton outlined above a well defined prescription, one has to be morespecific about the meaning of the Gaussian integration over the kernel φmV

−1mnφn, i.e. the

integration variables can be real or complex, and V must be a positive matrix (which isusually the physical situation).

⊲ There is some freedom as to the choice of the integration variable. For example, the factorof 1/4 in front of the Gaussan weight φmV

−1mnφn was introduced for mere convenience (viz.

to generate a coupling φmρm free of numerical factors). If one does not like to invert thematrix kernel Vmn, one can scale φm → (V φ)m, whereupon the key formula reads

e−ρmVmnρn =

∫

Dφe−14φmVmnφn−iφmVmnρn .

⊲ Exercise. Show that the passage from the Lagrangian to the Hamiltonian formulation ofthe Feynman path integral can be interpreted as a Hubbard–Stratonovich transformation.

5Here we have assumed that the matrix V is symmetric. If it is not, we can apply the relationρmVmnρn ≡ ρTV ρ = 1

2

[ρT (V + V T )ρ

]to symmetrize the interaction.



α β

γ δ

α β

γ δ

α β

γ δ

Figure 5.2: On the different channels of decoupling an interaction by Hubbard-Stratonovichtransformation. Left: decoupling in the ‘density’ channel; middle: decoupling in the ‘pairing’ or‘Cooper’ channel; and right: decoupling in the ‘exchange’ channel.

As defined, the Hubbard-Stratonovich transformation is exact. However, to make it a mean-ingful operation, it must be motivated by some physical considerations. In our discussion above,we split up the interaction by choosing ραβ as a composite operator. However, there is clearlysome arbitrariness with this choice. Why not, for example, pair the fermion–bilinears accordingto (ψαψδ)(ψγψδ), or otherwise? The three inequivalent choices of pairing up operators are shownin Fig. 5.2 where, as usual, the wavy line with attached field vertices represents the interaction,and the dashed ovals indicate how the field operators are paired.

The version of the transformation discussed above corresponds to the left diagram of thefigure. That type of pairing is sometimes referred to as decoupling in the direct channel. Thedenotation becomes more transparent if we consider the example of the spinfull electron-electroninteraction,

Sint =1

2

∫

dτ

∫

d3r d3r′ ψσ(r, τ)ψσ′ (r′, τ)V (r− r′)ψσ′(r

′, τ)ψσ(r, τ),

i.e. here α = β = (r, τ, σ), γ = δ = (r′, τ, σ′), and Vαβγδ = V (r − r′). The ‘direct’ decouplingproceeds via the most obvious choice, i.e. the density operator ρ(r, τ) = ψσ(r, τ)ψσ(r, τ). Onespeeks about decoupling in a ‘channel’ because, as will be elucidated below, the propagator ofthe decoupling field can be interpreted in terms of two Green function lines tied together bymultiple interactions, a sequential object reminiscent of a ‘channel’.

However, more important than the terminology is the fact that there are other choices forρ. Decoupling in the exchange channel is generated by the choice ραγ ∼ ψαψδ where, inthe context of the Coulomb interaction, the reversed pairing of field operators is reminiscent ofthe exchange contraction generating Fock–type contributions. Finally, one may decouple in theCooper channel, ρ = ψαψγ , ρβγ = ρ†γβ. Here, the pairing field is conjugate to two creationoperators. Below we will see that this type of decoupling is tailored to problems involvingsuperconductivity.

The remarks above may convey the impression of a certain arbitrariness inherent in theHubbard-Stratonovich scheme. Indeed, the ‘correct’ choice of decoupling can only be motivatedby physical reasoning, not by plain mathematics. Put differently, the transformation as such isexact, no matter what channel we choose. However, later, we will want to derive an effectivelow energy theory based on the decoupling field. In cases where one has accidentally decoupledin an ‘unphysical’ channel, it will be difficult, if not impossible to distill a meaningful lowenergy theory for the field φ conjugate to ρ. Although the initial model still contains the fullmicroscopic information (by virtue of the exactness of the transformation) it is not amenable tofurther approximation schemes.

In fact, one is frequently confronted with situations where more than one Hubbard-Stratonovichfield is needed to capture the full physics of the problem. To appreciate this point, consider the



Coulomb interaction in momentum space.

Sint[ψ, ψ] =1

2

∑

p1,...,p4

ψσp1ψσ′p3V (p1 − p2)ψσ′p4ψσp2 δp1−p2+p3−p4 . (5.3)

In principle, we can decouple this interaction in any one of the three channels discussed above.However, ‘interesting’ physics is usually generated by processes where one of the three unboundmomenta entering the interaction vertex is small. Only these interaction processes have a chanceto accumulate an overall collective excitation of low energy (cf. many of the examples to follow).It may be instructive to imagine the situation geometrically: In the three dimensional cartesianspace of free momentum coordinates (p1, p2, p3) entering the vertex, there are three thin layerswhere one of the momenta is small, (q, p2, p3), (p1, q, p3), (p1, p2, q), |q| ≪ |pi|. (Why not make allmomenta small? Because that would be in conflict with the condition that the Green functionsconnecting to the vertex be close to the Fermi surface.) One will thus often choose to break downthe full momentum summation to a restricted summation over the small–momentum sublayers:

Sint[ψ, ψ] ≃ 1

2

∑

p,p′,q

(

ψσpψσp+qV (q)ψσ′p′ψσ′p′−q − ψσpψσ′p+qV (p′ − p)ψσ′p′+qψσp′ −

ψσpψσ′−p+qV (p′ − p)ψσp′ψσ′−p′+q)

.

Now, each of these three contributions has its own predestined choice of a slow decoupling field.The first term should be decoupled in the direct channel ρd,q ∼ ∑

p ψσpψσp+q, the second in

the exchange channel ρx,σσ′q ∼ ∑

p ψσpψσ′p+q, and the third in the Cooper channel ρc,σσ′q ∼∑

p ψσpψσ′−p+q. One thus winds up with an effective theory that contains three independentslow Hubbard–Stratonovich fields. (Notice that the decoupling fields in the exchange and in theCooper channel explicitly carry a spin–structure.)

After this digression on the principles of the Hubbard–Stratonovich transformation, let us

now return to the discussion of the electron gas.

——————————————–At the expense of introducing a second field, the Hubbard-Stratonovich transformation

provides an action quadratic in the fermion fields. In this case, the fermion integrationcan be undertaken exactly. Making use of the Gaussian functional integral (4.19), oneobtains

Z =

∫

Dφ e−18π

P

q φqq2φ−q det

[

∂τ +p2

2m− µ+ ieφ

]2

,

where the factor of two accounts for the spin degeneracy.The standard procedure to deal with the determinants generated at intermediate stages

of the manipulation of a field integral is to simply re–exponentiate them. This is achievedby virtue of the identity,

ln det A = tr ln A (5.4)

valid for arbitrary (non–singular) operators A.6 Thus, the quantum partition function

6Eq. (5.4) is readily established by switching to an eigenbasis whereupon one obtains ln det A =∑

a ln ǫa = tr ln A, where ǫa are the eigenvalues of A and we have used the fact that the eigenvalues of

ln A are ln ǫa.



takes the form Z =∫Dφ e−S[φ], where

S[φ] =1

8π

∑

q

φqq2φ−q − 2tr ln

[

∂τ +p2

2m− µ+ ieφ

]

. (5.5)

This is as far as purely formal exact manipulations can carry us. We have managed totrade the integration over the interacting Grassmann field ψσ for an integration over anauxiliary field φ; a field that we believe encapsulates the relevant degrees of freedom ofthe model. This completes steps 1, 2, and 3 of the general programme outlined above.

Ordinarily, the next step in the programme is to subject the action to a stationaryphase analysis, i.e. to seek solutions of the set of saddle-point equations such that

∀q = (q 6= 0, ω) :δS[φ]

δφq

!= 0 .

Such a solution φ(x, t) ↔ φq is commonly referred to as a mean-field. This terminologycan be understood by inspection of the argument of the ‘tr ln’ above. The structurep2/2m − µ + ieφ, where φ is a fixed configuration (to be determined by solving thesaddle–point equations), resembles the Hamiltonian operator of particles subject to somebackground potential, or ‘mean’ field. The notation on the left hand side of the saddle-point equations indicates that our original interaction V (q) and, therefore, the decouplingfield φ do not possess a zero momentum mode (a consequence of charge neutrality).

However, in the present case, since the interaction is considered weak, we may an-ticipate that the solution to the saddle-point variation is the trivial one, viz. φ = 0— an assumption that we may check self-consistently. In this case, step 4 of the gen-eral programme may be considered as achieved and we may turn to explore fluctuationsaround φ = 0. Since the mean-field solution vanishes, it makes no sense to introduce newnotation, i.e. we will denote the fluctuations again by the symbol φ.

As regards the first term in the action (5.5), it has already a quadratic form. Thelogarithmic contribution can be expanded as if we were dealing with a function (again, aconsequence of the trace), i.e. setting

G−1 ≡ G−10 + ieφ,

G−10 ≡ ∂τ + p2

2m− µ is the momentum and frequency diagonal operator whose matrix

elements give the free Green function of the electron gas, we may express

tr ln G−1 = tr ln(G−10 + ieφ) = tr ln G−1

0 + tr ln(1 + ieG0φ)

= tr ln G−10 + ietr (G0φ) +

e2

2tr (G0φG0φ) + . . . .

Being φ–independent, the first term generates an overall constant multiplying the func-tional integral, viz. a constant that must describe the non–interacting content of thetheory. Indeed, one may note that e2tr ln G−1

0 = e−2tr ln G0 = (det G−10 )2 ≡ Z0 is just the

partition function of the non-interacting electron gas. Linear in φ, the second term of theexpansion must, by virtue of the mean-field analysis, vanish. (Afterall, we are expandingaround an extremum! To this end, one may note that tr (G0φ) =

∑

qG0(q)φq=0 = 0.)



The third term is the interesting one. Remembering that φ couples to the theory as apotential, this term describes how potential flucutations are affected by the presence ofthe electron gas, i.e. it must encode the screening of the electromagnetic field by theelectron degrees of freedom.

To resolve this connection, let us make the momentum dependence of the second–orderterm explicit (exercise7):

e2

2tr (G0φG0φ) =

e2

2βL3

∑

p,q

G0(p+ q)φqG0(p)φ−q =e2

4

∑

q

Π(q)φqφ−q ,

where, setting ξp = p2

2m− µ,

Π(q) =∑

p

G0(p)G0(p+ q) =2

βL3

∑

ωn,p

1

−iωn + ξp

1

−iωn − iωm + ξp+q

Collecting together the bare photon action with this expansion, to leading order in e,the quantum partition function takes the form

Z = Z0

∫

Dφ e−S[φ],

where the effective action for the electromagnetic field φ is given by

S[φ] =1

2

∑

q

′ 1

D(q)|φq|2 +O(e4)

This result has a clear physical interpretation: the interaction of the electron gas withthe electromagnetic field induces a modified or screened Coulomb interaction (seeFig. 5.3),

D(ωm,q) =1

ǫ(ωm,q)

4π

q2, ǫ(ωn,q) = 1 − 4πe2

q2Π(ωm,q)

where ǫ(ωm,q) is the energy and momentum dependent effective dielectric function.This perturbative result, which is known in the literature as the Random Phase Ap-proximation (RPA), amounts to treating the long-range part of the Coulomb inter-action as an “external” polarisation field, and the correction to the dielectric function,(4πe2/q2)Π(ωm,q), is known as the screened polarisability. To explore the dielectricproperties of the interacting electron gas it is necessary to understand the frequency andmomentum dependence of the density-density response function (5.6). To do so we mustfirst learn how to perform Matsubara frequency summations.

⊲ Info. Frequently, in working with imaginary time field integrals one often needs toperform Matsubara frequency summations. At low temperatures it can be argued that

7Hint: Consider the strategic incorporation of the resolution of identity 1 =∑

q |q〉〈q| underneath thetrace.



mq,

4πe 2

q2

k,ωn

ωmωω

n

+

+k+q,

ωmq,χ( )

=

-1

ωmq,=

=

++ ...

+

Figure 5.3: The modified screened Coulomb interaction, D(ωm,q) can be viewed as thesummation of an infinite ‘diagrammatic’ series expansion in the interaction: The bareCoulomb interaction vertex is ‘dressed’ by repeated particle-hole excitations of the electrongas. The corresponding summation of the geometric series (known as a Dyson equation)is shown schematically and can be compared to Eq. (5.6).

the spacing 2π/β between neighbouring Matsubara frequencies scales to zero which legitimateschanging the summations into integrals. However, as we have seen previously, there exists analternative way of evaluating the sums which not only keeps the finite temperature content ofthe theory but is also more efficient even in the limit of zero temperature.

Referring to our ealier discussion for details, the basic idea behind the standard scheme ofevaluating frequency summations is to introduce a complex auxiliary function g(z) that hassimple poles at z = iωn. The sum

∑

n f(iωn) one wishes to compute then emerges as the sum ofresidues obtained by integrating the product gf along a suitably chosen path C in the complexplane. The choice of both the function g and the integration contour C depend on the structureof the sum in which one is interested (convergence and analyticity properties of f , etc.). In thepresent case, the screened polarisability (5.6),

g(z) =β

exp(βz) + 1, f(z) ≡ 1

z − µ+ ǫ

1

z − µ+ ǫ′,

and C is a circular contour of infinite radius in the complex plane (see Fig. 5.4). We are thereforeled to consider the integral

I ≡∫

Cdz

β

exp(βz) + 1

1

z − µ+ ǫ

1

z − µ+ ǫ′. (5.6)

Two important observations can be made without explicit computation: (i) The integral exists(the integrand decays sufficiently fast in all directions for |z| → ∞), and (ii) I = 0 (the reasonbeing that for |z| → ∞, the product fg < z−γ , where γ > 1). Thus the integral of fg over acircle (radius ∼ z) scales to zero as the radius → ∞. On the other hand, the integral along Cgives the sum over the residues of all enclosed poles. The function fg has poles at the Matsubarafrequencies iωn and poles on the real axis at z = µ− ǫ and z = µ− ǫ′. Hence,

0 =

∫

Cdz(fg)(z) = 2πi

(∑

n

Res (fg)(iωn) + Res (fg)(µ− ǫ) + Res (fg)(µ− ǫ′)

)

.

At the Matsubara frequencies, f is analytic and g has, by construction, unit residue. Thus

Res (fg)(iωn) = −f(iωn).



c

Poles at Matsubarafrequencies

z

Pole of f

Figure 5.4: Complex integration contour employed in calculating the sum (5.6).

At z = µ− ǫ, f has a simple pole, whereas g is analytic, i.e.

Res (fg)(µ− ǫ) =g(µ− ǫ)

(ǫ′ − ǫ).

Combined with the pole at z = µ− ǫ′, and making use of the identity eiβωn = 1, we obtain

Π(ωm,q) =2

L3

∑

k

nF(ξk) − nF(ξk+q)

iωm + ξk − ξk+q

(5.7)

where we have made use of the fact that, on the real axis, the auxiliary function g is proportionalto the Fermi-Dirac distribution function,

nF(ξk) =1

eβ(µ−ǫk) + 1. (5.8)

——————————————–To analyse the screened Coulomb interaction D(ωn,q) we consider Eq. (5.7) for the

density-density response function, and divide our consideration into two limits:

⊲ Static Limit (|ωn| ≪ kF |q|/m)

For frequencies small as compared to the momentum transfer (and temperaturesT ≪ µ), the response function converges to the static limit

Π(0,q)

2≃L3

∑

k

q · ∂knF(ξk)

q · ∂kξk≃∫

d3k

(2π)3∂ξknF(ξk) =

∫ ∞

0

dǫν(ǫ)∂ǫnF(ǫ− µ) ≃ −ν(µ),

where here we have made use of the fact that ξk depends only on |k|, and we havedeployed the continuum limit, 1

L3

∑

k 7→∫

d3k(2π)3

=∫∞0dǫν(ǫ) with ν(ǫ) ≡ 1/|∇kξk| =

2mk/(2π)2 the density of states. From this result, one obtains the screened Coulombinteraction

D(q) =1

q2

4π+ 2e2ν(µ)



When transformed back to real space, one obtains the effective static screenedCoulomb interaction

D(r) =e2

|r|e−|r|/λTF

where λTF = (8πe2ν(µ))1/2 defines the Thomas-Fermi screening length. Physi-cally, the bare Coulomb interaction is screened by the collective fluctuations of theelectron gas.

⊲ High Frequency Limit (|ωn| ≫ kF |q|/m)

By contrast, in the high frequency limit, the density-density response function takesthe form

Π(ωm,q) ≃ − 2

L3

∑

k

q · ∂knF(ξk)

iωm − q · ∂kξk≃ −

∫d3k

(2π)3

2

iωm

(

1 +q · kimωm

)

q · ∂knF(ξk)

by parts= −

∫d3k

(2π)3

2q2

mω2m

nF(ξk) = − 1

(2π)3

4

3πk3

F

2q2

mω2m

= − N

2L3

2q2

mω2m

= − nq2

mω2m

,

where n = N/Ld denotes the total number density. Applying the analytic continu-ation to real frequencies, iωn → ω + i0, we obtain

limmω/kF |q|→∞

D(ω,q) =4πe2

q2

[

1 − ω2p

ω2

]−1

,

where ωp = 4πe2n/m represents the Plasma frequency. At high frequenciesthe dielectric response of the system is sensitive to the plasma oscillations of thebackground electron charge. In particular, for ω = ωp, the collective excitationsbecome undamped.

⊲ Ground State Energy

Finally, from the partition function, it is possible to obtain an estimate of the groundstate energy of the interacting electron gas.

limβ→∞

Z ∼ e−βEg.s..

In the RPA approximation, performing the functional integral over the Gaussianaction in φ, we obtain

Eg.s. = Eg.s.(e = 0) − 1

2β

∑

ωn,q

lnD(ωn,q).

where Eg.s. ≡ − lnZ0/β = 3nµ/5 is the ground state energy of the free electron gas.This was the formula derived by Gell-Mann and Bruckner from which, after someextensive algebra (exercise!), one obtains the high density (rs ≪ 1) expansion

Eg.s. = n

(2.21

r2s

− 0.916

rs+ 0.622 ln rs − 0.142

)

Ryd.



This concludes our preliminary analysis of the screening properties of the weaklyinteracting electron gas. By employing the coherent state path integral, we were(implicitly) able to establish the stability of the non-interacting ground state, deter-mine the modified screened Coulomb interaction, and identify plasma oscillations.In the following section, we will use the path integral to study the weakly interactingBose gas and the phenomenon of superfluidity.

5.3 Bose–Einstein Condensation and Superfluidity

Previously, we have considered the influence of weak Coulomb interaction on the proper-ties of the electron gas. In the following, our goal will be to consider the phases realisedby a weakly interacting Bose gas. To this end, let us introduce the quantum partitionfunction Z =

∫D(ψ, ψ) e−S[ψ,ψ], where

S[ψ, ψ] =

∫

ddr

∫

dτ[

ψ(r, τ)(∂τ + H0 − µ)ψ(r, τ) +g

2(ψ(r, τ)ψ(r, τ))2

]

. (5.9)

Here ψ represents a complex field subject to the periodic boundary condition ψ(r, β) =ψ(r, 0). The functional integral Z describes the physics of a system of bosonic particles ind–dimensions subject to a repulsive contact interaction of strength g > 0. For the momentthe specific structure of the one–body operator H0 need not be specified. The mostremarkable phenomena displayed by systems of this type are Bose–Einstein condensationand superfluidity. However, contrary to a widespread belief, these two effects do notmutually depend on each other: Superfluidity can arise without condensation and vice

versa. We begin our discussion with the more elementary of the two phenomena.

5.3.1 Bose–Einstein Condensation

As may be recalled from elementary statistical mechanics, at sufficiently low temperatures,the ground state of a bosonic system can involve the condensation of a macroscopicfraction of particles into a single state. This phenomenon, predicted in a celebrated workby Einstein8 is known as Bose–Einstein condensation. To see how this phenomenon isborn out of the functional integral formalism, let us temporarily switch off the interactionand turn to the basis in which the one–particle Hamiltonian is diagonal. Expressed in thefrequency representation, the partition function of the non–interacting system is given by,

Z0 ≡ Z∣∣∣g=0

=

∫

D(ψ, ψ) exp

[

−∑

an

ψan (−iωn + ǫa − µ)ψan

]

.

8

Albert Einstein 1879–1955: 1921 Nobel Laureate in Physics “for his ser-vices to theoretical physics, and especially for his discovery of the law ofthe photoelectric effect”. His work on the low temperature behaviour ofthe bosonic quantum gas is published in A. Einstein, Quantentheorie des

einatomigen idealen Gases, Sitzungsber. Preuss. Akad. Wiss. 1925, 14(1925).


5.3. BOSE–EINSTEIN CONDENSATION AND SUPERFLUIDITY 159

Without loss of generality, we may assume that the eigenvalues ǫa ≥ 0 are positive witha ground state ǫ0 = 0.9 (In contrast to the fermionic systems discussed above, we shouldnot have in mind low energy excitations superimposed on high energy microscopic degreesof freedom. Here, everything will take place in the vicinity of the ground state of themicroscopic single–particle Hamiltonian.) Furthermore, we note that, to ensure stability,the chemical potential determining the number of particles in the system must be negativefor, otherwise, the Gaussain weight corresponding to the low–lying states ǫa < −µ wouldchange sign resulting in an ill–defined theory.

From our discussion of section 4.2.1 we recall that the number of particles in thesystem is set by the relation (kB = 1)

N(µ) = −∂F∂µ

= T∂

∂µlnZ = T

∑

na

1

iωn − ǫa + µ=∑

a

nB(ǫa) ,

where, as usual, nB(ǫ) = (eβ(ǫ−µ) − 1)−1 denotes the Bose distribution. For a givennumber of particles, this equation determines the temperature dependence of the chemicalpotential, µ(T ). As the temperature is reduced, the distribution function controllingthe population of individual states decreases. Since the number of particles must bekept constant, this scaling must be counter–balanced by a corresponding increase in thechemical potential.

T

µ

0 Tc

Figure 5.5: Schematic plot showing the variation of the chemical potential as a function oftemperature. Note that, for T < Tc, the chemical potential remains pinned at zero.

Below a certain critical temperature Tc, even the maximum value of the chemicalpotential, µ = 0, will not suffice to keep the distribution function nB(ǫa6=0) large enough to

accomodate all particles in the states of non–vanishing energy, viz.∑

a>0 nB(ǫa)|µ=0T<Tc≡

N1 < N . I.e. below the critical temperature, the chemical potential stays constant atµ = 0 (see the figure). As a result, a macroscopic number of particles N − N1 mustaccumulate in the single–particle ground state: Bose–Einstein condensation.

⊲ Exercise. For a three–dimensional free particle spectrum, ǫk = ~2k2/2m, show that the

critical temperature is set by Tc = c0~2

ma2 , where a = ρ−1/3 is the average interparticle spacing,

and c0 is a constant of order unity. Show that for temperatures T < Tc, the density of particles

in the condensate (k = 0) is given by ρ0(T ) = ρ[1 − ( TTc)3/2].

⊲ Info. Since its prediction in the early 20s, the phenomenon of Bose–Einstein condensationhas been a standard component of undergraduate texts. However, it took some seven decades

9The chemical potential µ can always be adjusted so as to meet this condition.



before the condensation of bosonic particles was directly10 observed in experiment. The reasonfor this delay is that the critical condensation temperature of particles that are comfortablyaccessible to experiment — atoms — is absurdly low.

In 1995 the groups of Cornell and Wieman at Colorado University and, soon after, Ketterleat MIT succeeded in cooling a system of rubidium atoms down to temperatures of 20 billionths(!)of a Kelvin.11 To reach these temperatures, a gas of rubidium atoms was caught in a magnetictrap, i.e. a configuration of magnetic field gradients that couple to the magnetic moments ofthe atoms so as to keep the system spatially localized (see the schematic).

Figure 5.6: Spectroscopic images of a gas of atoms at 400nK (left), 200nK (middle), and 50nK(right). The peak in the density distribution signals the onset of condensation. Courtesy ofJILA institute, University of Colorado

The gas of atoms was then brought to a temperature of O(10−5)K — still much too hotto condense — by ‘laser cooling’; crudely speaking, a technique where atoms, subjected to asuitably adjusted ray of monochromatic light, may transmit more of their kinetic energy to thephotons than they get back. To lower the temperature still further, the principle of ‘evaporativecooling’ was applied: By lowering the potential well of the trap, a fraction of the atoms, namelythose with large kinetic energy, is allowed to escape. The remaining atoms have a low kineticenergy and, therefore, a low temperature. What sounds like a simple recipe actually representsa most delicate experimental procedure. (For example, if the trap potential is lowered toostrongly, all atoms escape and there is nothing left to condense. If, on the other hand, trappingis too strong, the atoms remain too hot, etc.) However, after more than a decade of intensiveexperimental preparation, the required temperatures have been reached.

Spectroscopic images of the Bose–Einstein condensation process are shown in the figure

(courtesy of the JILA institute, University of Colorado) for three values of temperature (400

nK, 200 nK, and 50 nK from left to right). The peak in the density distribution signals the onset

of condensation. On lowering the temperature, one may observe the transition to a condensed

phase by monitoring the formation of a peak in the density distribution. The preparation of

10Here, by ‘direct’ we refer to the controlled preparation of a state of condensed massive bosonicparticles. There are numerous ‘indirect’ manifestations of condensed states, e.g. the anomalous propertiesof the Helium liquids at low temperatures, or of Cooper–pair condensates in superconductors.

11M. H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E. A. Cornell, Observation of

Bose–Einstein Condensation in a Dilute Atomic Vapor, Science 269, 198 (1995).



a Bose condensed state of matter was recognized with the award of the 2001 Nobel prize in

physics. Since 1995, research on atomic condensates has blossomed into a broad arena of re-

search. Already, it is possible to prepare complex states of Bose condensed matter such as atomic

vortices in rotating Bose–Einstein condensates, condensates in different dimensionalities, or even

an artificial crystalline state of matter. A detailed discussion of these interesting developments

is beyond the scope of the present text.

——————————————–With this background, let us now try to understand how the phenomenon of Bose–

Einstein condensation can be implemented into the functional integral representation.Evidently, the characteristics of the condensate will be described by the zero field compo-nent ψ0(τ). The problem with this zero mode is that, below the condensation transition,its action apears to be unbound: both the chemical potential and the eigenvalue are zero.This means that the action of the zero Matsubara component ψ0,0 vanishes. We will dealwith this difficulty in a pragmatic way. That is, we will not treat ψ0(τ) as an integrationvariable but rather as a time–independent Lagrange multiplier to be used to fix the num-ber of particles below the transition. More precisely, we introduce a reduced action of theform

S0[ψ0, ψ0] = −ψ0βµψ0 +∑

a6=0,n

ψan (iωn + ǫa − µ)ψan ,

where we did not yet set µ = 0 (since we still need µ as a differentiation variable). Tounderstand the rational behind this simplification one may note that

N = −∂µF0|µ=0− = T∂µ lnZ0|µ=0− = ψ0ψ0 + T∑

a6=0,n

1

iωn − ǫa= ψ0ψ0 +N1 (5.10)

determines the number of particles. According to this expression, ψ0ψ0 = N0 sets thenumber of particles in the condensate. Now, what enables us to regard ψ0 as a time–independent field? Remembering the construction of the path integral, we note thatthe introduction of time–dependent fields, or ‘time slicing’ was necessitated by the factthat the operators appearing in the Hamiltonian of a quantum theory do not, in gen-eral, commute. (Otherwise we could have decoupled the expression tr (e−β(H−µN)(a†,a)) ≃∫d(ψ, ψ)e−β(H−µN)(ψ,ψ) in terms of a single coherent state resolution, i.e. a ‘static’ con-

figuration). Reading this observation in reverse, we conclude that the dynamic content ofthe field integral represents the quantum character of a theory. (Alluding to this fact, thetemporal fluctuations of field variables are often referred to as quantum fluctuations.)Conversely, a static approximation in a field integral ψ(τ) = ψ0 = const. amounts to

replacing a quantum degree of freedom by its classical approximation.

(In order to distinguish them from quantum, fluctuations in the ‘classical’ static sector ofthe theory are called thermal fluctuations.) To justify the approximation of a0 ↔ ψ0

by a classical object, notice that, upon condensation, N0 = 〈a†0a0〉 will assume ‘macro-scopically large’ values. On the other hand, the commutator [a0, a

†0] = 1, continues to be

of O(1). It thus seems to be legitimate to neglect all commutators of the zero operatora0 in comparison with its expectation value — a classical approximation.12

12Notice the similarity of that reasoning to the arguments employed in connection with the semi–



Now, we are still left with the problem that the ψ0–integration appears to be un–defined. The way out is to remember that the partition function should extend over thosestates that contain an (average) number of N particles. That is, Eq. (5.10) has to beinterpreted as a relation that fixes the modulus ψ0ψ0 so as to adjust the appropriate valueof N . (For a more rigorous discussion of the choice of the thermodynamic variables in thepresent context, we again refer to Ref.[1].)

5.3.2 The Weakly Interacting Bose Gas

Now, with this background, let us restore the interaction focussing on a small but finitecoupling constant g. To keep the discussion concrete, we specialize to the case of a freesingle-particle system, H0 = p2/2m. (Notice that the ground state wavefunction of thissystem describes a spatially constant zero momentum state.) By adiabatic continuity weexpect that much of the picture developed above will survive generalization to non–zerointeraction strengths. In particular, the ground state, which in the case under consider-ation corresponds to a temporally and spatially constant mode ψ0, will continue to bemacroscopically occupied. Under these circumstances, the dominant contribution to theaction will again come from the classical ψ0 sector:

TS[ψ0, ψ0] = −µψ0ψ0 +g

2Ld(ψ0ψ0)

2 . (5.11)

Crucially, the stability of the action is now guaranteed by the interaction vertex, no matterhow small is g > 0 (see the schematic plot of the action in the figure). Accordingly,we will no longer treat ψ0 as a fixed parameter but rather as an ordinary intergrationvariable. Integration over all field components will produce a partition function Z(µ)that depends parametrically on the chemical potential. As usual in statistial physics, thelatter can then be employed to fix the particle number. (Notice that, vis–a–vis aspectsof thermodynamics, the interacting system appears to behave more ‘naturally’ than itsideal, non–interacting approximation. This reflects a general feature of bosonic systems;interactions ‘regularize’ a number of pathological features of the ideal gas.)

Returning to the ψ0–integration, we observe that, for low enough temperatures, theproblem is an ideal candidate for saddle–point analysis. Variation of the action withrespect to ψ0 obtains

ψ0

(

−µ+g

Ldψ0ψ0

)

= 0.

This equation is solved by any constant complex field configuration ψ0 with modulus|ψ0| =

√

µLd/g ≡ γ. Inspite of its innocent appearance, this equation reveals muchabout the nature of the system:

classical treatment of spin systems in the limit of large S (section 2.2.4). Unfortunately, the actualstate of affairs with the classical treatment of the condensate is somewhat more complex than the simpleargument above suggests. (For a good discussion, see Ref. [1].) However, the net result of a more thoroughanalysis, i.e. an integration over all dynamically fluctuating components ψ0,n6=0, shows that the treatmentof ψ0 as classical represents a legitimate approximation.



Figure 5.7: The action S[ψ0, ψ0] shown as a function of real and imaginary part of thecondensate field (part removed for clarity). The most important features of the actionare (a) the existence of a degenerate minimum determined by the set of solutions of theequation ∂|ψ0|S = 0, and (b) the large amplitude asymptotics ∼ βg|ψ0|4 stabilizing theψ0-integration.

⊲ For µ < 0 (i.e. above the condensation threshold of the non–interacting system),the equation exhibits only the trivial solution ψ0 = 0. This means that no stablecondensate amplitude exists.

⊲ Below the condensation threshold (i.e. for µ ≥ 0),13 the equation is solved by anyconfiguration with |ψ0| = γ ≡

√

µLd/g. (Notice that ψ0ψ0 ∝ Ld, reflecting themacroscopic population of the ground state.)

⊲ The equation couples only to the modulus of ψ0. I.e. the solution of the stationaryphase equation is continuously degenerate: Each configuration ψ0 = γ exp(iφ), φ ∈[0, 2π] is a solution.

For our present discussion, the last of the three aspects mentioned above is the mostimportant. It raises the question as to which of the configurations ψ0 = γ exp(iφ) is the‘right’ one?

Without loss of generality, we may chose ψ0 = γ ∈ R as a reference configuration forour theory. This choice amounts to selecting a particular minimum lying in the ‘mexicanhat’ profile of the action shown above. However, it is clear that an expansion of the actionaround that minimum will be singular: Fluctuations ψ0 → ψ0 + δψ that do not leave theazimuthally symmetric well of degenerate minima do not change the action and, therefore,have vanishing expansion coefficients. As a result, in the present situation, we will not beable to implement a simple scheme viz. ‘saddle–point plus quadratic fluctuations’. (Thereis nothing that constrains the deviations δψ to be small.) The integral over fluctuationsaround the mean–field configuration has to be undertaken in a more careful way.

⊲ Info. The mechanism encoutered here is one of spontaneous symmetry breaking.To understand the general principle, consider an action S[ψ] with a global continuous symmetry

13Due to the stabilization of the zero mode integration by the interaction constant, µ ≤ 0 is no longera strict condition.



under some transformation g (not to be confused with the afore mentioned coupling constantof the Bose gas): Specifically, the action remains invariant under a global transformation of thefields such that, ∀i ∈M : ψi → gψi, whereM is known as the “base manifold”, i.e. S[ψ] = S[gψ].The transformation is ‘continuous’ in the sense that g takes values in some manifold, typicallya group G.

Examples: The action of a Heisenberg ferromagnet is invariant under rotation of all spins si-multaneously by the same amount, Si → gSi. In this case, g ∈ G = O(3), the three–dimensionalgroup of rotations. The action of the displacement fields u describing elastic deformations of asolid (phonons) is invariant under simultaneous translation of all displacements ui → ui + a,i.e. the symmetry manifold is the the d–dimensional translation group G ≃ R

d. In the exampleabove, we encountered a U(1) symmetry under phase multiplication ψ0 → eiφψ0. This phasefreedom expresses the global gauge symmetry of quantum mechanics under transformationby a phase, a point we will discuss in more detail below.

Now, given a theory with globally G invariant action, two scenarios are conceivable: Ei-ther the ground states share the invariance properties of the action or they do not. The twoalternatives are illustrated in the figure for the example of the Bose system. For µ > 0, theaction S[ψ0, ψ0] has a single ground state at ψ0 = 0. This state is trivially symmetric underthe action of G = U(1). However, for negative µ, i.e. in the situation discussed above, thereis an entire manifold of degenerate ground states, defined through the relation |ψ0| = γ. Theseground states transform into each other under the action of the gauge group. However, none ofthem is individually invariant.

With the other examples mentioned above, the situation is similar. For symmetry groupsmore complex than the one–dimensional manifold U(1), the ground states will, in general, beinvariant under transformation by the elements of a certain subgroup H ⊆ G (that includes thetwo extremes H = 1 and H = G). For example, below the transition temperature, the groundstate of the Heisenberg magnet will be given by (domainwise) aligned configurations of spins.Assuming that the spins are oriented along the z–direction, the ground state is invariant underthe abelian subgroup H ⊂ O(3) containing all rotations around the z–axis. However, invarianceunder the full rotation group is manifestly broken. Solids represent states where the translationsymmetry is fully broken, i.e. all atoms collectively occupy a fixed pattern of spatial positionsin space, H = 1, etc.

Inspite of the undeniable existence of solids, magnets, and Bose condensates of definite phase,the notion of a ground state that does not share the full symmetry of the theory may appearparadoxical, or at least ‘unnatural’. For example, even if any particular ground state of the‘Mexican hat’ potential shown in the figure above ‘breaks’ the rotational symmetry, shouldn’tall these states enter the partition sum with equal statistical weight, such that the net outcomeof the theory is again fully symmetric?

To understand why symmetry breaking is a ‘natural’ and observable phenomenon, it isinstructive to perform a gedanken experiment: To this end, consider the partition function of aclassical14 ferromagnet,

Z = tr(

e−β(H−h·P

i Si))

,

where H is the rotationally invariant part of the energy functional and h represents a weakexternal field. (Alternatively, we can think of h as an internal field, caused by a slight struc-tural imperfection of the system.) In the limit of vanishing field strength, the theory becomes

14The same argument can be formulated for the quantum magnet.



manifestly symmetric. Symbolically,

limN→∞

limh→0

Z −→ rot. sym. ,

where the limitN → ∞ serves as a mnemonic indicating that we consider systems of macroscopicsize. However, keeping in mind the fact that the model ought to describe a physical magneticsystem, the order of limits taken above appears questionable. Since the external perturbationcouples to a macroscopic number of spins, a more natural description of an ‘almost’ symmetricsituation would be

limh→0

limN→∞

Z −→ ?

The point is that the two orders of limits lead to different results. In the latter case, for any h,the N → ∞ system is described by an explicitly symmetry broken action. No matter how smallthe magnetic field, the energetic cost to rotate N → ∞ spins against the field is too high, i.e. theground state |S〉 below the transition temperature will be uniquely aligned, Si ‖ h. When wethen send h → 0 in a subsequent step, that particular state will remain the observable referencestate of the system. Although, formally, a spontaneous thermal fluctuation rotating all spinsby the same amount |S〉 → |gS〉 would not cost energy, that flucutation can be excluded byentropic reasoning.15 (By analogy, one rarely observes kettles crashing into the kichen wall as aconsequence of a concerted thermal fluctuation of the water molecules!)

However, the appearance of non–trivial ground states is just one manifestation of sponta-neous symmetry breaking. Equally important, residual fluctuations around the ground statelead to the formation of soft modes (sometimes known as massless modes), i.e. field config-urations φq whose action S[φ] vanishes in the limit of long wavelengths q → 0. Specifically, thesoft modes formed on top of a symmetry broken ground state are called Goldstone modes.As a rule, the presence of soft modes in a continuum theory has important phenomenologicalconsequences. To understand this point, notice that the general structure of a soft mode actionis given by

S[φ] =∑

q,i

φq

[ci1|qi| + ci2q

2i

]φ−q + O(φ4, q3) , (5.12)

where ci1,2 are coefficients. The absence of a constant contribution to the action (i.e. a contri-

bution that does not vanish in the limit q → 0) signals the existence of long–ranged power–law

correlations in the system. As we will see shortly, the vanishing of the action in the long wave-

length limit q → 0 further implies that the contribution of the soft modes dominates practically

all observable properties of the system.

——————————————–

5.3.3 Superfluidity

As we have seen, the theory of the weakly interacting superfluid to be discussed belowwas originally conceived by Bogoliubov, then in the language of second quantisation.16 Inthe following, we will reformulate the theory in the language of the field integral starting

15Note that this (overly) simple picture in fact breaks down in dimensions d ≤ 2, cf. our discussion ofthe thermal fluctuations of the ferromagnet in chapter 2.

16Bogoliubov, N. N., On the Theory of Superfluidity, J. Phys. (USSR) 11, 23 (1947), (reprinted in D.Pines, The Many–body Problem, W. A. Benjamin, New York, 1961).



with the action of the weakly interacting Bose gas (5.9). Focussing on temperaturesbelow Tc, (µ > 0), let us expand the theory around the particular mean–field groundstate ψ0 = ψ0 = (µLd/g)1/2 = γ. (Of course, any other state lying in the ‘Mexicanhat’ minimum of the action would be just as good.) Notice that the quantum groundstate corresponding to the configuration ψ0 is unconventional in the sense that it cannothave a definite particle number. The reason is that, according to the correspondenceψ ↔ a between coherent states and operators, respectively, a non–vanishing functionalexpectation value of ψ0 is equivalent to a non–vanishing quantum expectation value 〈a0〉.Assuming that, at low temperatures, the thermal average 〈. . .〉 will project onto theground state |Ω〉, we conclude that 〈Ω|a0|Ω〉 6= 0, i.e. |Ω〉 cannot be a state with a definitenumber of particles.17

The symmetry group U(1) acts on this state by multiplication, ψ0 → eiφψ0 and ψ0 →e−iφψ0. Knowing that the action of a weakly modulated field φ(r, τ) will be massless, letus introduce coordinates

Re ψ

Im ψ

φ

δρ

S = extr.

Figure 5.8: Schematic diagram showing the coordinates of the massive (δρ) and massless (δφ)fluctuations.

ψ(r, τ) = [ρ0 + δρ(r, τ)]1/2eiφ(r,τ),

ψ(r, τ) = [ρ0 + δρ(r, τ)]1/2e−iφ(r,τ),

where ρ0 ≡ γ2 = ψ0ψ0 is the condensate density. Evidently, the variable δρ parameterizesdeviations of the field ψ(r, τ) from the extremum. These excursions are energeticallycostly, i.e. δρ will turn out to be a massive mode. Also notice that the transformation ofcoordinates (ψ, ψ) → (δρ, φ), viewed as a change of integration variables, has a Jacobianof unity.

⊲ Info. As we are dealing with a (functional) integral, there is a lot of freedom as to the

choice of integration parameters. (I.e. in contrast to the operator formulation, there is no a

priori constraint for a transform to be ‘canonical’.) However, physically meaningful changes of

representation will usually be canonical transformations, in the sense that the corresponding

17However, as usual with grand canonical descriptions, in the thermodynamic limit, the relative uncer-tainty in the number of particles, (〈N2〉 − 〈N〉2)/〈N〉2 will become vanishingly small.



transformations of operators would conserve the commutation relations. Indeed, one may con-

firm that the operator transformation a(r) ≡ ρ(r)1/2 eiφ(r), a†(r) ≡ e−iφ(r) ρ(r)1/2, fulfills this

criterion (exercise).

——————————————–We next substitute the density–phase relation into the action and expand to second

order around the reference mean–field. Ignoring gradients acting on the density field (incomparison with the ‘potential’ cost of these fluctuations), we obtain

S[δρ, φ] ≈∫

ddr

∫

dτ

[

iρ∂τφ+ρ0

2m(∂φ)2 +

gδρ2

2

]

. (5.13)

The first term of the action has the canonical structure ‘momentum × ∂τ (coordinate)’indicative of a canonically conjugate pair. The second term measures the energy cost ofspatially varying phase flucutations. Notice that fluctuations with φ(r, τ) = const. donot incur an energy cost — φ is a Goldstone mode. Finally, the third term records theenergy cost of massive fluctuations from the potential minimum. Eq. (5.13) represents theHamiltonian version of the action, i.e. an action comprising coordinates φ and momentaδρ. Gaussian integration over the field δρ leads us to the Lagrangian form of the action(exercise):

S[φ] ≈ 1

2

∫

ddr

∫

dτ

[1

g(∂τφ)2 +

ρ0

m(∂φ)2

]

. (5.14)

Comparison with Eq. (1.2) identifies this action as the familiar the d–dimensional oscil-lator. Drawing on the results of chapter 1 (see, e.g., Eq. (1.15)), we find that the en-ergy ωk carried by elementary excitations of the system scales linearly with momentum,ωk = |k|ρ0/mg.

Let us now discuss the physical ramifications of these results. The actions (5.13) and(5.14) describe the phenomenon of superfluidity. To make the connection between thefundamental degree of freedom of a superfluid system, the supercurrent, and the phasefield explicit, let us consider the quantum mechanical current operator

j(r, τ) =i

2m

[(∇a†(r, τ))a(r, τ) − a†(r, τ)∇a(r, τ)

] fun. int−→

→ i

2m

[(∇ψ(r, τ))ψ(r, τ) − ψ(r, τ)∇ψ(r, τ)

]≈ ρ0

m∇φ(r, τ), (5.15)

where the arrow indicates the functional integral correspondence of the operator descrip-tion and we have neglected all contributions arising from spatial fluctuations of the densityprofile. (Indeed, these — massive — fluctuations describe the ‘normal’ contribution tothe current flow.)

⊲ Info. Superfluidity is one of the most counterintuitive and fascinating phenomena

displayed by condensed matter systems. Experimentally, the most straightforward access to

superfluid states of matter is provided by the Helium liquids. Representative of many other

effects displayed by superfluid states of Helium, we mention the capability of thin films to flow

up the walls of a vessel (if the reward is that on the outer side of the container a low lying basin

can be reached — the fountain experiment) or to effortlessly propagate through porous media



that no normal fluid may penetrate.

——————————————–The gradient of the phase variable is therefore a measure of the (super)current flow

in the system. The behaviour of that degree of freedom can be understood by inspectionof the stationary phase equations — alias, the Hamilton or Lagrange equations of motion— associated with the actions (5.13) or (5.14). Turning to the Hamiltonian formulation,one obtains (exercise)

i∂τφ = −gδρ, i∂τδρ =ρ0

m∂2φ = ∇ · j.

The second of these equations represents (the Euclidean time version) of a continuityequation. A current flow with non–vanishing divergence is accompanied by dynamicaldistortions in the density profile. The first equation tells us that the system adjusts tospatial fluctuations of the density by a dynamical phase fluctuation. The most remarkablefeature of these equations is that they possess steady state solutions with non–vanishingcurrent flow. Setting ∂τφ = ∂τδρ = 0, we obtain the conditions δρ = 0 and ∇ · j = 0, i.e.below the condensation temperature, a configuration with a uniform density profile cansupport a steady state divergenceless (super)current. Notice that a ‘mass term’ in theφ action would spoil this property, i.e. within our present approach, the phenomenon ofsupercurrent flow is intimately linked to the Goldstone mode character of the φ field.

⊲ Exercise. Add a ficticious mass term to the φ–action (viz. δL = mφ2) and explore its

consequences. How do the features discussed above present themselves in the Lagrange picture?

It is very instructive to interpret the phenomenology of supercurrent flow from a dif-ferent, more microscopic perspective. Steady state current flow in normal environmentsis prevented by the mechanism of energy dissipation, i.e. particles constituting thecurrent flow scatter off imperfections inside the system thereby converting part of theirenergy into the creation of elementary excitations. (Macroscopically, the conversion ofkinetic energy into the creation of excitations manifests itself as heat production.) Ap-parently, this mechanism is inactivated in superfluid states of matter, i.e. the current flowis dissipationless.

How can the dissipative loss of energy be avoided. Trivially, no energy can be ex-changed if there are no elementary excitations to create. In reality, this means that theexcitations of the system are energetically high–lying such that the kinetic energy storedin the current–carrying particles is insufficient to create them. But this is not the situ-tation that we encounter in the superfluid! As we saw above, there is no energy gapseparating the quasi–particle excitations of the system from the ground state. Rather,the dispersion ω(k) vanishes linearly as k → 0. However, there is an ingenuous argu-ment due to Landau showing that a linear excitation spectrum indeed suffices to stabilizedissipationless transport:

⊲ Info. Consider the flow of some fluid through a pipe (cf. Fig. 5.9 top left). To be concrete,let us assume that the flow occurs at a uniform velocity V. Taking the mass (of a certain portionof the fluid) to be M , the current carries a total kinetic energy E1 = MV2/2. Now, suppose weview the situation from the point of view of the fluid, i.e. we perform a Galileian transformation



into its own rest frame (see Fig. 5.9, top right). From the perspective of the fluid, the walls ofthe pipe appear as though they were moving with velocity −V. Now, suppose that frictionalforces between fluid and the wall lead to the creation of an elementary excitation of momentump and energy ǫ(p), i.e. the fluid is no longer at rest but carries kinetic energy. After a Galileiantransformation back to the laboratory frame, one finds that the energy of the fluid after thecreation of the excitation is given by (excercise)

E2 =MV2

2+ p ·V + ǫ(p) .

Now, since all of the energy needed to manufacture the excitation must have been provided by

V

–V

p

Figure 5.9: Top left: Flow of a fluid through a rough pipe. Top right: The same viewed fromthe rest frame of the fluid. Bottom left: Dissipative creation of a (quasi–particle) excitation.Bottom right: The same viewed from the laboratory frame.

the liquid itself, energy conservation requires that E1 = E2, or −p · V = ǫ(p). Since p · V >

−|p||V|, this condition can only be met if |p||V| > ǫ(p). While systems with a ‘normal’ gapless

dispersion, ǫ(p) ∼ p2 are compatible with this energy–balance relation (i.e. no matter how

small |V|, quasi–particles of low momentum can always be excited), both gapped dispersions

ǫ(p)p→0−→ const. and linear dispersions are incompatible if V becomes smaller than a certain

critical velocity V∗. Specifically for a linear dispersion ǫ(p) = v|p|, the critical velocity is

given by V∗ = v. For currents slower than that, the flow is necessarily dissipationless.

——————————————–Let us conclude our preliminary discussion of the weakly interacting Bose gas with a

very important remark. Superficially, Eqs. (5.13) and (5.14) suggest that we have managedto describe the long–range behaviour of the condensed matter system in terms of a freeGaussian theory. However, one must recall that φ is a phase field, defined only modulo2π. (In Eqs. (5.13) and (5.14) this condition is understood implicitly. At this point, itis perhaps worth reiterating that when dealing with Goldstone modes it is important tokeep the underlying geometry in mind and not too tightly focus on a specific coordinaterepresentation.) The fact that φ is defined only up to integer multiples of 2π manifestsitself in the formation of the most interesting excitations of the superfluid; vortices, i.e.phase configurations φ(r, τ) that change by a multiple of 2π as one moves around a certainreference coordinate, the vortex centre. Existing in parallel with harmonic phonon–like



excitations discussed above, these excitations lead to a wealth of observable phenomena.However, leaving such effects aside, let us turn to the discussion of another prominentsuperfluid, the condensate of Cooper pairs, more generally known as the superconductor.

5.4 Superconductivity

The electrical resistivity of many metals and alloys drops suddenly to zero when thespecimen is cooled to a sufficiently low temperature. This phenomenon, which goes bythe name superconductivity, was first observed by Kammerlingh Onnes in Leiden in1911, three years after he first liquefied Helium.18 Perhaps more striking, a superconductorcooled below its transition temperature in a magnetic field expels all magnetic flux fromits interior. This phenomenon of perfect diamagnetism is known as the Meissner effectand is characteristic of superconductivity.

Figure 5.10: The levitation of a magnet above a (high temperature) superconductor dueto the expulsion of magnetic flux.

The superconducting state is an ordered state of the conduction electrons of the metal.The nature and origin of the ordering was explained by Bardeen, Cooper and Schrieffer.19

At low temperatures, the presence of an attractive pairwise interaction can induce aninstability of the electron gas towards the formation of bound pairs of time-reversed states

18

Kammerlingh Onnes 1853-1926(left, photographed with van derWaals): 1913 Nobel Laureatein Physics for his investigationson the properties of matter atlow temperatures which led, in-

ter alia to the production of liq-uid helium.

19J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 106, 162 (1957); 108, 1175 (1957).

John Bardeen 1908-1991 (left), Leon N.Cooper 1930- (centre), and J. RobertSchrieffer 1931- (right): 1972 Nobel Lau-reate in Physics for their jointly developedtheory of superconductivity. (Bardeenwas also recipient of the 1956 Nobel Lau-reate in Physics for his research on semi-conductors and discovery of the transistoreffect.


5.4. SUPERCONDUCTIVITY 171

k ↑ and −k ↓ in the vicinity of the Fermi surface. We have already seen (in section 2.2.6)how the exchange of lattice vibrations or phonons can induce an attractive interactionof electrons within the Debye frequency ωD of the Fermi surface. Being made up of twoelectrons, these composite objects, known as Cooper pairs behave as bosons. At lowtemperatures, these bosonic degrees of freedom form a condensate which is responsiblefor the remarkable properties of superconductors such as perfect diamagnetism.

To explore the phenomenology of the Cooper instability of the electron gas, we willadopt a simplified model known as the pairing or reduced Hamiltonian

H =∑

kσ

ǫknkσ − g∑

kk′

c†k↑c†−k↓c−k′↓ck′↑

Although, strictly speaking, a realistic model of attraction would involve a more compli-cated momentum-dependent interaction such as the one obtained from the considerationof the electron-phonon interaction in section 2.2.6, the simple pairing interaction cap-tures the essential physics. More importantly, to simplify our discussion, we will take theelectrons to be otherwise non-interacting. In fact, the presence of a repulsive Coulombinteraction of the electrons plays a crucial role in the controlling properties of the super-conductor, a point to which we will return later.

k’

-k

k

-k’

µ

D2ω

Figure 5.11: Schematic diagram showing the Fermi surface of the electron gas. Theattractive interaction mediated by the exchange of phonons allows electrons within theDebye frequency ωD of the Fermi surface to pair.

Before turning to the field theoretic formulation, we will begin by investigating amean-field theory of the BCS transition from the stand point of the second quantisation.

5.4.1 Mean-Field Theory of Superconductivity

In its present form, the Hamiltonian explicitly involves a two-body interaction of theelectrons. As such, it seems infeasible to develop an exact many-body treatment of the



Hamiltonian. Instead we will seek an approximation which renders the Hamiltonian bi-linear in the electron operators and, therefore, tractable. As usual in physics, our methodrelies on the expected structure of the ground state wavefunction. In particular, antici-pating that electrons in time-reversed states pair, let us suppose that

∆ = g∑

k

〈g.s.|c−k↓ck↑|g.s.〉, ∆∗ = g∑

k

〈g.s.|c†k↑c†−k↓|g.s.〉

acquires a non-zero expectation value in the ground state. Here ∆ represents an orderparameter becoming non-zero in the condensed phase and therefore signalling the tran-sition to the superconducting state. At first sight, a non-zero expectation value ∆ looksstrange: such a result would imply that the ground state wavefunction of the supercon-ducting condensate is not an eigenstate of particle number (while one can see that theHamiltonian commutes with N). However, later, we will see that in the grand canoni-cal ensemble the ground state wavefunction is a superposition of states involving manyparticles but strongly peaked around the thermodynamic density N/Ld.

To develop the mean-field approximation, let us set

g∑

k

c−k↓ck↑ = ∆+

small︷︸︸︷(

g∑

k

c−k↓ck↑ − ∆

)

and keep only terms which depend up to quadratic order in the electron operators. Addingthe chemical potential, the ‘mean-field’ Hamiltonian takes the form

H − µN ≃∑

k

[∑

σ

ξk︷︸︸︷

(ǫk − µ) c†kσckσ −(

∆∗c−k↓ck↑ + ∆c†k↑c†−k↓

) ]

+|∆|2g

known as the Bogoluibov or Gor’kov Hamiltonian. In this simplified form, it isinteresting to note that the Hamiltonian does not now conserve particle number. Instead,pairs of particles are born and annihilated out of the vacuum.

To bring the mean-field Hamiltonian to a diagonal form, it is convenient to recast itin a Nambu spinor representation defining

Ψ†k = ( c†k↑ c−k↓ ) , Ψk =

(ck↑c†−k↓

)

after which the Hamiltonian takes the form (exercise: recall the fermionic anticommuta-tion relations of the electron operators)

H − µN =∑

k

[

Ψ†k

(ξk −∆

−∆∗ −ξk

)

Ψk + ξk

]

+|∆|2g

Now, being bilinear in the electron operators, the mean-field Hamiltonian can be broughtto a diagonal form by employing the unitary transformation

χ†k ≡

(α†

k↑α−k↓

)

=

(cos θk sin θksin θk − cos θk

)(c†k↑c−k↓

)

≡ Uψ†k,



(under which the anticommutation relations of the new electron operators αkσ are main-tained: exercise). Note that the notation is purely symbolic: α†

k↑ involves a superpo-

sition of c†k↑ and c−k↓. Choosing ∆ to be real,20 and setting tan(2θk) = −∆/ξk, i.e.

cos(2θk) = ξk/λk, sin(2θk) = −∆/λk, where λk = (∆2 + ξ2k)

1/2, the transformed Hamilto-nian takes the form (exercise)

H − µN =∑

k

λk

(

α†k↑αk↑ − α−k↓α

†−k↓

)

+∑

k

ξk +∆2

g

=∑

kσ

λkα†kσαkσ +

∑

k

(ξk − λk) +∆2

g

This result shows that the elementary excitations or quasi-particle states, known as “Bo-goluibons”, created by α†

kσ, have a minimum energy ∆, the energy gap.To determine the ground state wavefunction one simply has to identify the state which

is annihilated by all the quasi-particle annihilation operators αkσ. This condition is metuniquely by the state

|g.s.〉 =∏

k

αk↑α−k↓|Ω〉 ∝∏

k

(

cos θk − sin θkc†k↑c

†−k↓

)

|Ω〉

where |Ω〉 represents the vacuum state, and

2 sin2 θk = 1 − ξkλk

Physically, in the limit ∆ → 0, sin2 θk → θ(µ− ǫk), and the ground state collapses to thefilled Fermi sea with chemical potential µ. As ∆ becomes non-zero, states in the vicinityof the Fermi surface rearrange themselves into a bound state condensate and lower theirenergy.

2θk

ζk

Sin

0

2θkCos

Figure 5.12: Schematic diagram showing the variation of the occupancy of the momemtumbasis states in the ground state; sin2 θk represents the occupancy of the k states. Notethat the wavefunction of the ground state condensate involves the occupation of basisstates with momentum in excess of pF =

√2mµ.

An estimate of the corresponding ground state energy obtains

Eg.s. ≡ 〈g.s.|H − µN |g.s.〉 =∑

k

(ξk − λk) +∆2

g,

20One may show that the phase of ∆ may be chosen arbitrarily, a fact to which we will return later.



a result which always lowers the energy below the non-interacting g = 0 theory (exercise).Finally, to determine the scale of the order parameter we have to determine self-

consistently the order parameter, viz.,

∆ = g∑

k

〈g.s.|c−k↓ck↑|g.s.〉 = −g∑

k

sin θk cos θk

=g

2

∑

k

∆

(∆2 + ξ2k)

1/2≃ gL3∆

2ν(µ)

∫ ωD

−ωD

dξ

(∆2 + ξ2)1/2= gL3∆ν(µ) sinh−1(ωD/∆)

where∑

k →∫dξ ν(ξ) and ν(ξ) denotes the density of states. (It is left as an exercise

to show that a minimisation of the g.s. energy obtains the same self-consistent equationfor the order parameter.) Here we have assumed that the pairing interaction g extendsover an energy scale set by ωD. Physically, for pairing mechanisms which derive from theexchange of phonons, this energy scale is set by the corresponding Debye frequency, themaximum energy phonon. Finally rearranging this equation, one obtains

∆ =ωD

sinh(1/gL3ν(µ))≃ 2ωD exp

[

− 1

gL3ν(µ)

]

This completes our formal investigation of the BCS transition from the mean-fieldHamiltonian. How does the same transition emerge from the corresponding field theory?In the following, we will develop a theory of superconductivity from the coherent statepath integral for the quantum partition function.

5.4.2 Superconductivity from the Path Integral

To investigate the BCS transition within the framework of the coherent state path integral,it is convenient to abandon the long-ranged pairing Hamiltonian considered above andintroduce a space-local attractive interaction contained within the BCS Hamiltonian,

HBCS =

∫

ddr

[∑

σ

c†σ(r)p2

2mcσ(r) − gc†↑(r)c

†↓(r)c↓(r)c↑(r)

]

.

Expressed in the form of the coherent state path integral, the corresponding quantumpartition function takes the form

Z =

∫

a.p.b.c.

D(ψ, ψ) exp

−∫ β

0

dτ

∫

ddr

[∑

σ

ψσ

(

∂τ +p2

2m− µ

)

ψσ − gψ↑ψ↓ψ↓ψ↑

]

,

where ψ(r, τ) represent anticommuting or Grassmann fields and the nemonic a.p.b.c.denotes anti-periodic boundary conditions. As usual the quartic interaction of the fieldsprevents the partition function from being evaluated explicitly. Moreover, anticipating theexistence of a transition of the electron gas to a condensed phase in which electrons in thevicinity of the Fermi surface are paired, we can expect that a perturbative expansion inthe coupling constant g will be inadequate. Motivated by the mean-field theory discussed



above, we will instead introduce a bosonic field ∆ to decouple the interaction and whichwill have the physical significance of the complex order parameter.

The decoupling is arranged using a Hubbard-Stratonovich transformation

egR

dτR

ddrψ↑ψ↓ψ↓ψ↑ =

∫

D(∆∗,∆) exp

−∫

dτ

∫

ddr

[1

g|∆|2 −

(∆∗ψ↓ψ↑ + ∆ψ↑ψ↓

)]

where ∆(r, τ) represents a dynamically fluctuating bosonic or complex field with a sym-metry that reflects that of the bilinear ψ↓ψ↑. Taking ∆ to be homogeneous in space andtime, the quantum Hamiltonian corresponding to the action coincides with that of themean-field Hamiltonian considered in the previous section. Motivated by that analysis,we turn to the Nambu spinor representation

Ψ = ( ψ↑ ψ↓ ) , Ψ =

(ψ↑ψ↓

)

.

wherein the quantum partition function assumes the form

Z =

∫

D(ψ, ψ)

∫

D(∆∗,∆) exp

−∫

dτ

∫

ddr

[1

g|∆|2 + ΨG−1Ψ

]

,

G−1 =

([G

(p)0 ]−1 −∆

−∆∗ [G(h)0 ]−1

)

,

where [G(p)0 ]−1 = ∂τ +(ǫp − µ) and [G

(h)0 ]−1 = ∂τ − (ǫp − µ) represents the non-interacting

Green function of the particle and hole respectively, and G is known as the Gor’kovGreen function.

Being Gaussian in the fermionic fields, the functional integral over the Grassmannfields can be performed straightforwardly, and yields the formal expression

Z =

∫

D(∆∗,∆) exp

[

−∫

dτ

∫

ddr1

g|∆|2 + ln det G−1

]

,

where we have written detG−1 = exp[ln detG−1]. By introducing a Hubbard-Stratonovichdecoupling of the local interaction, we have succeeded in expressing the quantum partitionfunction as a path integral over an auxiliary bosonic field ∆. Further progress is possibleonly within some approximation. Empirically, we know that the superconducting tran-sition is second order, i.e. the order parameter ∆ develops a non-zero expectation valuebelow a critical temperature Tc growing continuously from zero. At temperatures T ≪ Tc,spatial and temporal fluctuations around the expectation value ∆ can be treated as small.In this limit, the action can be treated within a mean-field approximation where the par-tition function is dominated by the saddle-point configuration of ∆. The saddle-pointanalysis, which leads to a self-consistent equation for ∆ known as the Gap Equation,is left as an exercise in Problem Set 4. Instead we will focus on temperatures T ∼ Tc inwhich an effective action for ∆ can be obtained.

In the vicinity of the transition temperature Tc, the order parameter ∆ is expected tobe small. We are therefore at liberty to look for a perturbative expansion in powers of ∆.Setting

G−1 = G−10

[

1 − G0

(0 ∆

∆∗ 0

)]

,



where, by definition, G0 ≡ G(∆ = 0), and expanding the action to second order in ∆,21

the quantum partition function takes the form (exercise)

Z = Z0

∫

D(∆∗,∆)e−S, S =∑

ωn,q

[1

g+ Π(ωn,q)

]

|∆ωn,q|2 +O(∆4), (5.16)

with (see Fig. 5.13a)

Π(ωm,q) =1

βLd

∑

ωn,k

G(p)0 (ωn,k)G

(h)0 (ωn + ωm,k + q).

n ωm-k-q, −ω

ωnk,-q, −ωm -q, −ω

-

m

(b)(a)

Figure 5.13: Diagrammatic representation of (a) the response function Π(ωm,q), and (b)the quartic vertex.

An instability of the electron gas towards the formation of a paired condensate issignalled by the appearance of a non-zero expectation value of the anomalous average〈ψ↑ψ↓〉 (i.e. ∆ 6= 0). Moreover, intuitively, we would expect the action to be minimisedby a spatially and temporally uniform field configuration of ∆. Applying this Ansatz, weare led to consider a gradient expansion of the action in powers of ∆.

Neglecting temporal fluctuations altogether, a gradient expansion in powers of q ob-tains

Π(ωm,q) = Π(0, 0) +1

2q2

K > 0︷︸︸︷

∂|q|Π(0, 0) +O(iωm,q4),

where, by symmetry, we have made use of the fact that Π = Π(q2). Substituting thisexpansion into Eq. (5.16), and returning to the real space representation, we obtain thestatic effective action

S[∆] = β

∫

ddr

[t

2|∆|2 +

K

2|∂∆|2 + u|∆|4 + · · ·

]

(5.17)

where t/2 = g + Π(0, 0) represents an effective ‘chemical potential’ for ∆, and u > 0represents the constant coefficient associated with the quartic vertex (see Fig. 5.13b).(The calculation of the coefficients K and u is left as an exercise in Matsubara frequencysummations! However, in the following, we will not need to know their form explicitly.)

21Here we have made use of the identity ln det G−1 = tr ln G−1 to form the expansion

tr ln G−1 = tr ln G−1

0+

∞∑

n=1

1

ntr

[

G0

(0 ∆

∆∗ 0

)]n



Re

S

Im

[∆]

[∆][∆]

|∆|

Tc

Figure 5.14: Dependence of the effective action S0 on the complex mean-field order pa-rameter ∆. At t > 0, fluctuations of ∆ are energetically unfavourable whilst for t < 0, theaction becomes unstable towards the formation of a non-zero expectation value of ∆. Alsoshown is the expectation value ∆ of the order parameter as a function of temperature.

5.4.3 Gap Equation

Applying a mean-field approximation, i.e. focusing on the spatially homogeneouscomponent ∆0, the effective action takes the form

S[∆0]

βLd=t

2|∆0|2 + u|∆0|4. (5.18)

In particular, when the effective chemical potential t becomes negative, the expectationvalue of 〈∆0〉 becomes finite (see Fig. 5.14). Summing over momenta, and making use ofthe identity 1

Ld

∑

k =∫∞−∞ dξν(ξ) where, as usual, ν(ξ) denotes the density of states at

energy ξ, one obtains

Π(0, 0) = − 1

βLd

∑

ωn,k

1

ω2n + (ǫk − µ)2

≃ − 1

β

∑

ωn

∫ ∞

−∞dξν(ξ + µ)

ω2n + ξ2

≃ −∑

ωn

πν(µ)

βωn.

Recalling that the attractive interaction of the electrons was mediated by the exchangeof phonons, we note that the Matsubara summation should be cut-off at the scale of theDebye frequency. Setting ωD = (2nmax + 1)π/β, we obtain

Π(0, 0) ≃ −ν(µ)

nmax∑

n=−nmax

1

2n + 1≃ −2ν(µ)

∫ nmax

0

dn

2n+ 1≃ −ν(µ) ln(βωD).

Equating this result with 1/g, we deduce that the electron gas becomes unstable towardsthe formation of a pair condensate when

T < Tc ≡ ωD exp

[

− 1

ν(µ)g

]



Substituting this result into the expression for t we find that, in the vicinity of Tc,

t = 2ν(µ) ln

(T

Tc

)

≃ 2ν(µ)

(T − TcTc

)

,

i.e. one may think of the parameter t as a ‘reduced temperature’.

Finally, taking the partition function to be dominated by the minimum of the mean-field action (5.18) Z ∼ exp[−S0(|∆|)] (i.e. applying a saddle-point approximation), wefind a spontaneous breaking of the U(1) symmetry of the complex order parameter — i.e.∆ has a magnitude (see Fig. 5.14b),

|∆| =

0 t > 0,√

t/4u t < 0.

with arbitrary but constant phase. This situation is reminiscent of the Heisenberg fer-romagnet: the complex order parameter is isomorphic to a ‘two-component’ spin wherethe phase degree of freedom is mirrored in the orientation of the moment. Accordingto the Mermin-Wagner theorem, the breaking of the continuous U(1) symmetry shouldbe accompanied by the appearance of massless Goldstone modes. The stability of themean-field solution towards fluctuations is governed by the effective action S[∆] (5.17).

To summarise, the quantum partition function of an electron gas subject to a localattractive pairing interaction has been cast in the form of a quantum field theory involvinga complex scalar field ∆(r, τ) whose expectation value is connected to the anomalousaverage 〈ψ↑ψ↓〉. A gradient expansion of the effective action in powers ∆ reveals aninstability of the electron gas towards the formation of a spatially and temporally uniformpair condensate. Fluctuations of the order parameter ∆ around its mean-field expectationvalue are described by a low energy effective action (5.17).

In fact, interpreted as an effective Free energy, this result might have been guessedon purely phenomenological grounds: indeed, identifying the anomalous average as anappropriate order parameter, ∆ Eq. (5.17) is consistent with a gradient expansion ofthe Free energy (in powers of ∆) compatible with symmetry and the observed temperaturedependences. The phenomenology of superconductivity (expressed by the free energy of(5.17) and known as the Ginzburg-Landau Theory) anticipated the microscopic theoryof BCS.

5.4.4 †Superconductivity: Anderson-Higgs Mechanism

⊲ Info. So far, our analysis of the quantum partition function associated with the BCS Hamil-tonian is incomplete. Indeed, no reference has yet been made to the characteristic or definingproperties: superconductivity, and perfect diamagnetism. To discover such phenomena withinour theory we have to generalise our approach to accommodate an electromagnetic field. At themicroscopic level, we can do so by incorporating into the BCS Hamiltonian the canonical mo-mentum p → p−eA/c, and introducing the classical action associated with the electromagneticfield: L = −FµνFµν/4, where Fµν = ∂µAν − ∂νAµ represents the electromagnetic field tensor.Leaving the formal derivation as an exercise, we can, in the spirit of Ginzburg-Landau theory,



guess the appropriate form of the resulting (time-independent) action:

S = β

∫

ddr

[t

2|∆|2 +

K

2|(∂ + i2eA/c)∆|2 + u|∆|4 +

1

2(∂ ×A)2

]

(5.19)

This result can be inferred from gauge invariance of the order parameter under the local U(1)transformation ψ → eieφ(r)/cψ, ∆(r) → e2ieφ(r)/c∆(r) — see below — (or indeed obtained fromfirst principles by generalising the derivation presented above). The apparent ‘doubling of theelectric charge’ can be interpreted as reflecting the effective charge of a Cooper pair.

Gauge Invariance: In the presence of the electromagnetic field, the quantum partitionfunction Z =

∫DA

∫D[∆,∆∗]e−S exhibits a gauge invariance under the transformation (exer-

cise)

A 7→ A′ = A− ∂φ, ∆ 7→ ∆′ = e2ieφ/c∆.

That is, under such a transformation, the action remains invariant. Therefore, by gauge fixing∂φ = A, the phase of the order parameter can be eliminated from the action and the effectiveaction takes the form

S = β

∫

ddr

[t

2|∆|2 +

K

2(∂|∆|)2 − m2

ν

2A2 + u|∆|4 +

1

2(∂ ×A)2

]

,

where m2ν = 4e2K|∆|2/c2. As a result, we find that the massless phase degrees of freedom

φ have disappeared from the action! They have been subsumed into the longitudinal modeof the vector field A, which has itself become massive. This is an example of the celebratedAnderson-Higgs mechanism: below the transition temperature, the Goldstone bosons (inthis case φ) and the gauge field (in this case the electromagnetic field — the photon) conspireto create massive excitations, and the massless excitations are unobservable.

It is instructive to interpret this result from the saddle-point or mean-field equations ofthe motion of the order parameter and vector potential. Minimising the effective action (5.19)with respect to spatial variations of ∆ and A, one obtains the Gross-Pitaevskii equations

(exercise)

[

−K (∂ + 2ieA/c)2 + t+ 4u|∆|2]

∆ = 0

∂ × (∂ × A) ≡ j −m2νA, j = 2i

e

cK(∆∂∆ − ∆∂∆).

Substituting the order parameter by its homogeneous mean field value and differentiatingthe second equation, one finds that ∂ × (∂ × B) = −m2

νB, from which it follows that

(∂2 −m2

ν

)B = 0,

where B = ∂ × A. This result, known as the first London Equation, admits B = 0 as theonly constant spatially uniform solution. In a uniform superconductor the magnetic field is zero— the Meissner effect. At the edge of the superconductor, this equation can be integrated togive B ∼ e−mνx showing the field to penetrate a distance m−1

ν — the Penetration depth —into the sample.

That these equations imply superconductivity can be inferred from the time derivative ofthe current ∂tj = m2

νE, where E denotes an external electric field (the second London equation).If a uniform field is applied for a time t0, a current m2

νEt0 builds up. This current remains even



if the electric field is subsequently switched off. This contrasts with conventional conductorswhere there is a relaxation of the current.

Superconductivity is destroyed by a sufficiently strong magnetic field. An estimate of thecritical field can be made by studying the Gross-Pitaevskii equation

−K (∂ + 2ieA/c)2 ∆(r) = t∆(r)

This equation is formally equivalent to a Schrodinger equation describing a particle of charge2e and mass m = 1/2K in a uniform magnetic field. The lowest Landau level is defined by thecondition t/K = 2eBc/c. This defines the highest field where superconductivity can occur.

Finally, we remark that, to expel a magnetic field from a sample we require an energy

of B2/2 per unit volume to resist the magnetic pressure. This must be compensated by the

condensation energy S/βLd. If the threshold field is smaller than the critical field Bc (Type

II) magnetic field penetrates the sample in the form of flux tubes. At low temperatures the

latter arrange themselves in a hexagonal configuration known as an Abrikosov vortex lattice.

Superconductors where the situation is opposite are known as Type I.

——————————————–

Figure 5.15: Bitter patter of an Abrikosov vortex lattice.

This concludes our formal discussion of the instability of the electron gas. For the sakeof clarity we avoided a detailed discussion of the physical manifestations of the Cooperinstability of the electron gas — it was used here merely as a vehicle for illustrating thegeneral approach of the coherent state path integral. However, before leaving this section,a few remarks concerning the question of universality are in order.

5.4.5 Statistical Field Theory: Ferromagnetism Revisited

The motivation that stands behind the phenomenology of Ginzburg-Landau theory hasprofound implications that go beyond its application to superconductivity pervading all



areas of physics. To illustrate the generality of the concept, let us temporarily leavebehind superconductivity and consider the classical equilibrium statistical mechanics of a‘one-component’ or Ising ferromagnet (i.e. spin degrees of freedom can take only twovalues: Si = ±1). Our previous considerations in chapter 2 have emphasised that, whenviewed microscopically, the development of magnetic moments on the atomic lattice sitesof a crystal and the subsequent ordering of the moments is a complex process involving thecooperative behaviour of many interacting electrons. However, at first sight, this pictureseems to be at odds with the empirical observation that thermodynamic properties ofdifferent macroscopic ferromagnetic systems seem to be the same — e.g. temperaturedependence of the specific heat, susceptibility, etc. Moreover, the thermodynamic criticalproperties of completely different physical systems, such as an Ising ferromagnet and aliquid at its boiling point, show the same dependence on, say, temperature. What is thephysical origin of this Universality?

Suppose we take a ferromagnetic material and measure some of its material propertiessuch as its magnetisation. Dividing the sample into two roughly equal halves, keeping theinternal variables like temperature and magnetic field the same, the macroscopic proper-ties of each piece will then be the same as the whole. The same holds true if the processis repeated. But eventually, after many iterations, something different must happen be-cause we know that the magnet is made up of electrons and ions. The characteristiclength scale at which the overall properties of the pieces begins to differ markedly fromthose of the original defines a correlation length. It is the typical length scale overwhich the fluctuations of the microscopic degrees of freedom are correlated.

Now experience tells us that a ferromagnet may abruptly change its macroscopic be-haviour when the external conditions such as the temperature or magnetic field are varied.The points at which this happens are called critical points, and they mark a phasetransition from one state to another. In the ferromagnet, there are essentially two waysin which the transition can occur (see Fig. 5.16). In the first case, the two states oneither side of the critical point (spin up) and (spin down) coexist at the critical point.Such transitions, involve discontinuous behaviour of thermodynamic properties and aretermed first-order. (c.f. melting of a three-dimensional solid) The correlation length atsuch a first-order transition is generally finite.

In the second case, the transition is continuous, and the correlation length becomeseffectively infinite. Fluctuations become correlated over all distances, which forces thewhole system to be in a unique, critical, phase. The two phases on either side of thetransition (paramagnetic and ferromagnetic) must become identical as the critical pointis approached. Therefore, as the correlation length diverges, the magnetisation goessmoothly to zero. The transition is said to be second-order.

The divergence of the correlation length in the vicinity of a second order phase transi-tion suggests that properties near the critical point can be accurately described within aneffective theory involving only long-range collective fluctuations of the system. This invitesthe construction of a phenomenological Hamiltonian or Free energy which is constrainedonly by the fundamental symmetries of the system — Ginzburg-Landau theory. Al-though the detailed manner in which the material properties and microscopic couplingsof the ferromagnet influence the parameters of the effective theory might be unknown,qualitative properties such as the scaling behaviour are completely defined.



T

2

c

S

H

T

1

Figure 5.16: Phase diagram of the Ising ferromagnet showing the average magnetisation Sas a function of magnetic field H and Temperature T . Following trajectory 1 by changingthe magnetic field at constant temperature T < Tc, the sample undergoes a first orderphase transition from an average ‘spin-up’ phase to an average ‘spin-down’. By changingthe temperature at fixed zero magnetic field, the system undergoes a second order phasetransition at T = Tc where the average magnetisation grows continuously from zero. Thissecond order transition is accompanied by a spontaneous symmetry breaking in which thesystem chooses to be in either an up or down-spin phase. (Contrast this phase diagramwith that of the liquid-gas transition — magnetisation S → density ρ, and magnetic fieldH → pressure.) The circle marks the region in the vicinity of the critical point wherethe correlation length is large as compared to the microscopic scales of the system, andGinzburg-Landau theory applies.

Following this philosophy, the Ginzburg-Landau theory of the Ising ferromagnet isdefined by gradient expansion of the effective Free energy in powers of the order parameter,the local magnetisation S(r). Respecting the symmetry properties of the microscopicHamiltonian (translational and rotational invariance in the spatial degrees of freedom, up-down or Z2 invariance in the internal spin degrees of freedom, etc), the partition functionof the Ising ferromagnet can be expressed in the form of a functional field integral overdifferent spin configurations S(r)

Z =

∫

DS(r)e−βH[S(r)],

with the effective Ginzburg-Landau Hamiltonian or Free energy functional

βH [S(r)] =

∫

ddr

[t

2S2 +

K

2(∂S)2 + uS4 + · · ·+HS

]

.

At the mean-field level (i.e. neglecting fluctuations of the magnetisation field) a min-imisation of the effective Free energy (in the absence of an external magnetic field)

βH(S)

V=t

2S2 + uS4,

known as the Landau Free energy, leads to the average magnetisation

S =

0 t > 0,√

t/4u t < 0.



Thus a non-zero magnetisation develops when t < 0 identifying this parameter as thereduced temperature t = (T − Tc)/Tc.

The Ginzburg-Landau Free energy can be compared with that obtained from the mi-croscopic Hamiltonian for the superconductor above (5.17). Apart from the complexnature of the order parameter ∆, the action coincides. In fact, we might very well havecircumvented the analysis of the microscopic Hamiltonian, and written the Ginzburg-Landau Free energy on purely phenomenological grounds. The dependence of the param-eters K, u, etc., on the microscopic or material properties of the system would have beenunavailable, but the nature of the critical point and the physical properties associatedwith the transition would have been accessible. Indeed, as mentioned above much of thephenomenology of conventional superconductivity was developed and understood in thisframework even prior to the discovery of the BCS theory.

From this result, we can draw important conclusions: Critical properties in the vicin-ity of a both classical and quantum second order phase transitions fall into a limitednumber of universality classes defined, not by detailed material parameters, but by thefundamental symmetries of the system. When we study the critical properties of the Isingtransition in a one-component ferromagnet, we learn about the nature of the liquid-gastransition! Similarly, in the jargon of statistical field theory, the superconductor, withits complex order parameter ∆ is in the same universality class as the two-component orXY Heisenberg ferromagnet. The analyses of critical properties associated with differentuniversality classes is the subject of Statistical field theory.



1.3 Problem Set

1.3.1 Questions on Collective Modes and Field Theories

1. In obtaining the spectrum of collective phonon excitations for the lattice Lagrangian (1.1),a continuum approximation was employed. However, since the degrees of freedom arecoupled linearly, the equations of motion can be solved explicitly, even for the discretemodel. By constructing the equations of motion, obtain the normal modes of the systemand obtain the exact eigenspectrum of phonon excitations. [Hint: Look for a wave-likesolution of the discrete equations of motion.] Identify the limit in which the spectrum ofthe discrete lattice model coincides with that obtained for the continuum approximationof the model. In what limit does the continuum approximation fail and why?

——————————————–

mA

mB

(n-1)a na (n+1)a

Figure 1.6: Lattice with two atoms of mass mA and mB per unit cell.

2. In lattices with two atoms (of di↵erent mass mA and mB) per unit cell (see Fig. 1.6) thespectrum of elementary phonon excitations splits into an acoustic and optic branch. Forthis model, show that the discrete lattice Lagrangian for a periodic system with 2 Nmasses can be written as

L =NX

n=1

mA

2((A)

n )2 +mB

2((B)

n )2 ks2

(A)

n+1

(B)

n

2

ks2

(B)

n (A)

n

2

.

Applying the Euler-Lagrange equation for the discrete model, obtain the classical equa-

tions of motion. Switching to the discrete Fourier representation (cf. Problem 1), (A/B)

k =1pN

PNn=1

eikna(A/B)

n where k = 2m/a (m integer), show that the exact eigenspectrum,

!k, can be obtained from the solution of the 2 2 secular equation for each k value

det

mA!2

k 2ks ks(1 + eika)ks(1 + eika) mB!2

k 2ks

= 0.

By finding an expression for the spectrum, obtain the asymptotic dependence as k ! 0.In this limit, describe qualitatively the symmetry of the normal modes.

——————————————–

3. Applying the Euler-Lagrange equation, obtain the equation of motion associated with theLagrangian densities:

1. L[] = m2

2 ksa2

2(@x)

2 m

2!22

2. L[] = m2

2

2

@2

x2


1.3. PROBLEM SET 15

3. L[] = m2

2 m

2!22

44

4. L[i] =nX

i=1

m

22

i 1

2ksa

2(@xi)2

5. L[] = m

2||2 1

2ksa

2|@x|2

[Note that in 5. the field is complex.] Suggest a physical significance of the last termin 1. What is the e↵ect of this term on the excitation spectrum of the correspondingquantum Hamiltonian? Starting with the Lagrangian 2., obtain the Hamiltonian density.

——————————————–

4. Following the discussion in the lectures, a periodic one-dimensional quantum elastic chainof length L is expressed by the Hamiltonian

H =

Zdx

1

2m2 +

ksa2

2(@x)

2

where the field operators obey the canonical commutation relationsh

(x), (x0)i= i~(x x0).

(a) Defining the Fourier representation,k

k 1

L1/2

Z L

0

dxeikx

(x)(x)

,

(x)(x)

=1

L1/2

Xk

e±ikx

k

k,

whereP

k represents the sum over all quantised quasi-momenta k = 2m/L, m 2 Z, show

that the field operators obey the commutation relations [k, k0 ] = i~kk0 .(b) In the Fourier representation, show that the Hamiltonian takes the form

H =Xk

1

2mkk +

ksa2

2k2kk

.

(c) Defining

ak r

m!k

2~

k + i

1

m!kk

where !k = a(ks/m)1/2|k| = v|k| show that the field operators obey the canonical com-

mutation relations [ak, a†k0 ] = kk0 , and [ak, ak0 ] = 0.

(d) Finally, with this definition, show that the Hamiltonian can be expressed in the form

H =Xk

~!k

a†kak +

1

2

.

——————————————–



1.3.2 Answers

1. Applying the Euler-Lagrange equation dt(∂φnL) − ∂φn

L = 0 to the discrete La-grangian of the lattice model, one finds that the N equations of motion take theform of a three-term difference equation,14

mφn = ks (φn+1 − 2φn + φn−1) .

As in the continuum theory, the latter can be brought to diagonal form by turningto the Fourier representation. Applying the Ansatz

φn(t) =1√N

∑

k

e−i(ωkt−kna)φk

where the discrete quasi-momenta k = 2πm/Na take values from the range m =[−N/2, N/2] (i.e. the first Brillouin zone), we find [mω2

k−2ks(1−cos(ka))]φk = 0.From this equation one obtains the dispersion relation

ωk = 2

√

ks

m| sin(ka/2)|.

For k → 0, this result collapses to the linear dispersion relation ωk = v|k| obtainedfrom the continuum theory. This can be understood simply by comparing the wave-length of the lattice vibration λ = 2π/k with the lattice spacing a. When λ ≫ a,the relative displacement of the atomic sites is small and the continuum approxima-tion is justified. When λ ∼ a, the relative displacement is large and the continuumtheory becomes inapplicable.

——————————————–

2. Applying the Euler-Lagrange equations we obtain the 2 × N coupled equations ofmotion

mAφ(A)n = ks

(

φ(B)n − 2φ(A)

n + φ(B)n−1

)

,

mBφ(B)n = ks

(

φ(A)n+1 − 2φ(B)

n + φ(A)n

)

.

14Note that, looking for stationary solutions φn(t) = e−iωt, the difference equation can be written as

the eigenvalue equation mω2φn =∑

N

mMnmφn, where

M = ks

. . . . . . . . . 0−1 2 −1

−1 2 −10 . . . . . . . . .

.

The latter has eigenfunctions corresponding to the discrete Fourier transform: i.e. writing the eigen-value equation as mω2|n〉 = M |n〉, the normalised eigenfunctions |k〉, indexed by the wavenumberk ∈ [−π/a, π/a], are given by φn(k) ≡ 〈n|k〉 = 1

√

Neikna. Later, in the next chapter, we will see

that these eigenfunctions (of the discrete lattice Laplacian) will present a convenient basis for expansionof more complicated lattice models.


1.3. PROBLEM SET 19

Applying the Ansatz φ(A/B)n = 1√

N

∑

k e−i(ωkt−kna)φ(A/B)k , one obtains

(mAω2

k − 2ks ks(1 + e−ika)ks(1 + eika) mBω2

k − 2ks

) (φ

(A)k

φ(B)k

)

= 0.

Diagonalizing the 2 × 2 matrix one recovers the secular equation shown in thequestion and from which we obtain the dispersion relation (see Fig. 1.9)

ω(±)k = ω0

[

1 ±(

1 − 4mAmB

(mA + mB)2sin2(ka/2)

)1/2]1/2

,

where ω0 =√

ks/µ and µ = 1/(m−1A + m−1

B ) denotes the reduced mass.

π/a

m =mB A

ωk

k/a−π

B.Z.

Acoustic

Optic

Figure 1.9: Spectrum of the two-atom discrete linear chain. Note that when mA = mB

we recover the spectrum of the single atom chain with period a/2. For mA 6= mB, a gapopens at the Brillounin zone boundary. The lower energy band is known as the acousticbranch where atoms in each unit cell move in phase. The higher energy optic branchinvolves atoms in each cell moving in antiphase.

An expansion in the limit k → 0 yields

ω(±)k → ω0

√2

(

1 − mAmB

8(mA + mB)2(ka)2

)

+ O(k4)√

mAmB

2

1

(mA + mB)|ka| + O(k3)

from which we deduce that the lower branch describes acoustic phonons with a lineardispersion relation, while the optic phonons are massive with a quadratic spectrum.

——————————————–

3. To complete this problem, one must generalise the Euler-Lagrange equation derivedin lectures. E.g., for case 2., a variation of the action obtains

δS =

∫ [

L(φ + ǫη, ∂2xφ + ǫ∂2

xη) − L(φ, ∂2xφ)

]

= ǫ

∫[η∂φL + ∂2

xη∂∂2xφL

]= ǫ

∫

η[−dt∂φL + d2

x

(∂∂2

xφL)]



from which we obtain the Euler-Lagrange equation

−dt∂φL + d2x

(∂∂2

xφL)

= 0.

Applied to the Lagrangian functionals at hand, one finds the equation of motion

1. φ − ksa2

m∂2

xφ + ω2φ = 0,

2. φ +κ

m∂4

xφ = 0,

3. φ + ω2φ +η

mφ3 = 0

4. φi −ksa

2

m∂2

xφi = 0.

Finally, turning to case 5., it is necessary to generalise the Euler-Lagrange equationto account for complex fields. Since the real and imaginary parts fluctuate indepen-dently, we can consider a variation of each independently. Separating φ = φ′ + iφ′′

into its real and imaginary parts, and applying a variation to each component, oneobtains

φ′ − ksa2

m∂2

xφ′ = 0, φ′′ − ksa

2

m∂2

xφ′′ = 0.

In fact, this result shows that the components φ and the complex conjugate φ∗ canbe treated as independent. A variation of the action with respect to φ∗ obtains

5. φ − ksa2

m∂2

xφ = 0.

Physically, one can interpret the last term in 1. as a harmonic lattice bindingpotential. Its effect is to render the fluctuations massive, viz. ωk =

√ω2 + v2k2.

For case 2. the momentum conjugate to the field φ is given by π = ∂φL = mφ. TheHamiltonian corresponding to case 2. is given by

H = πφ − L =π2

2m+

ksa2

2(∂2

xφ)2.

——————————————–

4. (a) Using the definition provided, together with the canonical commutation relationsof the field operators, one obtains

[πk, φk′] =1

L

∫ L

0

dx

∫ L

0

dx′eikx−ik′x′

−i~δ(x − x′)︷︸︸︷

[π(x), φ(x′)] = −i~

δkk′

︷︸︸︷

1

L

∫ L

0

dx ei(k−k′)x′

= −i~δkk′ .

(b) Again, using the definition, the kinetic component of the Hamiltonian takes theform

∫ L

0

dxπ2

2m=

∑

kk′

1

L

∫ L

0

dx e−i(k+k′)x 1

2mπkπk′ =

∑

k

1

2mπkπ−k.


1.3. PROBLEM SET 21

Similarly, applied to the potential component one obtains the Hamiltonian as ad-vertised.

(c) Using the commutation relation derived above, one finds

[ak, a†k′] =

i

2~

(

[φ−k, π−k′] − [πkφk′])

= δkk′, [a†k, a

†k′] = 0.

(d) Inverting the expression for the field operators, one finds

φk =

(~

2mωk

)1/2 (

ak + a†−k

)

, πk = i

(m~ωk

2

)1/2 (

a†k − a−k

)

.

Using the identity

1

2m(πkπ−k + π−kπk) +

ksa2

2k2

(

φkφ−k + φ−kφk

)

=1

2~ωk

(

a†kak + aka

†k + a†

−ka−k + a−ka†−k

)

.

together with the commutation relations, after summation over k, we obtain theharmonic oscillator Hamiltonian.

——————————————–



2.4 Problem Set

2.4.1 Questions on the Second Quantisation

1. (a) Starting with the commutation relation for bosonic creation a† and annihilation oper-ators a,

a, a†

= 1, show that

[a†a, a] = a, [a†a, a†] = a†.

Using this result, show that, if |↵i represents an eigenstate of the operator a†a witheigenvalue ↵, a|↵i is also an eigenstate with eigenvalue (↵1) (unless a|↵i = 0). Similarly,show that a†|↵i is an eigenstate with eigenvalue (↵+ 1).

(b) If |↵i represents a normalised eigenstate of the operator a†a with eigenvalue ↵ for all↵ 0, show that

a|↵i =p↵|↵ 1i , a†|↵i =

p↵+ 1|↵+ 1i .

[Hint: consider the norm of the state.] Defining |i the normalised vacuum state, annil-iated by the operator a, show that |ni = 1p

n!(a†)n|i is a normalised eigenstate of a†a

with eigenvalue n.

As an additonal exercise, consider the generalisation of parts (a) and (b) to the case offermionic operators a.

——————————————–

2. Starting from first principles, show that the second quantised representation of the one-body kinetic energy operator is given by

T =

Z L

0

dx a†(x)p2

2ma(x).

[Hint: it may be helpful to start with the Fourier representation in which the one-bodykinetic energy operator is diagonal and carefully transform to the real space basis.]

——————————————–

3. Transforming to the Fourier basis, show that the non-interacting three-dimensional cubiclattice tight-binding Hamiltonian,

H(0) = tXhmni

c†mcn + h.c.

,

assumes a diagonal form. Here hmni denotes the sum over all neighbouring sites and h.c.is short-hand for the Hermitian conjugate.

——————————————–

4. Show that the Holstein-Primako↵ transformation,

S = a†2S a†a

1/2

, S+ = (S)† , Sz = S a†a ,

is consistent with the quantum spin algebra [S+, S] = 2Sz. [Hint: you may prove thisresult without explicitly expansion of the square root!]

——————————————–


2.4. PROBLEM SET 43

5. Confirm that the bosonic commutation relations of the operators ↵ and ↵† are preservedby the Bogoluibov transformation,

↵↵†

=

cosh sinh

sinh cosh

aa†

.

How and why is this transformation related to the Lorentz transformation?——————————————–

6. (a) Making use of the spin commutation relation, [S↵m, S

n ] = imn↵Sm (~ = 1), apply

the identity i ˙Si = [Si, H], to express the equation of motion of a spin in a nearest neighbourspin S one-dimensional Heisenberg ferromagnet, H = J

Pm Sm · Sm+1

.

(b) Interpreting the spins as classical vectors, and taking the continuum limit, show thatthe equation of motion of the hydrodynamic modes takes the form

S = Ja2S @2S,

where a denotes the lattice spacing. [Hint: in transferring to the continuum limit, apply

a Taylor expansion to the spins viz. Sm+1

= Sm + a@Sm + a2

2

@2Sm + · · ·.](c) Confirm that the equation of motion is solved by the Ansatz, S(x, t) = (c cos(kx !t), c sin(kx !t),

pS2 c2), and determine the dispersion. Sketch a ‘snapshot’ configu-

ration of the spins in the chain.——————————————–

7. †Valence Bond Solid: Starting with the spin 1/2 Majumdar-Ghosh Hamiltonian

HMG

=4|J |3

NXn=1

Sn · Sn+1

+1

2Sn · Sn+2

+

N |J |2

,

where the total number of sites N is even, and SN+1

= S1

, show that the two dimer or

valence bond states |±i = QN/2

n=1

1p2

(| "i2n | #i

2n±1

| #i2n | "i

2n±1

), are exact

ground states, i.e. |+

i describes the state where neighbouring spins on sites 2n and2n+ 1 are in a total spin singlet (S = 0) state. [Hint: recast the Hamiltonian in terms ofthe total spin of a triad Jn = Sn1

+ Sn + Sn+1

, and consider what this representationimplies.] Consider what would happen if the total number of sites was odd.

——————————————–

8. Su-Shrie↵er-Heeger Model: Polyacetylene consists of bonded CH groups forming anisomeric long chain polymer. According to molecular orbital theory, the carbon atomsare expected to be sp2 hybridised suggesting a planar configuration of the molecule. Anunpaired electron is expected to occupy a single p-orbital which points out of the plane.The weak overlap of the p-orbitals delocalise the electrons into a -conduction band (cf.benzene) — see Fig. 2.13a. Therefore, according to the nearly free electron theory, onemight expect the half-filled conduction band of a polyacetylene chain to be metallic.However, the energy of a half-filled band of a one-dimension system can always be loweredby imposing a periodic lattice distortion known as thePeirels instability (see Fig. 2.13b).



(c)

(a)

(b)

Figure 2.13: (a) Schematic diagram showing the -bonding orbitals in long chain polyacetylene.(b) One of the configurations of the Peirels distorted chain. The double bonds represent theshort links of the lattice. (c) A topological defect separating a two domains of the ordered phase.

One can think of an enhanced probability of finding the electron on the short bond wherethe orbital overlap is stronger — the “double bond”. The aim of this problem is to explorethe instability.

(a) At its simplest level, the conduction band of polyacetylene can be modelled by a simpleHamiltonian, due to Su-Shrie↵er-Heeger, in which the hopping matrix elements of theelectrons are modulated by the lattice distortion of the atoms. Taking the displacementof the atomic sites from the equilibrium separation a 1 to be un, and treating theirdynamics as classical, the e↵ective Hamiltonian takes the form

H = tNX

n=1

X

(1 + un)hc†ncn+1 + h.c.

i+

NXn=1

ks2(un+1

un)2 ,

where, for simplicity, the boundary conditions are taken to be periodic. The first termdescribes the hopping of electrons between neighbouring sites with a matrix element mod-ulated by the periodic distortion of the bond-length, while the last term represents theassociated increase in the elastic energy. Taking the lattice distortion to be periodic,un = (1)n↵, and the number of sites to be even, diagonalise the Hamiltonian. [Hint:the lattice distortion lowers the symmetry of the lattice. The Hamiltonian is most easilydiagonalised by distinguishing the two sites of the sublattice — i.e. doubling the size ofthe elementary unit cell — and transforming to the Fourier representation.] Show thatthe Peierls distortion of the lattice opens a gap in the spectrum at the Fermi level of thehalf-filled system.

(b) By estimating the total electronic and elastic energy of the half-filled band (i.e. anaverage of one electron per lattice site), show that the one-dimensional system is alwaysunstable towards the Peirels distortion. To complete this calculation, you will need theapproximate formula for the (elliptic) integral,Z /2

/2dk

1

1 ↵2

sin2 k

1/2 ' 2 + (a

1

b1

ln↵2)↵2

where a1

and b1

are (unspecified) numerical constants.†(c) For an even number of sites, the Peierls instability has two degenerate configurations(see Fig. 2.13b) — ABABAB.. and BABABA... Comment on the qualitative form of theground state lattice configuration if the number of sites is odd (cf. Fig. 2.13c).

——————————————–


2.4. PROBLEM SET 45

9. In the Schwinger boson representation quantum mechanical spin is expressed in termsof two bosonic operators, a and b, in the form S+ = a†b, S = (S+)†, Sz = 1

2

(a†a b†b).

(a) Show that this definition is consistent with spin commutation relations [S+, S] = 2Sz.

(b) Using the bosonic commutation relations, show that

|S,mi = (a†)S+mp(S +m)!

(b†)Smp(S m)!

|i,

is compatible with the definition of an eigenstate of the total spin operator S2 and Sz.Here |i denotes the vacuum of the Schwinger bosons, and the total spin S defines thephysical subspace |na, nbi : na + nb = 2S.

——————————————–

10. †The Jordan-Wigner Transformation: So far we have shown how the algebra ofquantum mechanical spin can be expressed using boson operators. Here we show that arepresentation for spin 1/2 can be obtained in terms of Fermion operators. Specifically,let us represent an up spin as a particle and a down spin as the vacuum |0i, viz. | "i |1i = f †|0i, and | #i |0i = f |1i. In this representation the spin raising and loweringoperators are expressed in the form S+ = f † and S = f , while Sz = f †f 1/2.

(a) With this definition, confirm that the spins obey the algebra [S+, S] = 2Sz.

However, there is a problem: spins on di↵erent sites commute while fermion operatorsanticommute, e.g. S+

i S+

j = S+

j S+

i , but f†i f

†j = f †

j f†i . To obtain a faithful spin represen-

tation, it is necessary cancel this unwanted sign. Although a general procedure is hard toformulate, in one dimension, this can be achieved by a non-linear transformation, viz.

S+

l = f †l e

iP

j<l

nj , S

l = eiP

j<l

njfl, Sz

l = f †l fl 1

2.

Operationally, this seemingly complicated transformation has a straightforward interpre-tation: in one dimension, the particles can be ordered on the line. By counting the numberof particles ‘to the left’ we can assign an overall phase of +1 or 1 to a given configurationand thereby transmute the particles into a fermions. (Put di↵erently, the exchange to twofermions induces a sign change which is compensated by the factor arising from the phase— the ‘Jordan-Wigner string’.)

(b) Using the Jordan-Wigner representation, show that S+

mSm+1

= f †mfm+1

.

(c) For the spin 1/2 anisotropic quantum Heisenberg spin chain, the spin Hamiltonianassumes the form

H = Xn

JzS

znS

zn+1

+J?2

S+

n Sn+1

+ Sn S

+

n+1

.

Turning to the Jordan-Wigner representation, show that the Hamiltonian can be cast inthe form

H = Xn

J?2

f †nfn+1

+ h.c.+ Jz

1

4 f †

nfn + f †nfnf

†n+1

fn+1

.

(d) The mapping above shows that the one-dimensional quantum spin 1/2 XY-model (i.e.Jz = 0) can be diagonalised as a non-interacting theory of spinless fermions. In this case,show that the spectrum assumes the form k = J? cos ka.

——————————————–



2.4.2 Answers

1. (a) Making use of the commutation relations for bosons, one finds

a†aa = a(a†a − 1), a†aa† = a†(1 + a†a)

from which the results follow. Using these results, one finds that, providing a|α〉 6= 0,

a†a a|α〉 = a(a†a − 1)|α〉 = (α − 1)a|α〉a†a a†|α〉 = a†(1 + a†a)|α〉 = (1 + α)a†|α〉

(b) If |α〉 is a normalised eigenstate of a†a with eigenvalue α, the norm of statecreated by the action of the creation operator is given by

||a†|α〉|| ≡√

〈α|aa†|α〉 =√

〈α|a†a + 1|α〉 =√

α + 1.

Similarly, the norm of state created by the action of the annihilation operator isgiven by

||a|α〉|| ≡√

〈α|a†a|α〉 =√

α.

Therefore, if we define |α + 1〉 and |α − 1〉 as the normalised eigenstates of theoperator a†a with eigenvalue α + 1 and α − 1 respectively, one finds

a|α〉 =√

α|α − 1〉a†|α〉 =

√α + 1|α + 1〉

Defining as the vacuum |Ω〉 the normalised state that is annihilated by the operatora, an application of the result above shows the state |n〉 = (1/

√n!)(a†)n|Ω〉 to be a

normalised eigenstate of a†a with eigenvalue n.——————————————–

2. The kinetic energy operator is diagonal in the momentum basis. Following the anal-ysis in the text, the corresponding second quantised one-body operator is given byT =

∑

pp2

2ma†

pap. Transforming to the coordinate representation, ap = L−1/2∫ L

0dxeipx/~a(x),

one obtains

T =1

L

∑

p

p2

2m

∫ L

0

dy

∫ L

0

dxeip(x−y)/~a†(y)a(x).

Expressing factor p2 as a derivative of the exponential factor, and integrating byparts, one obtains the required result.

——————————————–


2.4. PROBLEM SET 57

3. A Hamiltonian which is translationally invariant is easily diagonalised in the Fourierrepresentation. Setting c†

mσ = 1Nd/2

∑

keik·mc†

kσ, the Hamiltonian takes the form

H(0) =

B.Z.∑

k

∑

σ

ǫkc†kσckσ,

where ǫk = −t∑

i=x,y,z eik·ei = −2t∑

i=x,y,z cos(k · ei), with the sum running overneighbouring lattice vectors ei, and the lattice spacing is taken to be unity.

——————————————–

4. Substituting the definition of the spin raising and lowering operators using theHolstein-Primakoff transformation, the commutator is obtained as

1

2S[S+, S−] =

(

1 − a†a

2S

)1/21 + a†a︷︸︸︷

aa†

(

1 − a†a

2S

)1/2

− a†

(

1 − a†a

2S

)

a

=

(

1 − a†a

2S

)

+ a†a

(

1 − a†a

2S

)

− a†a +a†a†aa

2S= 1 − a†a

S.

With Sz = S − a†a, we obtain the required commutation relation [S+, S−] = 2Sz.——————————————–

5. By symmetry, the maximal exchange energy that can be recovered is obtained whenthe spins are maximally anti-aligned, i.e. at 120o to each other. Using the spin

(a) (b)

orientation of a single triangle, the two-dimensional triangular lattice can be tessi-lated with all spins aligned at 120o to the neighbours. Notice that this configurationrepresents just one of an infinite degenerate manifold of ground states obtained byglobal rotation of the spins.

——————————————–



6. To confirm the validity of the Bogoluibov transformation let us consider the com-mutation relations of α:27

[α, α†

]=

[cosh θ a + sinh θ a†, cosh θ a† + sinh θ a

]

= cosh2 θ[a, a†

]+ sinh2 θ

[a†, a

]= cosh2 θ − sinh2 θ = 1.

as required.——————————————–

7. (a) Making use of the equation of motion iSi = [Si, H], and the commutationrelation [Sα

i , Sβj ] = iδijǫ

αβγSγi , we obtain

iSi = JiSi ×(

Si+1 + Si−1

)

(b) Interpreting the spins as classical vectors, and applying the Taylor expansionSi+1 = Si + a∂Si + (a2/2)∂2Si + · · ·, we obtain the classical equation of motion

S = Ja2S× ∂2S.

Substituting, we find that S =(c cos(kx − ωt), c sin(kx − ωt),

√S2 − c2

), satisfies

the equation of motion with ω = J(ka)2√

S2 − c2.

Figure 2.14: Spin-wave dispersion.

(c) The corresponding spin wave solution has the precessional form shown in Fig. 2.14.——————————————–

27More formally, one may prove the relation as follows: Suppose we define a two component operatora = (a, a†). If a obeys bosonic commutation relations, [aµ, a†

ν] = gµν where the diagonal matrix has

elements g = diag(1,−1). If we define an operator transformation Λ such that αµ = Λµνaν (summationconvention implied), then the condition that the commutation relations are preserved requires

gµν

!= [αµ, α†

ν] = ΛµηΛ∗

νγ[aη, a†

γ] = ΛµηΛ∗

νγgηγ ,

i.e. the admissable transformations fulfil the condition that gµν = ΛµηΛ∗νγ

gηγ . Such transformations, thatpreserve the “metric” g = diag(1,−1), belong to the group of Lorentz transformations. In the context ofbosonic operators, they are termed Bogoluibov transformations. Note that, for fermionic systems, withc = (c, c†), [cµ, c†

ν] = δµν . In this case, we require that δµν = ΛµηΛ∗

νγδηγ , i.e. Λ belongs the class of

Unitary transformations: Λ†Λ = 1


2.4. PROBLEM SET 59

8. Defining the total spin on a triad Jn = Sn−1 + Sn + Sn+1, the Hamiltonian can berecast in the form

HMG = |J |N∑

n=1

P3/2(n − 1, n, n + 1),

where P3/2(n − 1, n, n + 1) = (J2n − 3/4)/3 annihilates any state with total spin

J = 1/2 of the triad. Since in any three sites, two of the spins are in a singlet,there can be no components of J = 3/2 on any triad. Therefore the dimer statesare eigenstates of zero energy. Now since P3/2 is positive definite, these states mustbe the ground states.

——————————————–

9. (a) Since each unit cell is of twice the dimension of the original lattice, we begin byrecasting the Hamiltonian in the sublattice form

H = −t

N/2∑

m=1,σ

(1 + α)

[a†

mσbmσ + h.c.]+ (1 − α)

[b†mσam+1σ + h.c.

]+

Nksα2

2.

Switching to the Fourier basis, am =√

2/N∑

k e−2ikmak (similarly bm), where ktakes N/2 values uniformly on the interval [−π/2, π/2], the Hamiltonian takes theform

H =Nksa

2

2α2

−t∑

kσ

( a†kσ b†kσ )

(0 (1 + α) + (1 − α)e−2ik

(1 + α) + (1 − α)e2ik 0

)(akσ

bkσ

)

.

Diagonalising the 2 × 2 Hamiltonian, we obtain the spectrum

ǫ(k) = ±2t[cos2 k + α2 sin2 k

]1/2.

Reassuringly, in the limit α → 0, we obtain the cosine spectrum of the undistortedproblem, while in the limit α → 1, pairs of monomers become decoupled and weobtain a massively degenerate bonding and antibonding spectrum.

(b) According to the formula given in the text, the total shift in energy is given by

δǫ = −2t(a1 − b1 ln α2)α2 +Nksα

2

2

Maximising the energy gain with respect to α, one finds that the stable configurationis found when

α2 = exp

[a1

b1− 1 − Nks

4tb1

]



(c) If the number of sites is odd, the Peirels distortion is inevitably frustrated — aconfiguration that starts ABABAB must finish as BABABA. The result is that thepolymer chain must accommodate a topological excitation. The excitation is saidto be topological because the defect can not be removed by a smooth continuousdeformation — it is like a dislocation line in a crystal. Its effect on the spectrum ofthe model is to introduce a state that lies within the band gap of the material.

The consideration of an odd number of sites forces a topological defect into thesystem. However, even if the number of sites is even, one can create low energytopological excitations of the system either by doping (see fig. 2.13c), or by the cre-ation of excitons, particle-hole excitations of the system. Indeed, such topologicalexcitations can dominate the transport properties of the system.

As a footnote, one should add that the particular model considered above is some-what over-simplified. It seems likely that Coulomb interactions play a dominantrole in driving the Peirels instability in Polyacetylene. However, the qualitativeinterpretation of the existence of topological excitations is born out by experiment.

——————————————–

10. (a) Using the commutation relation for bosons, one finds.

[S+, S−] = a†b b†a − b†a a†b = a†abb† − aa†b†b

= a†a(b†b + 1) − (a†a + 1)b†b = a†a − b†b = 2Sz.

(b) Using the identity

S2 = (Sz)2 +1

2

(

S+S− + S−S+)

=1

4(na − nb)

2 +1

2

(a†b b†a + b†a a†b

)

=1

4(na − nb)

2 + nanb +1

2(na + nb) ,

one finds that

S2|S, m〉 =[m2 + (S + m)(S − m) + S

]|S, m〉 = S(S + 1)|S, m〉

as required. Similarly, one finds

Sz|S, m〉 =1

2(na − nb)|na = S + m, nb = S − m〉

=1

2[(S + m) − (S − m)] |S, m〉 = m|S, m〉,

showing |S, m〉 to be an eigenstate of the operator Sz with eigenvalue m.

As with the Holstein-Primakoff representation, the Schwinger boson represents yetanother representation of quantum mechanical spin. Which representation is mostconvenient for the analysis of quantum spin models depends sensitively on the natureof the microscopic Hamiltonian.

——————————————–


2.4. PROBLEM SET 61

11. (a) Using the fermionic anticommutation relations, one finds

[S+, S−]− = [f †, f ]− = f †f − ff †

= 2f †f − 1 = 2Sz.

(b) Using the fact that the number operators on different sites commute, one finds

S+mS−

m+1 = f †meiπ

P

j<m nje−iπP

l<m+1nlfm+1 = f †

me−iπnmfm+1 = f †mfm+1

where here we have made use of the fact that, for fermionic particles f †me−iπnm ≡ f †

m.

(c) The fermion representation is simply obtained by substitution.

(d) With Jz = 0, the spin Hamiltonian assumes the form of a non-interacting tight-binding Hamiltonian

H = −J⊥

2

∑

n

(f †

nfn+1 + h.c.).

This Hamiltonian, which has been encountered previously, is diagonalised in theFourier space after which one obtains the cosine band dispersion.

——————————————–


3.5. QUESTIONS ON THE PATH INTEGRAL 115

3.5 Questions on the Path Integral

1. Quantum Harmonic Oscillator: As emphasized in lectures, the quantum har-monic oscillator provides a valuable arena in which to explore the Feynman pathintegral and methods of functional integration. Along with a small number of otherprecious examples, the path integral may be computed exactly, and the Feynmanpropagator explored rigorously.

(a) Starting with the Feynman path integral, show that the propagator for theone–dimensional quantum Harmonic oscillator, H = p2/2m + mω2q2/2, takes theform


( mω

2πi~ sinωt

)1/2

exp

[

i

2~mω

(

[

q2I + q2

F

]

cotωt−2qIqFsinωt

)]

.

Suggest why the propagator varies periodically on the time interval t, and explainthe origin of the singularities at t = nπ/ω, n = 1, 2, . . .. Taking the frequencyω → 0, show that the propagator for the free particle is recovered.

(b) Show that, the wavepacket ψ(q, t = 0) = (2πa)−1/4 exp[−q2/4a] remains Gaus-sian at all subsequent times. Obtain the width a(t) as a function of time.

(c) Semiclassical limit: Taking the initial wavepacket to be of the form

ψ(q, t = 0) = (2πa)−1/4 exp

[

i

~mvq −

1

4aq2

]

,

(which corresponds to a wavepacket centered at an initial position q = 0 with avelocity v) find the wavepacket at times t > 0, and determine the mean position,mean velocity, and mean width as a function of time.

2. Density Matrix: Focusing on the quantum harmonic oscillator, here we explorehow real–time dynamical information can be converted into quantum statisticalinformation.

Using the results of the previous question, obtain the density matrix ρ(q, q′) =

〈q|e−βH |q′〉 for the harmonic oscillator at finite temperature, β = 1/T (kB = 1).Obtain and comment on the asymptotics: (i) T ≪ ~ω and (ii) T ≫ ~ω. [Hint: Inthe high temperature case, be sure to carry out the expansion in ~ω/T to secondorder.]

3. Winding Numbers: In lectures, we considered the application of the Feynmanpath integral to model systems where trajectories could be parameterised in termsof their harmonic (Fourier) expansion. However, very often, one is interested in ap-plications of the path integral to spaces that are not simply connected. In this case,one must include classes of trajectories which can not be simply continued. Rather,trajectories are classified by their ‘winding number’ on the space. To illustrate thepoint, let us consider the application of the path integral to a particle on a ring.



(a) Starting with the Hamiltonian H = − 12I

∂2

∂θ2 , where θ denotes an angle variable,

show from first principles that the quantum partition function Z = tr e−βH is givenby

Z =∞

∑

n=−∞

exp

[

−βn2

2I

]

. (3.61)

(b) Formulated as a Feynman path integral, show that the quantum partition func-tion can be cast in the form

Z =

∫ 2π

0

dθ

∞∑

m=−∞

∫

θ(0)=θ

θ(β)=θ(0)+2πm

Dθ(τ) exp

[

−I

2

∫ β

0

dτ θ2

]

.

(c) Varying the Eulidean action with respect to θ, show that the path integral isminimised by the classical trajectories θ(τ) = θ+2πmτ/β. Parametrising a generalpath as θ(τ) = θ(τ) + η(τ), where η(τ) is a path with no net winding, show that

Z = Z0

∞∑

m=−∞

exp

[

−I

2

(2πm)2

β

]

, (3.62)

where Z0 represents the quantum partition function for a free particle with openboundary conditions. Making use of the free particle propagator, show that Z0 =√

I/2πβ.

(d) Finally, making use of Poisson’s summation formula,∑

m h(m) =∑

n

∫ ∞

−∞dφ h(φ)e2πinφ,

show that Eq. (3.62) coincides with Eq. (3.61).

4. †Particle in a Periodic Potential: In section 3.3.2 it was shown that the quan-tum probability amplitude for quantum mechanical tunneling can be expressed asa sum over instanton field configurations of the Euclidean action. By generalisingthis approach, the aim of the present problem is to explore quantum mechanicaltunneling in a periodic potential. Such an analysis allows us to draw a connectionto the problem of the Bloch spectrum.

(a) A quantum mechanical particle moves in a periodic lattice potential V withperiodicity a. Taking the Euclidean action for the instanton connecting two neigh-bouring minima to be Sinst., express the Euclidean time propagator G(ma, na; τ),with m and n integer, as a sum over instanton and anti–instanton field configura-tions.

(b) Making use of the identity δqq′ =∫ 2π

0ei(q−q′)θdθ/(2π) show that

G(ma, na; τ) ∼ eωτ/2

∫ 2π

0

dθ

2πe−i(n−m)θ exp

[

∆ǫτ

~2 cos θ

]

,

where our notation is taken from section 3.3.2.


3.5. QUESTIONS ON THE PATH INTEGRAL 117

(c) Keeping in mind that, in the periodic system, the eigenfunctions are Bloch statesψpα(q) = eipqupα(q) where upα(q + ma) ≡ upα(q) denotes the periodic part of theBloch function, show that the propagator is compatible with a spectrum of thelowest band α = 0, ǫp = ~ω/2 − 2∆ǫ cos(pa).



3.6 Answers

1. (a) Making use of the Feynman path integral, the propagator can be expressed asthe functional integral,


∫ q(t)=qF

q(0)=qI

Dq eiS[q]/~, S[q] =

∫ t

0

dtm

2

(

q2 − ω2q2)

.

The evaluation of the functional integral over field configurations q(t′) is facilitatedby parameterising the path in terms of fluctuations around the classical trajectory.Setting q(t′) = qcl(t

′) + r(t′) where qcl(t′) satisfies the classical equation of motion

mqcl = −mω2qcl, and applying the boundary conditions, one obtains the solutionqcl(t

′) = A sin(ωt′) +B cos(ωt′), with the coefficients B = qI and A = qF/ sin(ωt)−qI cot(ωt). Being Gaussian in q, the action separates as S[q] = S[qcl] + S[r], where

S[qcl] =mω2

2

[

(A2 −B2)sin(2ωt)

2ω+ 2AB

cos(2ωt) − 1

2ω

]

=mω

2

[

(q2I + q2

F ) cot(ωt) − 2qIqFsin(ωt)

]

.

Finally, integrating over the fluctuations and applying the identity z/ sin z =∏

∞

n=1(1−z2/π2n2)−1 one obtains the required result, periodic in t with frequency ω, and sin-gular at t = nπ/ω. In particular, a careful regularisation of the expression for thepath integral shows that

〈qF |e−iHt/~|qI〉 7→

δ(qF − qI) t = 2πn/ω,δ(qF + qI) t = π(2n+ 1)/ω.

Physically, the origin of the singularity is clear: The harmonic oscillator is pecu-liar in having a spectrum with energies uniformly spaced in units of ~ω. Notingthe eigenfunction expansion 〈qF |e−iHt~|qI〉 =

∑

n〈qF |n〉〈n|qI〉e−iωnt, this means thatwhen ~ω× t/~ = 2π× integer there is a coherent superposition of the states and theinitial state is recovered. Furthermore, since the ground state and its even integerdescendents are symmetric while the odd states are antisymmetric, it is straightfor-ward to prove the identity for the odd periods (exercise).

(b) Given the initial condition ψ(q, t = 0), the time evolution of the wavepacket can

be determined from the propagator as ψ(q, t) =∫

∞

−∞dq′〈q|e−iHt/~|q′〉ψ(q′, 0) from

which one obtains

ψ(q, t) = J(t)

∫

∞

−∞

dq′1

(2πa)1/4e−q′2/4ae

i~

mω2

“

[q2+q′2] cot(ωt)− 2qq′

sin(ωt)

”

where J(t) represents the time–dependent contribution arising from the fluctuationsaround the classical trajectory. Being Gaussian in q′, the integral can be performedexplicitly. Setting α = 1/2a−imω cot(ωt)/~, β = imωq/(~ sin(ωt)), and performingthe Gaussian integral over q′, one obtains

ψ(q, t) = J(t)1

(2πa)1/4

√

2π

αeβ2/2α exp

[

i

2~mωq2 cot(ωt)

]


3.6. ANSWERS 119

where β2/2α = −(1 + iκ cot(ωt))q2/4a(t). Rearranging terms, it is straightforward

to show that ψ(q, t) = (2πa(t))−1/4 exp[− q2

4a(t)]eiϕ(q,t), where a(t) = a[cos2(ωt) +

κ−2 sin2(ωt)], κ = 2amω/~ and ϕ(q, t) represents a pure phase.42 As required, underthe action of the propagator, the normalisation of the wavepacket is preserved. (Agraphical representation of the time evolution is shown in Fig. 3.13a.) Note that, ifa = ~/2mω (i.e. κ = 1), a(t) = a for all times — i.e. it is a pure eigenstate.

|ψ|2 |ψ|2

t t

q q

π/ω

(a) (b)

Figure 3.13: (a) Variation of a “stationary” Gaussian wavepacket in the harmonic oscillatortaken from the solution, and (b) variation of the moving wavepacket.

(c) Still of a Gaussian form, the integration can again be performed explicitly for thenew initial condition. In this case, we obtain an expression of the form above butwith β = i

~

mωsin(ωt)

(q− vω

sin(ωt)). Reading off the coefficients, we find that the position

and velocity of the wavepacket have the form q0(t) = (v/ω) sin(ωt), v(t) = v cos(ωt)coinciding with that of the classical dynamics. Note that, as above, the width a(t)of the wavepacket oscillates at frequency ω. (A graphical representation of the timeevolution is shown in Fig. 3.13b.)

2. The density matrix can be deduced from the general solution of the previous ques-tion. Turning to the Euclidean time formulation,

ρ(q, q′) = 〈q|e−βH|q′〉 = 〈q|e−(i/~)H(~β/i)|q′〉

=

(

mω

2π~ sinh(β~ω)

)1/2

exp

[

−mω2~

(

(q2 + q′2) coth(β~ω)− 2qq′

sinh(β~ω)

)]

.

(i) In the low temperature limit T ≪ ~ω (β~ω ≫ 1), coth(β~ω) → 1, sinh(β~ω) →eβ~ω/2, and

ρ(q, q′) ≃( mω

π~eβ~ω

)1/2

exp[

−mω2~

(q2 + q′2)]

= 〈q|n = 0〉 e−βE0 〈n = 0|q′〉 .

42For completeness, we note that ϕ(q, t) = − 12 tan−1( 1

κcot(ωt)) − κq

2

4acot(ωt)( a

a(t) − 1).



(ii) Using the relation coth(x)x≪1= 1/x−x/3+ . . . and 1/ sinh(x)

x≪1= 1/x−x/6+ . . .,

the high temperature expansion (T ≫ ~ω) of the density operator obtains

ρ(q, q′) ≃(

m

2πβ~2

)1/2

e−m(q−q′)2/2β~2

exp

[

−~βmω2

6~(q2 + q′

2+ qq′)

]

≃ δ(q − q′)e−βmω2q2

2 ,

i.e. one recovers the classical Maxwell–Boltzmann distribution!

3. (a) Solving the Schrodinger equation, the wavefunctions obeying periodic boundaryconditions take the form ψn = einθ/

√2π, n integer, and the eigenvalues are given by

En = n2/2I. Cast in the eigenbasis representation, the partition function assumesthe form (3.61).

(b) Interpreted as a Feynman path integral, the quantum partition function takesthe form of a propagator with

Z =

∫ 2π

0

dθ 〈θ|e−βH |θ〉 =

∫ 2π

0

dθ

∫

θ(β)=θ(0)=θ

Dθ(τ) exp

[

−∫ β

0

dτI

2θ2

]

.

The trace implies that paths θ(τ) must start and finish at the same point. However,to accommodate the invariance of the field configuration θ under translation by 2πwe must must impose the boundary conditions shown in the question.

(c) Varying the action with respect to θ we obtain the classical equation Iθ = 0.Solving this equation subject to the boundary conditions, we obtain the solutiongiven in the question. Evaluating the Euclidean action, one finds

∫ β

0

(∂τθ)2dτ =

∫ β

0

[

2πm

β+ ∂τη

]2

dτ = β

(

2πm

β

)2

+

∫ β

0

(∂τη)2dτ.

Thus, one obtains the partition function (3.62), where

Z0 =

∫

Dη(τ) exp

[

−I2

∫ β

0

(∂τη)2dτ

]

=

√

2πI

β.

denotes the free particle partition function. The latter can be obtained from directevaluation of the free particle propagator.

(d) Applying the Poisson summation formula with h(x) = exp[− (2π)2I2β

x2], one findsthat

∞∑

m=−∞

e−(2π)2Im2

2β =

∞∑

n=−∞

∫

∞

−∞

dφ e−(2π)2I

2βφ2+2πinφ =

√

β

2πI

∞∑

n=−∞

e−β

2In2

.

Multiplication with Z0 obtains the result.


3.6. ANSWERS 121

4. (a) In the double well potential, the extremal field configurations of the Euclideanaction involve consecutive sequences of instanton/anti–instanton pairs. However, inthe periodic potential, the q instantons and q′ anti–instantons can appear in anysequence provided only that q − q′ = n −m. In this case, the Feynman amplitudetakes the form

G(ma, na; τ) ∼∞

∑

q=0

∞∑

q′=0

δq−q′,n−m

q! q′!(τKe−Sinst./~)q+q′

(b) To evaluate the instanton summation, one may make use of the identity δq−q′,n−m =∫ 2π

0ei(q−q′−n+m)θdθ/(2π). As a result, one obtains

G(ma, na; τ) ∼ eωτ/2

∫ 2π

0

dθ

2πe−i(n−m)θ

∞∑

q=0

(τKeiθe−Sinst./~)q

q!

∞∑

q′=0

(τKe−iθe−Sinst./~)q′

q′!

∼ eωτ/2

∫ 2π

0

dθ

2πe−i(n−m)θ exp

[

∆ǫτ

~eiθ

]

exp

[

∆ǫτ

~e−iθ

]

from which one obtains the required result.

(c) Expanded in terms of the Bloch states of the lowest band of the periodic potentialα = 0, one obtains

G(ma, na; τ) =∑

p

ψ∗

p(ma)ψp(na)e−iǫpτ/~ =

∑

p

|up(0)|2eip(n−m)ae−iǫpτ/~

Interpreting θ = pa, and taking |up(0)|2 = const. independent of p, one can drawthe correspondence ǫp = ~ω/2 − 2∆ǫ cos(pa).



4.4 Questions on the Field Integral Method

1. Exercises on Fermion Coherent States: To practice the coherent state method,we begin with a few simple exercises on the fermionic coherent state which comple-ments the structures discussed in the main text.

Considering a fermionic coherent state |η〉, verify the following identities: (a) 〈η|a†i =〈η|ηi, (b) a†i |η〉 = −∂ηi

|η〉 and 〈η|ai = ∂ηi〈η|, (c) 〈η|ν〉 = exp[

∑i ηiνi], and (d)∫

d(η, η) dηi e−

P

iηiηi |η〉〈η| = 1F , where d(η, η) ≡

∏i dηidηi. Finally, (e) show that

〈n|ψ〉〈ψ|n〉 = 〈ζψ|n〉〈n|ψ〉, where |n〉 is an n-particle state in Fock space while |ψ〉is a coherent state.

2. Feynman path integral from the Functional Field Integral

The abstraction of the coherent state representation betrays the close similaritybetween the Feynman and coherent state path integrals. To help elucidate theconnection, the goal of the present problem is to confirm that the Feynman pathintegral of the quantum harmonic oscillator follows from the coherent state pathintegral.

Consider the simplest bosonic many-body Hamiltonian, H = ~ω(a†a + 12), where

a† creates ‘structureless’ particles, i.e. states in a one–dimensional Hilbert space.H can be interpreted as the Hamiltonian of a single oscillator degree of freedom.Show that the field integral for the partition function Z = tr [exp(−βH)] can bemapped onto the (imaginary–time) path integral of an harmonic oscillator by asuitable variable transformation. [Hint: Let yourself be guided by the fact that theconjugate operator pair (a, a†) is related to the momentum and coordinate operators(p, q) through a canonical transformation.]

3. Quantum Partition Function of the Harmonic Oscillator The following in-volves a practice exercise on elementary field integral manipulations, and infiniteproducts.

Compute the partition function of the harmonic oscillator Hamiltonian in the fieldintegral formulation. To evaluate the resulting infinite product over Matsubarafrequencies apply the formula x/ sin x =

∏∞

n=1 (1 − x2/(πn)2)−1. [Hint: The nor-malization of the result can be fixed by demanding that, in the zero temperaturelimit, the oscillator occupies its ground state.] Finally, compute the partition func-tion by elementary means and check your result. As an additional exercise, repeatthe same steps for the ‘fermionic oscillator’, i.e. with a, a† fermion operators. Hereyou will need the auxiliary identity cosx =

∏∞

n=1 (1 − x2

(π(n+1/2))2).


4.5. ANSWERS 143

4.5 Answers

1. Making use of the rules of Grassmann algebra,

(a) 〈η|a†i = 〈0| exp[−∑

j

aj ηj]a†i = 〈0|

∏

j

(1 − aj ηj)a†i = 〈0|(1 − aiηi)a

†i

∏

j 6=i

(1 − aj ηj)

= 〈0|aia†i

︸︷︷︸

=〈0|[ai,a†i]+=〈0|

ηi

∏

j 6=i

(1 − aj ηj) = 〈0|∏

j

(1 − aj ηj)ηi = 〈η|ηi ,

(b) a†i |η〉 = a†i (1 − ηia†i )

︸︷︷︸

=a†i=∂ηi

ηia†i=−∂ηi

(1−ηia†i)

∏

j 6=i

(1 − ηja†j)|0〉 = −∂ηi

∏

j

(1 − ηja†j)|0〉 = −∂ηi

|η〉 ,

〈η|ai = 〈0|∏

j 6=i

(1 − aj ηj) (1 − aiηi)ai︸︷︷︸

=ai=−∂ηiaiηi=∂ηi

(1−aiηi)

= ∂ηi〈0|

∏

j

(1 − aj ηj) = ∂ηi〈η| ,

(c) 〈η|ν〉 = 〈η|∏

i

(1 − νia†i )

︸︷︷︸

(1+a†iνi)

|0〉 = 〈η|∏

i

(1 + ηiνi)︸︷︷︸

exp[P

i ηiνi]

|0〉 = exp[∑

i

ηiνi] .

(d) To prove the completeness of fermion coherent states, we apply Schur’s lemma,

i.e. we need to show that [a(†)j ,

∫d(η, η) e−

P

i ηiηi |η〉〈η|] = 0.

a†j

∫

d(η, η) e−P

i ηiηi |η〉〈η| = −

∫

d(η, η) e−P

i ηiηi∂ηj|η〉〈η|

=

∫

d(η, η) ∂ηj

(e−

P

i ηiηi)

︸︷︷︸

=ηje−P

i ηiηi

|η〉〈η| =

∫

d(η, η) e−P

i ηiηi |η〉〈η|a†j

aj

∫

d(η, η) e−P

i ηiηi |η〉〈η| =

∫

d(η, η) e−P

i ηiηiηj︸︷︷︸

=−∂ηj(e−P

i ηiηi)

|η〉〈η|

=

∫

d(η, η) e−P

i ηiηi |η〉∂ηj〈η| =

∫

d(η, η) e−P

i ηiηi |η〉〈η|aj .

The constant of proportionality is fixed by taking the expectation value with thevacuum.

〈0|

∫

d(η, η) e−P

i ηiηi |η〉〈η|0〉 =

∫

d(η, η) e−P

i ηiηi = 1 .

(e) A general n-particle state is given as |n〉 = a†1 . . . a†n|0〉, 〈n| = 〈0|an . . . a1, where

we neglected a normalization factor. The matrix element 〈n|ψ〉, thus, reads

〈n|ψ〉 = 〈0|an . . . a1|ψ〉 = 〈0|ψn . . . ψ1|ψ〉 = ψn . . . ψ1 .

Similarly, we obtain 〈ψ|n〉 = ψ1 . . . ψn. Using these results,

〈n|ψ〉〈ψ|n〉 = ψn . . . ψ1ψ1 . . . ψn = ψ1ψ1 . . . ψnψn

= (ζψ1ψ1) . . . (ζψnψn) = (ζψ1) . . . (ζψn)ψn . . . ψ1 = 〈−ζψ|n〉〈n|ψ〉 .



2. In the coherent state representation, the quantum partition function of the oscillatorHamiltonian is expressed in terms of the path integral (~ = 1)

Z =

∫

D(φ, φ) exp

[

−

∫ β

0

dτ(φ∂τφ+ ωφφ

)]

, (4.42)

where φ(τ) denotes a complex scalar field, the constant factor e−βω/2 has been ab-sorbed into the measure of the functional integral D(φ, φ), and we have set thechemical potential µ = 0. The connection between the coherent state and Feynman

integral is established by the change of field variables, φ(τ) =(

mω2

)1/2(

q(τ) + ip(τ)mω

)

,

where p(τ) and q(τ) represent real fields. Substituting this representation in Eq. (4.42),and rearranging some terms by integrating by parts, the connection is established:Z =

∫D(p, q) exp[−

∫ β

0dτ(−ipq + p2

2m+ mω2

2q2)].

3. Making use of the Gaussian functional integral for complex fields, one obtains fromEq. (4.42) (~ = 1)

ZB ∼ det(∂τ + ω)−1 ∼∏

ωn

(iωn + ω)−1 ∼

∞∏

n=1

[(2nπ

β

)2

+ ω2

]−1

∼

∞∏

n=1

[

1 +

(βω

2πn

)2]−1

∼1

sinh(βω/2).

Now, in the limit of small temperatures, the partition function is dominated by theground state, limβ→∞ZB = exp[−βω/2], which fixes the constant of proportionality.Thus, Zb = [2 sinh(β~ω/2)]−1.

In the fermionic case, the Gaussian integration obtains a product over eigenvaluesin the numerator and we have to use fermionic Matsubara frequencies, ωn = (2n +1)π/β:

Zf ∼ det(∂τ + ω) ∼∏

ωn

(iωn + ω) ∼

∞∏

n=1

[((2n+ 1)π

β

)2

+ ω2

]

∼

∞∏

n=1

[

1 +

(βω

(2n+ 1)π

)2]

∼ cosh(βω/2) .

Fixing the normalization, one obtains Zf = 2e−βω cosh(βω/2). Altogether, theseresults are easily confirmed by direct computation, viz.

ZB = e−βω/2

∞∑

n=0

e−nβω =e−βω/2

1 − e−βω=

1

2 sinh(βω/2),

ZF = e−βω/2

1∑

n=0

e−nβω = e−βω/2(1 + e−βω) = 2e−βω cosh(βω/2) .



5.5 Problem Set

5.5.1 Questions on the Functional Field Integral

1. In chapter 4., the connection between the coherent state path integral and theFeynman path integral for a Harmonic oscillator was exposed. The aim of thisproblem is to extend this calculation to obtain the Lagrangian form of the pathintegral for a harmonic chain from the coherent state path integral. In chapter 1we derived a free scalar field theory for the classical Harmonic chain. Qauntisingthe classical theory, we showed that, in second quantised form, the Hamiltonian wasgiven by

H =∑

k

~ωk

(

a†kak +1

2

)

,

where ωk = ksa2k2/m. Absorbing the zero-point energy into the definition of the

functional integral measureD[ψk, ψk], the corresponding coherent state path integralfor the quantum partition function assumes the form

Z =

∫

D[ψk, ψk] exp

[

−

∫ β

0

dτ∑

k

(

ψk∂τψk + ~ωkψkψk

)

]

,

where ψk(τ) denotes a complex scalar field, and the chemical potential has been setto zero µ = 0.

(a) By applying the change of field variables,

ψk(τ) =(mωk

2~

)1/2[

qk(τ) +ipk(τ)

mωk

]

where we choose qk(τ) = q−k(τ) and pk(τ) = p−k(τ), show that the path integralcan be cast in the Hamiltonian form of the Feynman path integral.

(b) Integrating over the fields pk(τ), obtain the Lagrangian form of the path integralfor the quantum partition function. Returning to the real space representation, showthat the corresponding effective action takes the form

Z =

∫

Dq(x) exp

[

−

∫ β

0

dτ

∫ L

0

dx

(

1

2mq2 +

1

2ksa

2(∂xq)2

)]

.

——————————————–

2. Zero temperature gap equation: In our analysis of the field theory of the super-conductor, by taking the order parameter ∆ to be small, we restricted the validity ofthe field theory to vicinity of the critical temperature Tc. However, at zero temper-ature, we can obtain an estimate of the magnitude of the order parameter withoutresort to perturbative expansion.


5.5. PROBLEM SET 185

In the mean-field approximation, i.e. taking ∆ constant in space and time, showthat the effective action for the BCS superconductor at zero temperature can beexpressed in the form,22

S

βLd= −

∫

d3k

(2π)3

∫

dω

2πtr ln

(

−iω + ǫk − µ ∆∆∗ −iω − ǫk + µ

)

+|∆|2

g.

Setting ξk = ǫk − µ, show that the action can be rewritten as

S

βLd≃ −ν(µ)

∫

dξ

∫

dω

2πln

(

ω2 + |∆|2 + ξ2)

+|∆|2

g.

where ν(µ) denotes the density of states at the Fermi level. Minimising the actionwith respect to ∆, obtain an expression for the gap equation. For ωD ≫ |∆|, whereωD denotes the Debye frequency cut-off on the ω integral, show that

|∆| = 2ωD exp

[

−1

gν(µ)

]

.

——————————————–

3. Mean-Field Theory of the BCS Hamiltonian: This question involves the in-vestigation of the BCS Hamiltonian within the so-called mean-field approximation.It is also serves as useful revision of methods of second quantisation.

In the mean-field approximation, the microscopic Bogoluibov Hamiltonian can beexpressed in second quantised form

H − µN =∑

k

[

∑

σ

ξkc†kσckσ −

(

∆∗c−k↓ck↑ + ∆c†k↑c

†−k↓

)

]

+|∆|2

g,

where ξk = ǫk−µ and the complex order parameter is uniform and taken to be deter-mined self-consistently as ∆ = g

∑

k〈g.s.|c−k↓ck↑|g.s.〉 and ∆∗ = g

∑

k〈g.s.|c†

k↑c†−k↓|g.s.〉.

In the Nambu spinor representation

Ψ†k

= ( c†k↑ c−k↓ ) , Ψk =

(

ck↑

c†−k↓

)

,

the Hamiltonian can be recast in the form

H =∑

k

[

Ψ†k

(

ξk −∆−∆∗ −ξk

)

Ψk

+ ξk

]

+|∆|2

g.

(a) Taking the order parameter ∆ to be real, show that the quadratic Hamiltoniancan be diagonalised by a unitary transformation and gives

H =∑

kσ

λkα†kσαkσ +

∑

k

(ξk − λk) +∆2

g,

22Note that

1Ld

∑

k

L→∞

=∫

ddk(2π)d and

1β

∑

ωm

β→∞

=∫

dω2π



where(

α†k↑

α−k↓

)

=

(

cos θk sin θksin θk − cos θk

) (

c†k↑

c−k↓

)

and λk = (ξ2

k+ ∆2)1/2.

The elementary excitations of the superconducting system are known as Bogolui-

bons and are created by the operators α†k↑ and α†

−k↓. The result above show thatthe quasi-particles have a minimum energy gap of ∆.

(b) In terms of quasi-particle operators, the ground state of the system can bewritten in the form |g.s.〉 =

∏

kαk↑α−k↓|Ω〉, where |Ω〉 represents the vacuum state

— i.e. |g.s.〉 is annihilated by all quasi-particle operators αk↑ and α−k↓. Expanding,show that, up to a constant factor,

|g.s.〉 =∏

k

α−k↓αk↑|Ω〉 ∼∏

k

(


†−k↓

)

|Ω〉.

With this choice, confirm that this state is normalised. Obtain an expression forthe ground state energy.

(c) Expanding the bare electron operators in terms of the quasi-particle operators,or Bogoluibons, show that the self-consistency equation ∆ = g

∑

k〈g.s.|c

k↓ck↑|g.s.〉results in a self-consistent equation for ∆ — the BCS Gap equation.

——————————————–


5.5. PROBLEM SET 187

5.5.2 Answers

1. (a) Substituting the change of field variables, and rearranging some terms by inte-grating by parts, one obtains

Z =

∫

D(pk, qk) exp

[

−

∫ β

0

dτ∑

k

(

pkp−k

2m+

1

2mω2qkq−k −

i

~p−k∂τqk

)

]

.

With the change of variables τ = it/~, one obtains the Hamiltonian formulation ofthe path integral.

(b) Performing the Gaussian integral over the fields pk, one obtains

Z =

∫

Dqk exp

[

−

∫ β

0

dτ∑

k

(

1

2~2m|qk|

2 +1

2mω2

k|qk|2

)

]

.

With ω2

k = k2ksa2/m, turning to the real space representation, one obtains the

required result.——————————————–

2. Starting with the expression for the action given in the question, differentiating Swith respect to ∆ one obtains

1

βL3

dS

d∆= −ν(µ)

∫

dξ

∫

dω

2π

∆∗

ω2 + |∆|2 + ξ2+

∆∗

g.

Although the integral is formally divergent, recalling that the microscopic mech-anism derives from the phonon-mediated interaction, we introduce a high energycut-off at the Debye frequency, ωD. As a result one obtains

1

βL3

dS

d∆= −

ν(µ)

2

∫ ωD

−ωD

∆∗dξ

(ξ2 + |∆|2)1/2+

∆∗

g= −ν(µ)∆∗ sinh−1

(

ωD

|∆|

)

+∆∗

g

≃ −∆∗

[

ν(µ) ln

(

ωD

|∆|

)

−1

g

]

.

Setting dS/d∆ = 0, we obtain the zero temperature gap equation.——————————————–

3. (a) Defining the unitary transformation,

(

α†k↑

α−k↑

)

=

(

cos θk sin θk

sin θk − cos θk

) (

c†k↑

c−k↓

)

,

the quadratic Hamiltonian is diagonalised when tan(2θk) = −∆/ξk



(b) Taking the definition from the question

〈g.s.|g.s.〉 = 〈Ω|∏

k

(cos θk − sin θkc−k↓ck↑)(cos θk − sin θkc†k↑c

†−k↓)|Ω〉

=∏

k

(

cos2 θk + sin2 θk

)

〈Ω|Ω〉.

The corresponding ground state energy is given by

〈g.s.|H|g.s.〉 =∑

k

(ξk − λk) +∆2

g.

(c) Making use of the relation between the bare electron operators and the quasi-particle operators, the self-consistency equation yields

∆ =⟨

Ω∣

∣

∣

(

cos θk − sin θkc−k↓ck↑

)

c−k↓ck↑

(


†−k↓

)∣

∣

∣Ω

⟩

= −g∑

k

sin θk cos θk =g

2

∑

k

∆

(∆2 + ξ2

k)1/2

≃g∆

2ν(µ)

∫ ωD

−ωD

dξ

(∆2 + ξ2)1/2.

From this result we obtain

gν(µ) ln

(

2ωD

∆

)

= 1, ∆ = 2ωD exp

[

−1

gν(µ)

]

.

——————————————–


Con epts in Theoreti al Physi s

Con epts in Theoreti al Physi s

. Aim of ourse: to develop methods of

many-body & (low-energy) quantum eld theory

motivated by appli ations!

. Synopsis

. Relation to QFT & other Part III ourses?

. Prerequistes? (TPs et .)

. Le ture notes, problem sets & books

ourse web-page:

http://www.t m.phy. am.a .uk/~ bds10/

. Supervisions

Ben Simons

Room 518 (Mott)

e-mail: bds10 am.a .uk

1

How an we develop a

Quantum Theory of \Dense" Matter?

the \quantum many-body problem"...

. First prin iples?

a ording to some,

we already have a Theory of Everything...

...but learly the ability to redu e everything to

simple fundamental laws does not imply the ability

to start form those laws and re onstru t the

universe!

. Perturbation theory about non-intera ting

(i.e. free-parti le) referen e state?

ee tive, but limited in s ope...

onsider ee t of intera tions...

2

Robert Laughlin (Nobel le ture)

3

How do parti le intera tions in uen e properties

of quantum many-parti le system?

. Non-intera ting system ; ideal gas laws

e.g. free ele tron theory, Debye theory, et .

. Weak intera tion

...relies on prin iple of \adiabati ontinuity":

λ

0 1

labels of wavefun tions (quantum nos.)

1

more robust than wavefun tions themselves

; \quasi-parti le orresponden e"...

1

...whi h depend only on fundamental symmetries

(translation, rotation, et .) and dimensionality

4

How do parti le intera tions in uen e properties

of quantum many-parti le system?

. Strong intera tion ; symmetry breaking

(i.e. transitions to new phases of matter)

e.g. rystal, magnet, super uid...

ea h phase hara terised by

new \ olle tive ex itations"

2

e.g. ...

2

i.e. parti le-like ex itations

involving the olle tive motion of many elementary parti les

5

Broken symmetry & olle tive modes

I. Crystal: broken translational symmetry

; latti e vibrations (phonons)

ks

(n+1)ana(n-1)a

xn-1 φnm

II. Magnet: broken spin rotational symmetry

; spin waves (magnons)

III. Super uid | broken gauge symmetry

; phase mode

IV. Quark plasma | strong intera tion

broken gauge symmetry ; hadrons

et ., et .

...properties of olle tive ex itations usually

very dierent from elementary bare parti les

6

Heirar hi al view of matter

1

. Ea h phase of matter hara terised

and lassied by fundamental symmetries

e.g. translation, rotation, spin, gauge,...

. Within ea h phase, u tuations hara terised by

(parti le-like) ex itations, quasi-parti les, whose

properties mirror those of the free theory

e.g. ele tron/hole quasi-parti les,...

. Transitions between phases signalled by symmetry

breaking ; new olle tive quasi-parti le ex itations

(usually very dierent from the elementary bare

parti les)

e.g. fra tional quantum Hall liquid,...

This heirar hy lends itself to the ommon

language of quantum eld theory...

1

...familiar from high energy physi s

7

correspondenceQuasi-particle

Phase transitionBroken symmetry:

collective excitations

rearrangement of g.s. appearance of new

n.

n.

weak int

strong int

Hig

h en

ergy

phy

sics

involving "elementary"bare particles

Free field theory

Con

dens

ed m

atte

r ph

ysic

s

8

Quantum Theory of Matter

I. Mi ros opi Theory

(a) Many-parti le Hamiltonian...

en odes fundamental parti le intera tions

(b) ... ast as quantum eld theory

via quantum path or eld integral

( ) Mean-eld theory

broken symmetries ; global phase stru ture

(d) Low-energy u tuations

; quantum eld theory of olle tive ex itations

II. Phenomenology

Fundamental symmetries:

onstrain low-energy ee tive eld theory

i.e. Universality!

9

Aim of Course

...to develop basi theoreti al ma hinery

to address quantum many-body physi s:

. Method of se ond quantisation:

language of many-parti le quantum me hani s

. Fun tional eld integral methods

(a) Feynman path integral

(b) Coherent state path integral

. [Criti al phenomena

& methods of statisti al eld theory

10

Synopsis

. Colle tive ex itations | from parti les to elds: Linear

harmoni hain & free s alar eld theory; lassi al olle tive

modes, phonons; quantising the lassi al eld; fun tional

analysis [3

. Method of se ond quantisation: reation operators;

appli ations to phonons, Mott-Hubbard transition & quantum

magnetism; spin wave theory;

y

antiferromagnetism & weakly

intera ting Bose gas [5

. Fun tional methods: Feynman path integral; Fun tional

integration & saddle-point analyses; semi- lassi s, statisti al

me hani s; single & double well, instantons & tunnelling;

appli ations to soft matter & quantum fri tion. [5

. Fun tional eld theory: Grassmann Algebra; oherent state

path integral; quantum partition fun tion; Cooper instability

& BCS theory; Ginzburg-Landau & statisti al eld theory;

Gauge theory, Higgs me hanism & super ondu tivity. [5

. Relativisti quantum me hani s: symmetries &

y

Lie groups;

Klein-Gordon & Dira equation; free parti le; Klein paradox;

antiparti les; spin; oupling to lassi al EM eld;

y

parti les,

intera tions & gauge theories. [6

11

Super ondu tivity: A Quantum Phase

Transition of the Ele tron Gas

e e

. Ex hange of latti e vibrations (phonons) between ele trons

; (non-lo al) pairing intera tion

ee

ee

ee

eeee

eeee

. At low T , ele trons \ ondense" in pairs

f. Bose-Einstein ondensation

12

. Phase transition signalled by development of order parameter:

h

"

#

i = jje

i

possessing both amplitude & phase

Re

F

Im

[∆]

[∆][∆]

Tc

|∆|

. Breaking of (gauge) symmetry...

( f. ferromagnet, rystal, et .)

...a ompanied by low-energy olle tive phase u tuations

( f. spin waves, phonons, et .)

. Intera tion of phase eld with EM-eld

; photon mass(!) & super ondu tivity

( f. Higgs me hanism in ele troweak theory)

13

Consequen es

. Super ondu tivity:

H. K. Onnes, Commun. Phys. Lab. 12, 120 (1911)

. Meissner Ee t:

perfe t diamagneti response; ex lusion of magneti eld

14

Topologi al Ex itations

. Topologi al defe ts: vorti es ( f. dislo ations)

Vortex latti e

. Topologi al Phase Transition in Thin-lms

vortex plasma

dipole gas

( f. 2d melting)

15

From Mi ros opi s to Quantum Fields

e e

ee

ee

ee

eeee

eeee

1. Many-body Hamiltonian:

\se ond quantization"

2. Quantum eld integral:

oherent state path integral

3. Mean-eld: symmetry breaking

& BCS transition

4. Flu tuations: gauge eld theory

5. Topologi al phase transition

16

Coulomb Intera tion

e e

. Ele troni Phases of Matter:

Landau Fermi-liquid

Wigner rystal

Mott Insulator

Quantum magnetism & spin liquids

Quantum Hall uids

Heavy fermion super ondu tor

Luttinger liquid

Ex iton insulator & ondensate

17

Books

. Quantum Many-body theory

R. P. Feynman, Statisti al Me hani s, Benjamin, New York,

(1972).

N. Nagaosa, Quantum Field Theory in Condensed Matter

Physi s, Sringer 1999.

J. W. Negele and H. Orland, Quantum Many-Parti le

Systems, Addison-Wesley Publishing, 1988.

. Fun tional Field Integral methods

A. Das, Field Theory: A Path Integral Approa h, World

S ienti Publishing, (1993).

R. P. Feynman and A. R. Hibbs, Quantum Me hani s and

Path Integrals, M Graw-Hill, New York, (1965).

L. S. S hulman, Te hniques and Appli ations of Path

Integration, John Wiley & Sons, 1981.

. Relativisti Quantum Me hani s

J. D. Bjorken and S. D. Drell, Relativisti Quantum Me hani s,

M Graw-Hill (1964).

L. H. Ryder, Quantum Field Theory, Cambridge University

Press, 1996.

18

Lecture II 1

Lecture II: Collective Excitations: From Particles to Fields

Free Scalar Field Theory: Phonons

The aim of this course is to develop the machinery to explore the properties of quantum

systems with very large or infinite numbers of degrees of freedom. To represent such systems it

is convenient to abandon the language of individual elementary particles and speak about quantum

fields. In this lecture, we will consider the simplest physical example of a free or non-interacting

many-particle theory theory which will exemplify the language of classical and quantum fields.

Our starting point is a toy model of a mechanical system describing a classical chain of atoms

coupled by springs.

⊲ Discrete elastic chain

ks

(n+1)ana(n-1)a

xn-1 φnm

Equilibrium position xn ≡ na; natural length a; spring constant ks

Goal: to construct and quantise a classical field theoryfor the collective (longitundinal) vibrational modes of the chain

⊲ Discrete Classical Lagrangian:

L = T − V =

N∑

n=1

(

k.e.︷︸︸︷m

2x2

n −

p.e. in spring︷︸︸︷

ks

2(xn+1 − xn − a)2

)

assume periodic boundary conditions (p.b.c.) xN+1 = Na + x1 (and set xn ≡ ∂txn)

Using displacement from equilibrium φn = xn − xn

L =

N∑

n=1

(m

2φ2

n −ks

2(φn+1 − φn)2

)

, p.b.c : φN+1 ≡ φ1

In principle, one can obtain exact solution of discrete equation of motion — see PS I

However, typically, one is not concerned with behaviour on ‘atomic’ scales:

1. for such purposes, modelling is too primitive!anharmonic contributions

2. such properties are in any case ‘non-universal’

Aim here is to describe low-energy collective behaviour — generic, i.e. universal

In this case, it is often permissible to neglect the discreteness of the microscopic entities

of the system and to describe it in terms of effective continuum degrees of freedom.

Lecture Notes October 2005

Lecture II 2

φn φ(x)Continuum Limit

(n-1)a na (n+1)a

⊲ Continuum Lagrangian

Describe φn as a smooth function φ(x) of a continuous variable x;makes sense if φn+1 − φn ≪ a (i.e. gradients small)

φn → a1/2φ(x)

∣∣∣∣x=na

, φn+1 − φn → a3/2∂xφ(x)

∣∣∣∣x=na

,∑

n

−→1

a

∫ L=Na

0

dx

N.B. [φ(x)] = L1/2

Lagrangian functional︷︸︸︷

L[φ] =

∫ L

0

dx L(φ, ∂xφ, φ),

Lagrangian density︷︸︸︷

L(φ, ∂xφ, φ) =m

2φ2 −

ksa2

2(∂xφ)2

N.B. [L] = [energy]/[length]⊲ Classical action

S[φ] =

∫

dt L[φ] =

∫

dt

∫ L

0

dx L(φ, ∂xφ, φ)

• N -point particle degrees of freedom 7→ continuous classical field φ(x)

• Dynamics of φ(x) specified by functionals L[φ] and S[φ]

What are the corresponding equations of motion...?——————————————–

⊲ Hamilton’s Extremal Principle (HEP): (Revision)

Suppose classical point particle x(t) described by action S[x] =

∫

dt L(x, x)

HEP: configurations x(t) that are realised are those that extremise the actioni.e. for any smooth function η(t), the “variation”

δS[x] ≡ limǫ→0

1

ǫ(S[x + ǫη] − S[x]) = 0

to 1st order in ǫ, action must by stationary


Lecture II 3

x

t

x(t)

(t)εη

Extremal condition 7→ Euler-Lagrange equations of motion

S[x + ǫη] =

∫ t

0

dt L(x + ǫη, x + ǫη) =

∫ t

0

dt (L(x, x) + ǫη∂xL + ǫη∂xL) + O(ǫ2)

δS[x] =

∫

dt (η∂xL + η∂xL)by parts

=

∫

dt

= 0︷︸︸︷(

∂xL −d

dt(∂xL)

)

η = 0

Note: boundary terms vanish by construction

(The variationδL

δx= ∂xL −

d

dt(∂xL) is an example of functional differentiation.

A formal discussion of its legitimacy is included in the lecture notes.)

How does HEP generalise to continuum field x 7→ φ(x)?

x

t L

T

φ

εη

(x,t)

(x,t)

φ

Apply same extremal principle: φ(x, t) 7→ φ(x, t) + ǫη(x, t)with both φ and η both periodic in x

S[φ + ǫη] = S[φ] + ǫ

∫ t

0

dt

∫ L

0

dx(

mφη − ksa2∂xφ∂xη

)

+ O(ǫ2).

Integrating by parts boundary terms vanish by construction: ηφ|t0 = 0 = η∂xφ|L0

δS = −

∫ t

0

dt

∫ L

0

dx(

mφ − ksa2∂2

xφ)

η = 0

Since η(x, t) is an arbitrary smooth function, we must have

(m∂2

t − ksa2∂2

x

)φ = 0


Lecture II 4

-Φ Φ+ x=vtx=-vt

i.e. φ(x, t) obeys classical wave equation

General solutions are of the form: φ+(x + vt) + φ−(x − vt)where v = a

√

ks/m is sound wave velocity and φ± are arbitrary smooth functions

More generally, for the Lagrangian density L = L(φ, ∂xφ, φ),

∂L

∂φ−

d

dt

∂L

∂φ−

d

dx

∂L

∂(∂xφ)= 0

——————————————–

⊲ Comments

• Low-energy collective excitations — known as phonons — are lattice vibrationspropagating as sound waves to left or right at constant velocity v

• Trivial behaviour of model is a direct consequence of its simplistic definition:

Lagrangian is quadratic in fields 7→ linear equation of motion

Higher order couplings (i.e. interactions) 7→ dissipation and dispersion

• L is said to be a ‘free (i.e. non-interacting) scalar (i.e. one-component) field theory’

• In higher dimensions, field has vector components7→ transverse and longintudinal modes

Variational principle is example of functional analysis — a useful (but not essentialconcept for this course) — see lecture notes


Lecture III 5

Lecture III: Quantising the Classical Field

Having established that the low energy properties of the atomic chain are represented by afree scalar classical field theory, we now turn to the formulation of the quantum system.

⊲ Canonical Quantisation procedure: recall point particle mechanics

1. Define canonical momentum p = ∂xL

2. Construct Hamiltonian H = px − L(p, x)

3. Promote position and momentum to operators with canonical commutation relations

x 7→ x, p 7→ p, [p, x] = −i~, H 7→ H

1. Canonical momentum: natural generalisation to continuous field

π(x) ≡∂L

∂φ(x)

applied to atomic chain, π = ∂φ(mφ2/2) = mφ

2. Classical Hamiltonian:

H [φ, π] ≡

∫

dx

Hamiltonian density H(φ, π)︷︸︸︷[

πφ −L(∂xφ, φ)]

i.e. H(φ, π) =1

2mπ2 +

ksa2

2(∂xφ)2

3. Canonical Quantisation:

(a) promote φ(x) and π(x) to operators: φ 7→ φ, π 7→ π

(b) generalise the canonical commutation relations

[π(x), φ(x′)] = −i~δ(x − x′)

N.B. [δ(x − x′)] = [Length]−1 (exercise)

Operator-valued functions φ and π referred to as quantum fields

Comments: H represents a quantum field theoretical formulation of elastic chain, butnot yet a solution. In fact, the development of a variety of methods for the analysis ofquantum field theoretical models will represent major part of course. Here, objective ismerely to exemplify how physical information can be extracted from this particular model.


Lecture III 6

As with any fn, operator-valued fns. can be expressed as Fourier series expansion:

φ(x)π(x)

=1

L1/2

∑

k

e±ikx

φk

πk

,

φk

πk

≡1

L1/2

∫ L=Na

0

dx e∓ikx

φ(x)π(x)

∑

k runs over all quantised wavevectors k = 2πm/L, m ∈ Z

Exercise: confirm [πk, φk′] = −i~δkk′

Advice: Maintain strict conventions(!) — we will pass freely between real and Fourierspace (and we will not care to write a tilde in each case).

Hermiticity: φ†(x) = φ(x), implies φ†k = φ−k (similarly π). Using

∫ L

0

dx (∂φ)2 =∑

k,k′

(ikφk)(ik′φk′)

δk+k′,0︷︸︸︷

1

L

∫ L

0

dx ei(k+k′)x=∑

k

k2φkφ−k

(

=∑

k

k2|φk|2

)

H =∑

k

[ 1

2mπkπ−k+

mω2k/2

︷︸︸︷

ksa2

2k2 φkφ−k

]

ωk = v|k|, v = a(ks/m)1/2

In Fourier representation, ‘modes k’ decoupled

Comments:

• H provides explicit description of the low energy excitations of the system (waves)in terms of their microscopic constituents (atoms)

• However, it would be much more desirable to develop a picture where therelevant excitations appear as fundamental units...

to learn how, noting the structural similarity, let us digress and discuss/revise the...

⊲ Quantum Harmonic Oscillator (Revisited)

H =p2

2m+

1

2mω2q2

Although a single-particle problem, its property of equidistant

energy level separation, ǫn = ~ω(n + 1

2

)suggests alternative interpretation:

State with energy ǫn can be viewed as an “assembly” of n elementary, structureless (i.e.the only ‘quantum number’ is their energy ~ω), bosonic particles (state can be multiplyoccupied) each having an energy ~ω


Lecture III 7

ω

⊲ Formally, defining the ladder operators

a ≡

√mω

2~

(

x +i

mωp

)

, a† ≡

√mω

2~

(

x −i

mωp

)

canonical commutation relation [a, a†] = 1 (characteristic of bosons)

H = ~ω

(

a†a +1

2

)

If we find state |0〉: a|0〉 = 0 H|0〉 = ~ω2|0〉, i.e. |0〉 provides ground state

Using commutation relations, one may show |n〉 ≡1

(n!)1/2a†n|0〉

is (normalised) eigenstate with eigenvalue ~ω(n + 12)

Comments: a-representation affords a many-particle interpretation

• |0〉 represents ‘vacuum’, i.e. state with no particles

• a†|0〉 represents state with a single particle of energy ~ω

• a†n|0〉 is many-body state with n particles

i.e. a† is an operator that creates particles

• In ‘diagonal’ form H = ~ω(a†a + 12) simply counts number of particles,

i.e. a†a|n〉 = n|n〉, and assigns an energy ~ω to each

⊲ Returning to quantum harmonic chain, let us then introduce new representation:

ak ≡

√mωk

2~

(

φk +i

mωk

π−k

)

, a†k ≡

√mωk

2~

(

φ−k −i

mωk

πk

)

N.B. By convention, drop hat from operators a

with [ak, a†k′] =

i

2~

(−i~δkk′

︷︸︸︷

[π−k, φ−k′] −[φk, πk′])

= δkk′

i.e. bosonic commutation relations

⊲ And obtain (exercise — PS I)

H =∑

k

~ωk

(

a†kak +

1

2

)


Lecture III 8

Elementary collective excitations of quantum chain (phonons)

created/annihilated by bosonic operators a†k and ak

Spectrum of excitations is linear ωk = v|k| (cf. relativistic)

Lessons:

• Low-energy excitations of discrete model involve slowly varying collective modes;i.e. each mode involves many atoms

• Low-energy (k → 0) 7→ long-wavelength excitations— i.e. universal, insensitive to microscopic detail;

• This fact allows many different systems to be mapped onto a few (hopefully simple)classical field theories;

• Canonical quantisation procedure for point mechanics generalises toquantum field theory;

• Simplest model actions (such as the one considered here) are quadratic in the fields— known as free field theory;

• More generally, interactions non-linear eqs. of motionand interacting quantum field theories

⊲ Other examples? †Quantum Electrodynamics

EM field — specified by 4-vector potential A(x) = (φ(x),A(x)) (c = 1)

Classical action : S[A] =

∫

d4x L(A), L = −1

4FµνF

µν

Fµν = ∂µAν − ∂νAµ — EM field tensor

Classical equation of motion:

Euler − Lagrange eqns.︷︸︸︷

∂AαL − ∂β ∂L

∂(∂βAα)= 0 7→

Maxwell′s eqns.︷︸︸︷

∂αF αβ = 0

Quantisation of classical field theory identifies elementary excitations: photons

for more details, see handout, or go to QFT!


Lecture IV 9

Lecture IV: Second Quantisation

We have seen how the elementary excitations of the quantum chain can be presented

in terms of new elementary quasi-particles by the ladder operator formalism. Can this

approach be generalised to accommodate other many-body systems? The answer is provided

by the method of second quantisation — an essential tool for the development of interacting

many-body field theories. The first part of this section is devoted largely to formalism —

the second part to applications aimed at developing fluency.

Reference: see Feynman’s book on “Statistical Mechanics”

⊲ Notations and Definitions

Consider a single-particle Schrodinger equation:

H|ψλ〉 = ǫλ|ψλ〉

How can one construct a many-body wavefunction?Particle indistinguishability demands symmetrisation:

nλ nλ

0

1

1

1

0

1

2

3

ε λ

ε 4

ε 2

ε 3

ε 1

Fermions Bosons

ε 0

e.g. two-particle wavefunction for fermions i.e. particle 1 in state 1, particle 2...

ψF (x1, x2) ≡1√2(

state 1︷︸︸︷

ψ1 (

particle 1︷︸︸︷x1 )ψ2(x2) − ψ2(x1)ψ1(x2))

In Dirac notation:

|1, 2〉F ≡ 1√2

(|ψ1〉 ⊗ |ψ2〉 − |ψ2〉 ⊗ |ψ1〉)

⊲ General normalised, symmetrised, N -particle wavefunctionof bosons (ζ = +1) or fermions (ζ = −1)

|λ1, λ2, . . . λN 〉 ≡ 1√

N !∏∞

λ=0nλ!

∑

P

ζP |ψλP1〉 ⊗ |ψλP2

〉 . . .⊗ |ψλPN〉

• nλ — no. of particles in state λ(for fermions, Pauli exclusion: nλ = 0, 1, i.e. |λ1, λ2, . . . λN〉 is a Slater determinant)


Lecture IV 10

•∑

P : Summation over N ! permutations of λ1, . . . λNrequired by particle indistinguishability

• Parity P — no. of transpositions of two elements which brings permutation(P1,P2, · · · PN) back to ordered sequence (1, 2, · · ·N)

Evidently, “first quantised” representation looks clumsy!motivates alternative representation...

⊲ Second quantisation

Define vacuum state: |Ω〉, and set of field operators aλ and adjoints a†λ — no hats!

aλ|Ω〉 = 0,1

√∏∞

λ=0nλ!

N∏

i=1

a†λi|Ω〉 = |λ1, λ2, . . . λN〉

cf. bosonic ladder operators for phonons N.B. ambiguity of ordering?

Field operators fulfil commutation relations for bosons (fermions)

[

aλ, a†µ

]

−ζ= δλµ,

[

aλ, aµ

]

−ζ=

[

a†λ, a†µ

]

−ζ= 0

where [A, B]−ζ ≡ AB − ζBA is the commutator (anti-commutator)

• Operator a†λ creates particle in state λ, and aλ annihilates it

• Commutation relations imply Pauli exclusion for fermions: a†λa†λ = 0

• Any N -particle wavefunction can be generated by application of set ofN operators to a unique vacuum state

e.g. |1, 2〉 = a†2a†

1|Ω〉

• Symmetry of wavefunction under particle interchange maintained bycommutation relations of field operators

e.g. |1, 2〉 = a†2a†

1|Ω〉 = ζa†

1a†

2|Ω〉

(So, providing one maintains a consistent ordering convention,the nature of that convention doesn’t matter)

⊲ Fock space: Defining FN to be ‘linear span’ of all N -particle states |λ1, λ2, · · ·λN〉Fock space F is defined as ‘direct sum’ ⊕∞

N=0FN

• General state |φ〉 of the Fock space is linear combination of stateswith any number of particles

• Note that the vacuum state |Ω〉 (sometimes written as |0〉) is distinct from zero!


Lecture IV 11

+ a+a

a

0a

... F0

a

F1F2

⊲ Change of basis:

Using the resolution of identity 1 ≡ ∑

λ |λ〉〈λ|, we have

a†λ|Ω〉

︷︸︸︷

|λ〉 =∑

λ

a†λ|Ω〉︷︸︸︷

|λ〉〈λ|λ〉

i.e. a†λ

=∑

λ

〈λ|λ〉a†λ, and aλ =∑

λ

〈λ|λ〉aλ

E.g. Fourier representation: aλ ≡ ak, aλ ≡ a(x)

a(x) =∑

k

eikx/√L

︷︸︸︷

〈x|k〉 ak, ak =1√L

∫ L

0

dx e−ikxa(x)

⊲ Occupation number operator: nλ = a†λaλ measures no. of particles in state λe.g. (bosons)

a†λaλ(a†λ)

n|Ω〉 = a†λ

1 + a†λaλ︷︸︸︷

aλa†λ (a†λ)

n−1|Ω〉 = (a†λ)n|Ω〉 + (a†λ)

2aλ(a†λ)

n−1|Ω〉 = · · · = n(a†λ)n|Ω〉

Exercise: check for fermions


Lecture V 12

Lecture V: Second Quantised Representation of Operators

So far we have developed an operator-based formulation of many-particle states. However,

for this representation to be useful, we have to understand how the action of first quantised

operators on many-particle states can be formulated within the framework of the second

quantisation. To do so, it is natural to look for a formulation in the diagonal basis and

recall the action of the particle number operator. To begin, let us consider...

⊲ One-body operators, i.e. operators which address only one particle at a time

O1 =N∑

n=1

on, e.g. k.e. T =N∑

n=1

p2n

2m

• Suppose o diagonal in orthonormal basis |λ〉 e.g. o = p2/2m with |p〉 and op = p2/2mi.e. o =

∑∞

λ=0 |λ〉oλ〈λ|, oλ = 〈λ|o|λ〉

〈λ′1, · · ·λ

′N |O1|λ1, · · ·λN 〉 =

(N∑

i=1

oλi

)

〈λ′1, · · ·λ

′N |λ1, · · ·λN〉

= 〈λ′1, · · ·λ

′N |

∞∑

λ=0

oλnλ|λ1, · · ·λN〉,

Since this holds for any basis state, O1 =∞∑

λ=0

oλnλ =∞∑

λ=0

oλa†λaλ

i.e. in diagonal representation, simply count number of particles in state λand multipy by corresponding eigenvalue of one-body operator

Transforming to general basis (recall aλ =∑

ν〈λ|ν〉aν)

O1 =∑

λµν

〈µ|λ〉oλ〈λ|ν〉a†µaν =

∑

µν

〈µ|o|ν〉a†µaν

i.e. O1 scatters particle from state ν to µ with probability amplitude 〈µ|o|ν〉

⊲ Examples of one-body operators:

1. Total number operator: N =∫

dx a†(x)a(x) =∑

k a†kak

2. Electron spin operator: S =∑

αβ a†αSαβaβ , Sαβ = 〈α|S|β〉 = 1

2σαβ

where α =↑, ↓, and σ are Pauli spin matrices

σz =

(1 00 −1

)

7→ Sz =1

2(n↑ − n↓), σ+ = σx + iσy =

(0 10 0

)

7→ S+ = a†↑a↓


Lecture V 13

3. Free particle Hamiltonian

∑

p

p2

2ma†

papExercise

=

∫ L

0

dx a†(x)(−~

2∂2x)

2ma(x)

i.e. H = T + V =

∫ L

0

dx a†(x)

[p2

2m+ V (x)

]

a(x)

where p = −i~∂x

⊲ Two-body operators, i.e. operators which engage two-particles

E.g. symmetric pairwise interaction: V (x, x′) ≡ V (x′, x) (such as Coulomb)acting between two-particle states

V =1

2

∫

dx

∫

dx′ |x, x′〉V (x, x′)〈x, x′|

When acting on many-particle states,

V |x1, x2, · · ·xN〉 =1

2

N∑

n 6=m

V (xn, xm)|x1, x2, · · ·xN〉

How can one express V in second quantised form?

might guess that

V =1

2

∫

dx

∫

dx′ a†(x)a†(x′)V (x, x′)a(x′)a(x)

i.e. annihilation operators check for presence of particles at x and x’ — if they exist,asign the potential energy and then recreate particles in correct order (viz. statistics).Factor of two for double-counting.

check:

a†(x)a†(x′)a(x′)a(x)|x1, x2, · · ·xN 〉 = a†(x)a†(x′)a(x′)a(x) a†(x1) · · ·a†(xN)|Ω〉

=N∑

n=1

ζn−1δ(x − xn)a†(xn)

n(x′)︷︸︸︷

a†(x′)a(x′) a†(x1) · · ·a†(xn−1)a

†(xn+1) · · ·a†(xN )|Ω〉

=N∑

n=1

ζn−1δ(x − xn)N∑

m(6=n)

δ(x′ − xm)a†(xn) a†(x1) · · ·a†(xn−1)a

†(xn+1) · · ·a†(xN )|Ω〉

=

N∑

n,m6=n

δ(x − xn)δ(x′ − xm)|x1, x2, · · ·xN 〉


Lecture V 14

then multiplying by V (x, x′)/2, and integrate over x and x′ 7→ V

N.B.1

2

∫

dx

∫

dx′ V (x, x′)n(x)n(x′) does not reproduce the two-body operator

⊲ Turning to a non-diagonal basis

O2 =∑

λλ′µµ′

Oµ,µ′,λ,λ′a†µa†

µ′aλaλ′ , Oµ,µ′,λ,λ′ ≡ 〈µ, µ′|O2|λ, λ′〉

⊲ Applications of Second Quantisation

1. Phonons

Oscillator states |k〉 form a Fock space:for each mode k, an arbitrary state of excitation can be created from the vacuum

|k〉 = a†k|Ω〉, ak|Ω〉 = 0, [ak, a

†k′] = δkk′, H =

∑

k

~ωk

(

a†kak + 1/2

)

In this case, the Hamiltonian is diagonal: any state |k1, k2, · · ·〉 = a†k1

a†k2· · · |Ω〉 is an

eigenstate of H with eigenvalue ~ωk1+ ~ωk2

+ · · ·

2. Interacting Electron Gas

(i) Free-electron Hamiltonian

H(0) =∑

kσ

~2k2

2mc†kσckσ, [ckσ, c

†k′σ′ ] = δkk′δσσ′

also diagonal in plain wave basis

(ii) Two-body interactions:

H = H(0) +1

2

∫

dx

∫

dx′∑

σσ′

c†σ(x)c†σ′(x′)V (x − x′)cσ′(x

′)cσ(x)

N.B. off-diagonal in Fourier basis!∑

kk′q

∑

σσ′

V (q) c†kσc†k′σ′ck′+q,σck−q,σ

Feynman diagram:

’

σ’

σ

σV(q)

k−q

k

k’+q

k’

σ


Lecture V 15

⊲ Comments:

• Phonon Hamiltonian is example of ‘free field theory’:involves field operators at quadratic order but no higher...

• (whereas) electron Hamiltonian is typical of an interacting field theory:here there are two-body terms!

• As compared to free theories, analysis of interacting theories is infinitely harder...

⊲ To familiarise ourselves with the second quantisation,in the following lectures we will look at several case studies:

• ‘Atomic limit’ of strongly interacting electron gas:electron crystallisation and Mott transition

• Quantum magnetism

• Weakly interacting Bose gas

——————————————–


Lecture VI 16

Lecture VI: Tight-binding and the Mott transition

According to band picture of non-interacting electrons, a 1/2-filled band of states is metal-lic. But strong Coulomb interaction of electrons can lead to a condensation or crystallisa-tion of the electron gas into a solid, magnetic, insulating phase — Mott transition. Herewe employ the second quantisation to explore the basis of this phenomenon.

⊲ ‘Atomic Limit’ of crystalHow do atomic orbitals broaden into band states? Transparencies

ψ0

ψ1

ε0

ε1

V(x)

x

Es=1

s=0 ε0A

E

ε0B

ε1A

ε1B

(n−1)a

Eπ/ a

E

k0

a

x

(n+1)ana

Weak overlap of tightly bound states 7→ narrow band:Bloch states |ψks〉, band index s, k ∈ [−π/a, π/a]

Bloch states can be used to define †‘Wannier basis’cf. discrete Fourier decomposition

|ψns〉 ≡1√N

B.Z.∑

k∈[−π/a,π/a]

e−ikna|ψks〉, |ψks〉 ≡1√N

N∑

n=1

eikna|ψns〉, k =2π

Nam

ψn0(x)

(n−1)a

x

(n+1)ana

In ‘atomic limit’, Wannier states |ψns〉 mirror atomic orbital |s〉 on site n


Lecture VI 17

Field operators associated with Wannier basis:

c†nsσ|Ω〉︷︸︸︷

|ψns〉 =

∫

dx

c†σ(x)|Ω〉︷︸︸︷

|x〉ψns(x)︷︸︸︷

〈x|ψns〉

c†nsσ ≡∫

dx ψns(x)c†σ(x)

and using completeness (exercise)∑

ns ψ∗ns(x

′)ψns(x) = δ(x− x′)

c†σ(x) =∑

ns

ψ∗ns(x)c

†nsσ, [cnsσ, c

†n′s′σ′ ]+ = δσσ′δnn′δss′

i.e. operators c†nsσ/cnsσ create/annihilate electrons at site n in band s with spin σ

⊲ In atomic limit, bands are well-separated in energy. If electron densities are low,one may project onto lowest band s = 0

Transforming to Wannier basis, interacting electron Hamiltonian takes form

H =∑

mnσ

tmnc†mσcnσ +

∑

mnrsσσ′

Umnrsc†mσc

†nσ′crσ′csσ′

where “hopping” matrix elements: tmn = 〈ψm|H(0)|ψn〉 = t∗nm

and “interaction parameters”

Umnrs =1

2

∫

dx

∫

dx′ψ∗m(x)ψ∗

n(x′)e2

|x− x′|ψr(x′)ψs(x)

(For lowest band) representation is exact:but, in atomic limit, matrix elements decay exponentially with separation

(i) “Tight-binding” approximation:

tmn =

ǫ m = n−t mn neighbours0 otherwise

, H(0) ≃∑

nσ

ǫ c†nσcnσ − t∑

nσ

(

c†n+1σcnσ + h.c.)

In discrete Fourier basis: c†nσ =1√N

B.Z.∑

k∈[−π/a,π/a]

eiknac†kσ

−tN∑

nσ

(

c†n+1σcnσ + h.c.)

= −t∑

kk′σ

δkk′

︷︸︸︷

1

N

∑

n

ei(k−k′)na eikac†kσck′σ + h.c. = −2t∑

kσ

cos(ka) c†kσckσ

H(0) =∑

kσ

(ǫ− 2t cos ka)c†kσckσ

As expected, as k → 0, spectrum becomes free electron-like:ǫk → ǫ− 2t+ t(ka)2 + · · · (with m∗ = 1/2a2t)


Lecture VI 18

π/a

ε (k)

π/a k-

B.Z.

(ii) Interaction

• Focusing on lattice sites m 6= n:

1. Direct terms Umnnm ≡ Vmn — couple to density fluctuations:∑

m6=n Vmnnmnn

potential for charge density wave instabilities

2. Exchange coupling JFmn ≡ Umnmn (exercise — see lecture handout)

∑

m6=n,σσ′

Umnmnc†mσc

†nσ′cmσ′cnσ = −2

∑

m6=n

JFmn

(

Sm · Sn +1

4nmnn

)

, Sm =1

2c†mασαβcmβ

i.e. weak ferromagnetic coupling (JF > 0) cf. Hund’s rule in atoms

spin alignment 7→ symmetric spin state and asymmetric spatial state — lowers p.e.

But, in atomic limit, both tmn and JFmn exponentially small in separation |m− n|a

• ‘On-site’ Coulomb or ‘Hubbard’ interaction∑

nσσ′

Unnnnc†nσc

†nσ′cnσ′cnσ = U

∑

n

nn↑nn↓, U ≡ 2Unnnn

⊲ Minimal model for strong interaction: Hubbard Hamiltonian

H = −t∑

nσ

(c†n+1σcnσ + h.c.) + U∑

n

nn↑nn↓

...could have been guessed on phenomenological grounds

Transparencies on Mott-Insulators and the Magnetic State


Lecture VII 19

Lecture VII: Quantum Magnetism and the Ferromagnetic Chain

Following on from our investigation of the phonon and interacting electron system, we nowturn to another example involving bosonic degrees of freedom — the problem of quantummagnetism.

⊲ Spin S Quantum Heisenberg Magnet spin analogue of discrete harmonic chain

H = −JN∑

m=1

Sm · Sm+1

periodic boundary conditions Sn+N = Sn

N 1 2

Sign of exchange coupling J depends on material parameters: Coulomb interactiontends to favour ferromagnetism J > 0 (cf. Hund’s rule) while “superexchange” processesfavour antiferromagnetism J < 0.

Aim: To uncover the ground states and nature of low-energy (collective) excitations

⊲ Classical ground states?

• Ferromagnet: all spins aligned along a given (arbitrary) directioni.e. manifold of continuous degeneracy (cf. crystal)

• Antiferromagnet: (where possible) all neighbouring spins antiparallel — Neel state

⊲ Quantum ground states:

H = −J∑

m

[

SzmSz

m+1+

1

2(S+

mS−m+1 + S−

mS+m+1)

︷︸︸︷

SxmSx

m+1 + SymSy

m+1

]

where S± = Sx ± iSy denotes spin raising/lowering operator

• Ferromagnet: as classical, e.g. |g.s.〉 = ⊗Nm=1|Sz

m = S〉No spin dynamics in |g.s.〉, i.e. no zero-point energy! (cf. phonons)

Manifold of degeneracy explored by acting total spin lowering operator∑

m S−m

• Antiferromagnet: spin exchange interaction (viz. S+mS−

m+1) zero point fluctuationswhich, depending on dimensionality, may or may not destroy ordered ground state


Lecture VII 20

⊲Elementary excitations?

Formation of magnetically ordered state breaks continuous spin rotation symmetry

low-energy collective excitations (spin waves or magnons) — cf. phonons in a crystal

Example of general principle known as Goldstone’s theorem

However, as with lattice vibrations, ‘general theory’ is nonlinearfortunately, low-energy excitations described by free theory

To see this, for large spin S, it is helpful to switch to a spin representation in whichdeviations from |g.s.〉 are parameterised as bosons:

|Sz = S〉 |n = 0〉|Sz = S − 1〉 |n = 1〉|Sz = S − 2〉 |n = 2〉...

...|Sz = −S〉 |n = 2S〉

i.e. a maximum of n bosons per lattice site (“softcore” constraint)

For ferromagnetic ground state with spins oriented along z-axis,the ground state coincides with the vacuum state |g.s.〉 ≡ |Ω〉, i.e. am|Ω〉 = 0

Mapping useful when elementary spin wave excitation involves n ≪ 2S

⊲ Mapping of operators: Sz, S± = Sx ± iSy?with ~ = 1, operators obey quantum spin algebra

[Sα, Sβ] = iǫαβγ Sγ [S+, S−] = 2Sz, [Sz, S±] = ±S±

cf. bosons: [a, a†] = 1, n = a†a

According to mapping, Sz = S − a†a;therefore, to leading order in S ≫ 1 (spin-wave approximation),

S− ≃ (2S)1/2a†, S+ ≃ (2S)1/2a

In fact, exact equivalence provided by Holstein-Primakoff transformation

S− = a†(2S − a†a

)1/2, S+ = (S−)†, Sz = S − a†a

⊲ Applied to ferromagnetic Heisenberg spin S chain, ‘spin-wave’ approximation:

H = −JN∑

m=1

SzmSz

m+1 +1

2(S+

mS−m+1 + S−

mS+m+1)

= −J∑

m

S2 − S(a†mam − a†

m+1am+1) + S(ama†m+1 + a†

mam+1) + O(S0)

= −J∑

m

S2 − 2Sa†

mam + S(a†

mam+1 + h.c.)

+ O(S0)


Lecture VII 21

with p.b.c. Sm+N = Sm and am+N = am

To leading order in S, Hamiltionian is bilinear in Bose operators;diagonalised by discrete Fourier transform (exercise)

ak =1√N

N∑

n=1

eiknan, an =1√N

B.Z.∑

k

e−iknak, [ak, a†k′] = δkk′

noting∑

n ei(k−k′)n = Nδkk′

H = −JNS2 +B.Z.∑

k

ωka†kak + O(S0), ωk = 2JS(1 − cos k) = 4JS sin2(k/2)

cf. free-particle spectrum

Terms of higher order in S spin-wave interactions


Lecture VIII 22

Lecture VIII: Quantum Antiferromagnetism

We have seen that the quantum Heisenberg Ferromagnetic spin chain is characterised by a

magnetically ordered ground state with free particle-like elementary spin wave excitations

— magnons. What happens in the antiferromagnetic system?

⊲ Antiferromagnet Heisenberg spin S chain

H = J

N∑

m=1

Sm · Sm+1, J > 0, p.b.c. Sm+N = Sm

2N 1

Classical ground state (Neel) no longer an eigenstate— nevertheless, it serves as reference for spin-wave expansion

In this case, it is convenient to implement transformation in which spinson one sublattice, say B, are rotated through 180o about the x-axis,

i.e. SxB 7−→ Sx

B, SyB 7−→ −S

yB, Sz

B 7−→ −SzB

Note that transformation is said to be canonical:i.e. it respects the canonical commutation relations

H = −J∑

m

[

SzmSz

m+1 −1

2(S+

mS+m+1 + S−

mS−m+1)

]

In rotated frame, classical ground state is ferromagneticbut S−

mS−m+1 zero-point fluctuations (ZPF)

Applying spin wave approximation: Szm = S − a†

mam, S−m ≃ (2S)1/2a†

m, etc.

H = −NJS2 + JS∑

m

[

2a†mam + amam+1 + a†

ma†m+1

]

+ O(S0)

processes that do not conserve particle number! (ZPF)

Turning to Fourier representation: am = 1N1/2

∑

k e−ikmak, etc., and using

N∑

m=1

amam+1 =∑

kk′

δk+k′,0︷︸︸︷

1

N

N∑

m=1

e−i(k+k′)m e−ik =∑

k

aka−ke−ik ≡

∑

k

aka−k

γk = cos k︷︸︸︷

1

2(eik + e−ik)


Lecture VIII 23

H = −NJS(S + 1) + JS∑

k

( a†k a−k )

(1 γk

γk 1

) (ak

a†−k

)

+ O(S0)

To diagonalise H, might think of making unitary transformation. However, transfor-mation must preserve canonical commutation relations. Achieved by (exercis, PS II)

⊲ Bogoliubov transformation: (cf. Lorentz boost — preserves metric g = σz)(

ak

a†−k

)

=

(cosh θk − sinh θk

− sinh θk cosh θk

) (αk

α†−k

)

Off-diagonal terms removed by setting tanh(2θk) = γk

H = −NJS(S + 1) + JS∑

k

| sin k|(

α†kαk + α−kα

†−k

)

+ O(S0)

= −NJS(S + 1) + 2JS∑

k

| sin k|

[

α†kαk +

1

2

]

+ O(S0)

linear (cf. relativistic) excitation spectrum (cf. phonons, photons, etc.)

Experiment?

⊲ Average Magnetisation

• Do thermal fluctuations destroy magnetic order in ferromagnet?

〈M〉 = 〈g.s.|1

N

∑

i

Szi |g.s.〉 = S − 〈g.s.|

1

N

∑

i

a†iai|g.s.〉 = S − 〈g.s.|

1

N

∑

k

a†kak|g.s.〉

= S −

∫ddk

(2π)d

nB(T )︷︸︸︷

1

eωk/kBT − 1

kBT≫JS∼ S −

kBT

JS

∫ 1/a

0

kd−1dk1

k2

divergent for T 6= 0 in d ≤ 2

i.e. In d ≤ 2 long-range order destroyed by thermal fluctuations at any non-zerotemperature (example of general principle: Mermin-Wagner theorem)


Lecture VIII 24

• At T = 0, do ZPF destroy long-range order in antiferromagnet?

Referring to sublattice magnetisation (cf. 〈m〉 in rotated frame)

〈Ms.l.〉 = 〈g.s.|1

N

∑

i

Szi |g.s.〉 = S − 〈g.s.|

1

N

∑

k

a†kak|g.s.〉 = S −

1

N

∑

k

sinh2 θk

= S −

∫ddk

(2π)d

1

2

[(1 − γ2

k)−1/2 − 1

]∼

∫ 1/a

0

kd−1dk1

k

diverges in d = 1!

i.e. quantum fluctuations destroy antiferromagnetic order in d = 1 even at T = 0!spin liquid phase

⊲ Frustration

AF exchange interaction on “bipartite” lattice Neel orderingwhich, in d > 1, survives quantum ZPF

For non-bipartite lattice (such as triangular),AF exchange interaction is said to be frustrated...

Can ZPF lead to spin liquid in higher dimensions...? subject of current research!


Lecture IX 25

Lecture IX: Bogoliubov Theory of weakly interacting Bose gas

Although strong interaction effects can lead to the formation of novel ground states of the

electron system, the properties of the weakly interacting system mirror closely the trivial

behaviour of the non-interacting fermi gas. By contrast, even in the weakly interacting

system, the Bose gas has the capacity to form a Bose-Einstein condensed phase. The

aim of this lecture is to explore the nature of the ground state and the character of the

elementary excitation spectrum in the condensed phase.

⊲ Weakly interacting Bose gas

Consider a system of N bosons confined to a volume Ld

In the non-interacting system, at T = 0, all bosons are condensedin the lowest energy single-particle state, viz. |g.s.〉 = 1

N !(a†

0)N |Ω〉

Aim: How is ground state and spectrum of elementary excitations influencedby weak interaction?

H =∑

k

~2k2

2ma†kak+

HI︷︸︸︷

1

2

∫

ddx

∫

ddx′ a†(x)a†(x′)V (x − x′)a(x′)a(x)

V (x − x′) — pairwise particle interaction

In Fourier basis, with a(x) = 1

Ld/2

∑

k e−ik·xak and V (q) =∫

ddx e−iq·xV (x)

HI =1

2Ld

∑

k,k′,q

V (q) a†k+qa

†k′akak′+q

If interaction is weak, in condensed phase, one may assume that the lowest-lyingsingle-particle state is still macroscopically occupied, i.e. N0/N = O(1)

Therefore, since N0 = a†k=0

ak=0 = O(N) ≫ 1 and a0a†0 − a†

0a0 = 1, to a good

approximation, a0 and a†0 can be replaced by the ordinary c-number

√N0

Taking (for simplicity) V (q) = V const., i.e. a contact interaction,expansion in N0 obtains

HI =V

2LdN2

0 +V

LdN0

∑

k 6=0

[

a†kak + a†

−ka−k +1

2

(

a−kak + a†ka

†−k

)]

+ O(N00 )

cf. quantum AF in spin-wave approximation

⊲ Physical interpretation of components:

• V a†kak represents the ‘Hartree-Fock energy’ of excited particles interacting with con-

densate — Note that the contact nature of the interaction disguises the presence of

the direct and exchange contributions


Lecture IX 26

• V (a−kak + a†ka

†−k) represents creation or annihilation of excited particles from the

condensate; Note that, in this approximation, total no. of particles is not conserved

Finally, using the identity N = N0 +∑

k 6=0a†kak to trade N0 for N ,

H =V nN

2+

∑

k 6=0

[(ǫ0k + V n

)a†kak +

V n

2

(

a−kak + a†ka

†−k

)]

where n = N/Ld denotes number density and ǫ0k = ~2k2

2m

As with quantum AF, H diagonalised by Bogoluibov transformation:(

ak

a†−k

)

=

(cosh θk − sinh θk

− sinh θk cosh θk

) (αk

α†−k

)

Left as exercise to show that, when tanh(2θk) = V n/(ǫ0k + V n),

H =V nN

2− 1

2

∑

k 6=0

(ǫ0k + nV ) +

∑

k 6=0

ǫk︷︸︸︷[(

ǫ0k + V n

)2 − (V n)2

]1/2(

α†kαk +

1

2

)

In particular, for |k| → 0, spectrum of low-energy excitations scales as ǫ(k) ≃ ~c|k|with c = (V n/m)1/2

At high energies (k > k0 = mc/~), spectrum becomes free particle-like

⊲ †Ground state wavefunction: defined by condition αk|g.s.〉

Since Bogoluibov transformation can be written as αk = UakU−1 where (exercise)

U = exp

[∑

k 6=0

θk

2(a†

ka†−k − aka−k)

]

may infer g.s. |ΦV 〉 from non-interacting g.s. |Φ0〉 as |ΦV 〉 = U |Φ0〉

Proof: since, for V = 0, all particles are in the k = 0 state,

0 = ak 6=0|Φ0〉 = U−1

αk︷︸︸︷

UakU−1 U |Φ0〉

⊲ †Depletion of condensate due to interaction

N − N0

N=

1

N

∑

k 6=0

〈g.s.|a†kak|g.s.〉 =

1

N

∑

k 6=0

sinh2 θk =1

n

∫ddk

(2π)3sinh2 θk

exercise=

1

3π2nk3

0

i.e. ca. one particle per ‘coherence length’ ξ ∼ 1/k0


Lecture IX 27

Recast using scattering length (cf. TP2) of contact interaction V = 4π~2a/m,

N − N0

N=

8

3√

π(na3)1/2

⊲ †Ground state energy

E0 =V nN

2− 1

2

∑

k 6=0

(ǫ0k + nV − ǫk −

(nV )2

2ǫ0k

)

(where extra term controls unphysical UV divergencerequired by contact nature of potential)

E0

Ld=

n2V

2

[

1 +128

15√

π(na3)1/2

]

⊲ Experiment? transparencies

When cooled to T ∼ 2K, liquid 4He undergoes

transition to Bose-Einstein condensed state

Neutron scattering measurements can be used to infer spectrum of

collective excitations

In Helium, steric interactions are strong and at higher energy scales

an important second branch of excitations known as rotons appear

A second example of BEC is presented by ultracold atomic gases:

By confining atoms to a magnetic trap, time of flight measurements

can be used to monitor momentum distribution of condensate

Moreover, the perturbation imposed by a laser due to the optical

dipole interaction provides a means to measure the sound wave velocity


Lecture IX 28


Lecture X 29

Lecture X: Feynman Path Integral

Although the second quantisation provides a convenient formulation of many-body systems,it admits solution only for systems that are effectively free. In our choice of applications,we were careful to consider only those systems for which interaction effects could be consid-ered as small, e.g. large spin in quantum magnetism of weak interaction in the dilute Bosegas. Yet interactions can have a profound effect leading to transitions to new phases withelementary excitations very different from the bare particles. To address such phenom-ena, it is necessary to switch to a new formulation of quantum mechanics. However, todo so, it will be necessary to leave behind many-body theories and return to single-particlesystems.

⊲ Motivation:

• Alternative formulation of QM (cf. canonical quantisation)

• Close to classical construction — i.e. semi-classics easily retrieved

• Effective formulation of non-perturbative approaches

• Prototype of higher-dimensional field theories

⊲ Time-dependent Schrodinger equation

i~∂t|Ψ〉 = H|Ψ〉

Formal solution: |ψ(t)〉 = e−iHt/~|ψ(0)〉 =∑

n

e−iEnt/~|n〉〈n|ψ(0)〉

⊲ Time-evolution operator

|Ψ(t′)〉 = U(t′, t)|Ψ(t)〉, U(t′, t) = e−i~

H(t′−t)θ(t′ − t) N.B. Causal

• Real-space representation:

Ψ(q′, t′) ≡ 〈q′|Ψ(t′)〉 = 〈q′|U(t′, t)

∫

dq|q〉〈q|∧ |Ψ(t)〉 =

∫

dq U(q′, t′; q, t)Ψ(q, t),

where U(q′, t′; q, t) = 〈q′|e− i~H(t′−t)|q〉θ(t′ − t) — propagator or Green function

(

i~∂t′ − H)

U(t′ − t) = i~δ(t′ − t) N.B. ∂t′θ(t′ − t) = δ(t′ − t)

Physically: U(q′, t′; q, t) describes probability amplitude for particle to propagatefrom q at time t to q′ at time t′

⊲ Construction of Path Integral

Feynman’s idea: separate time evolution into N → ∞ discrete time steps ∆t = t/N

e−iHt/~ = [e−iH∆t/~]N


Lecture X 30

Then separate the operator content so that momentum operators stand to the leftand position operators to the right:

e−iH∆t/~ = e−iT∆t/~e−iV ∆t/~ +O(∆t2)

〈qF |[e−iH∆t/~]N |qI〉 ≃ 〈qF |∧e−iT∆t/~e−iV ∆t/~

∧ . . .∧e−iT∆t/~e−iV ∆t/~|qI〉

Inserting at ∧ resol. of id. =

∫∞

−∞

dqn

∫∞

−∞

dpn|qn〉〈qn|pn〉〈pn|, and using 〈q|p〉 =1√2π~

eiqp/~,

e−iV ∆t/~|qn〉〈qn|pn〉〈pn|e−iT∆t/~ = |qn〉e−iV (qn)∆t/~〈qn|pn〉e−iT (pn)∆t/~〈pn|,

and 〈pn+1|qn〉〈qn|pn〉 =1

2π~eiqn(pn−pn+1)/~


∫ N−1∏

n=1

qN =qF ,q0=qI

dqn

N∏

n=1

dpn

2π~exp

[

− i

~∆t

N−1∑

n=0

(

V (qn) + T (pn+1) − pn+1qn+1 − qn

∆t

)]

q I

qF

tn

p

N1 2

PhaseSpace

t

i.e. at each time step, integration over the classical phase space coords. (qn, pn)

Contributions from trajectories where (qn+1 − qn)pn+1 > ~ are negligible— motivates continuum limit


∫

q(t)=qF ,q(0)=qI

D(q, p)

︷︸︸︷∫ N−1∏

n=1

qN =qF ,q0=qI

dqn

N∏

n=1

dpn

2π~exp

[

− i

~

∫ t

0

dt′

︷︸︸︷

∆tN−1∑

n=0

(

H(q, p|t′=tn)︷︸︸︷

V (qn) + T (pn+1) −

pq|t′=tn︷︸︸︷

pn+1qn+1 − qn

∆t)]

⊲ Hamiltonian formulation of Feynman Path Integral:Propagator expressed as functional integral


∫

q(t)=qF ,q(0)=qI

D(q, p) exp[ i

~

Action︷︸︸︷

∫ t

0

dt′

Lagrangian︷︸︸︷

(pq −H(p, q))]


Lecture X 31

Quantum transition amplitude expressed as sum over all possible phase spacetrajectories (subject to appropriate b.c.) and weighted by classical action

⊲ Lagrangian formulation: for “free-particle” Hamiltonian H(p, q) = p2/2m+ V (q)


∫

q(t)=qF ,q(0)=qI

Dq e−(i/~)R t

0dt′V (q)

∫

Dp

Gaussian integral on p︷︸︸︷

exp

[

− i

~

∫ t

0

dt′(p2

2m− pq

)]

p2

2m− pq 7→ 1

2m

p′2

︷︸︸︷

(p−mq)2 −1

2mq2

Functional integral justified by discretisation


∫

q(t)=qF ,q(0)=qI

Dq exp

[i

~

∫ t

0

dt′(mq2

2− V (q)

)]

Dq ≡ limN→∞

(Nm

it2π~

)N/2 N−1∏

n=1

dqn


Lecture XI 32

Lecture XI: Statistical Mechanics and Semi-Classics

⊲ Connection of Path Integral to Classical Statistical Mechanics

Consider flexible string held under constant tension and confined to ‘gutter’ potential

x

u

V(u)

Potential energy stored in spring due to line tension:

from segment x to x + dx, dVT = T

extension︷︸︸︷

[(dx2 + du2)1/2 − dx]≃ T dx (∂xu)2/2

VT [∂xu] ≡∫

dVT =1

2

∫ L

0

dx T (∂xu(x))2

External (gutter) potential: Vext[u] ≡∫ L

0dx V [u(x)]

According to Boltzmann principle, equilibrium partition function

Z = tr(e−βF

)=

∫

Du(x) exp

[

−β

∫ L

0

dx

(T

2(∂xu)2 + V (u)

)]

cf. quantum mechanical transmission amplitude⊲ Mapping:

Z =

∫

b.c.

Dq(t) exp

[i

~

∫ t

0

dt′(

mq2

2− V (q)

)]

Wick rotation t → −iτ 7→ imaginary (Euclidean) time path integral

∫ t

0

idt′ (∂t′q)2 −→ −

∫ τ

0

dτ ′(∂τ ′q)2, −∫ t

0

idt′V (q) −→ −∫ τ

0

dτ ′V (q)

Z =

∫

b.c.

Dq exp

[

−1

~

∫ τ

0

dτ ′

(m

2(∂τ ′q)2 + V (q)

)]

N.B. change of relative sign!

(a) Classical partition function of one-dimensional systemcoincides with quantum mechanical amplitude

Z =

∫

dq 〈q|e−iHt/~|q〉∣∣∣t=−iτ

where time is imaginary, and ~ plays the role of temperature


Lecture XI 33

More generally, path integral for d-dimensional quantum systemcorresponds to classical statistical mechanics of d + 1-dimensional system

(b) Quantum partition function

Z = tr(e−βH) =

∫

dq 〈q|e−βH|q〉

i.e. Z can be interpreted through dynamical transition amplitude 〈q|e−iHt/~|q〉evaluated at imaginary time t = −i~β.

(c) In semi-classical limit (~ → 0), PI dominated by stationary configurations of actionS[p, q] =

∫dt(pq − H(p, q))

δS = S[p + δp, q + δq] − S[p, q]

=

∫

dt [δpq + pδq − δp∂pH − δq∂qH ] + O(δp2, δq2, δpδq)

=

∫

dt [δp (q − ∂pH) + δq (−p − ∂qH)] + O(δp2, δq2, δpδq)

i.e. Hamilton’s classical e.o.m.: q = ∂pH , p = −∂qH with b.c. q(0) = qI , q(t) = qF

(Similarly, with Lagrangian formulation : δS = 0 ⇒ (dt∂q − ∂q) L(q, q) = 0)

qq

I

qFα h

1/2

t

q

q(t)

Contributions to PI from fluctuations around classical paths?

Usually, exact evaluation of PI impossible — resort to approximation schemes...

⊲ Saddle-point and Stationary Phase analysis

Consider integral over single variable

I =

∫∞

−∞

dz e−f(z)

Integral dominated by minima of f(z); suppose unique i.e. f ′(z0) = 0


Lecture XI 34

Taylor expand around minimum: f(z) = f(z0) + (z − z0)

7→ 0︷︸︸︷

f ′(z0) +1

2(z − z0)

2f ′′(z0) + · · ·

I ≃ e−f(z0)

∫∞

−∞

dz e−(z−z0)2f ′′(z0)/2 =

√

2π

f ′′(z0)e−f(z0)

Example : Γ(s + 1) =

∫∞

0

dzzse−z =

∫∞

0

dz e−f(z), f(z) = z − s ln z

f ′(z) = 1 − s/z i.e. z0 = s, f ′′(z0) = s/z20 = 1/s

i.e. Γ(s + 1) ≃√

2πse−(s−s ln s) — Stirling’s formula

If minima not on contour of integration — deform contour through saddle-pointe.g. Γ(s + 1), s complex

What if exponent complex? Fast phase fluctuations cancellationi.e. expand around region of slowest (i.e. stationary) phase and use identity

∫∞

−∞

dz eiaz2/2 =

√

2π

aeiπ/4

⊲ Can we apply same approach to analyse the FPI?

Yes: but we we must develop new technology;basic tool of QFT — the Gaussian functional integral!


Lecture XII 35

Lecture XII: Applications of the Feynman Path Integral

⊲ Digression: Free particle propagator (exercise) cf. diffusion

Gfree(qF , qI ; t) ≡ 〈qF |e−ip2t/2m~|qI〉Θ(t) =

( m

2πi~t

)1/2

exp

[i

~

m(qF − qI)2

2t

]

Θ(t)

Difficult to derive from PI(!), but useful for normalization

⊲ Quantum Particle in a Single (Symmetric) Well: V (q) = V (−q)

V

q

ω

e.g. QM amplitude

G(0, 0; t) ≡ 〈0|e−iHt/~|0〉Θ(t) =

∫

q(t)=q(0)=0

Dq exp

[i

~

∫ t

0

dt′(

mq2

2− V (q)

)]

⊲ Evaluate PI by stationary phase approximation: general recipe

(i) Parameterise path as q(t) = qcl(t) + r(t) and expand action in r(t)

S[q + r] =

∫ t

0

dt′[m

2

qcl2 + 2qclr + r2

︷︸︸︷

(qcl + r)2 −

V (qcl) + rV ′(qcl) +r2

2V

′′

(qcl) + · · ·︷︸︸︷

V (qcl + r)]

= S[qcl] +

∫ t

0

dt′r(t′)

δS

δq(t′)=0

︷︸︸︷

[−mqcl − V ′(qcl)] +1

2

∫ t

0

dt′r(t′)

δ2S

δq(t′)δq(t′′)︷︸︸︷[−m∂2

t′ − V ′′(qcl)]

r(t′) + · · ·

(ii) Classical trajectory: mqcl = −V ′(qcl)

Many solutions — choose non-singular solution qcl = 0 (why?)i.e. S[qcl] = 0 and V ′′(qcl) = mω2 constant

G(0, 0; t) ≃

∫

r(0)=r(t)=0

Dr exp

[i

~

∫ t

0

dt′r(t′)m

2

(−∂2

t′ − ω2)r(t′)

]

N.B. if V was quadratic, expression trivially exact

More generally, qcl(t) non-trivial 7→ non-vanishing S[qcl] — see PS3

Fluctuation contribution? — example of a...

⊲ Gaussian functional integration: mathematical interlude


Lecture XII 36

• One variable Gaussian integral: (∫∞

−∞dv e−av2/2)2 = 2π

∫∞

0r dr e−ar2/2 = 2π

a

∫∞

−∞

dv e−a2v2

=

√

2π

a, Re a > 0

• More than one variable:∫

dv e−12vT Av = (2π)N/2detA−1/2

where A is +ve definite real symmetric N × N matrix

Proof: A diagonalised by orthogonal transformation: A = OTDO

Change of variables: w = Ov (Jacobian det(O) = 1) N decoupled

Gaussian integrations: vTAv = vTOTOAOTOv = wTDw =∑N

i diw2i

Finally,∏N

i=1 di = detD = detA

• Infinite number of variables; interpret vi 7→ v(t) as continuous field and

Aij 7→ A(t, t′) = 〈t|A|t′〉 as operator kernel∫

Dv(t) exp

[

−1

2

∫

dt

∫

dt′ v(t)A(t, t′)v(t′)

]

∝ (det A)−1/2

(iii) Applied to quantum well, A(t, t′) = − i~mδ(t − t′)(−∂2

t′ − ω2) and formally

G(0, 0; t) ≃ J det(−∂2

t′ − ω2)−1/2

where J absorbs various constant prefactors (im, ~, etc.)

What does ‘det’ mean? Effectively, we have expanded trajectories r(t′)

in eigenbasis of A subject to b.c. r(t) = r(0) = 0(−∂2

t − ω2)rn(t) = ǫnrn(t), cf. PIB

i.e. Fourier series expn: rn(t′) = sin(nπt′

t), n = 1, 2, . . . , ǫn = (nπ

t)2 − ω2

det(−∂2

t − ω2)−1/2

=

∞∏

n=1

ǫ−1/2n =

∞∏

n=1

((nπ

t

)2

− ω2

)−1/2

⊲ For V = 0, G = Gfree known — use to eliminate constant prefactor J

G(0, 0; t) =G(0, 0; t)

Gfree(0, 0; t)Gfree(0, 0; t) =

∞∏

n=1

[

1 −

(ωt

nπ

)2]−1/2

( m

2πi~t

)1/2

Θ(t)

Finally, applying identity∏

∞

n=1[1 − ( xnπ

)2]−1 = xsin x

G(0, 0; t) ≃

√mω

2πi~ sin(ωt)Θ(t)

(exact for harmonic oscillator)


Lecture XIII 37

Lecture XIII: Double Well Potential: Tunneling and Instantons

How can phenomena of QM tunneling be described by Feynman path integral?No semi-classical expansion!

⊲ E.g QM transition probability of particle in double well: G(a,−a; t) ≡ 〈a|e−iHt/~| − a〉

Potential

x

Invertedpotential

V(x)

⊲ Feynman Path Integral:

G(a,−a; t) =

∫ q(t)=a

q(0)=−a

Dq exp

[i

~

∫ t

0

dt′(m

2q2 − V (q)

)]

Stationary phase analysis: classical e.o.m. mq = −∂qV7→ only singular (high energy) solutions Switch to alternative formulation...

⊲ Imaginary (Euclidean) time Path Integral: Wick rotation t = −iτ

N.B. (relative) sign change! “V → −V ”

G(a,−a; τ) =

∫ q(τ)=a

q(0)=−a

Dq exp

[

−1

~

∫ τ

0

dτ ′(m

2q2 + V (q)

)]

Saddle-point analysis: classical e.o.m. mq = +V ′(q) in inverted potential!solutions depend on b.c.

(1) G(a, a; τ) qcl(τ) = a(2) G(−a,−a; τ) qcl(τ) = −a(3) G(a,−a; τ) qcl : rolls from −a to a

Combined with small fluctuations, (1) and (2) recover propagator for single well

Invertedpotential

-a a-V(x)

x

a

-a

x

τ1/ω

(3) accounts for QM tunneling and is known as an “instanton” (or “kink”)


Lecture XIII 38

⊲ Instanton: classically forbidden trajectory connecting two degenerate minima— i.e. topological, and therefore particle-like

For τ large, qcl ≃ 0 (evident), i.e. “first integral” mqcl2/2 − V (qcl) = ǫ

τ→∞→ 0

precise value of ǫ fixed by b.c. (i.e. τ)Saddle-point action (cf. WKB

∫dqp(q))

Sinst. =

∫ τ

0

dτ ′(m

2q2cl + V (qcl)

)

≃

∫ τ

0

dτ ′mq2cl =

∫ a

−a

dqclmqcl =

∫ a

−a

dqcl(2mV (qcl))1/2

Structure of instanton: For q ≃ a, V (q) = 12mω2(q − a)2 + · · ·, i.e. qcl

τ→∞≃ ω(qcl − a)

qcl(τ)τ→∞= a − e−τω, i.e. temporal extension set by ω−1 ≪ τ

Imples existence of approximate saddle-point solutionsinvolving many instantons (and anti-instantons): instanton gas

τ 1

a

-a

x

ττ 5 τ 4 τ 3 τ 2

⊲ Accounting for fluctuations around n-instanton configuration

G(a,±a; τ) ≃∑

n even / odd

Kn

∫ τ

0

dτ1

∫ τ1

0

dτ2 · · ·

∫ τn−1

0

dτn

An,cl.An,qu.︷︸︸︷

An(τ1, . . . , τn),

constant K set by normalisation

An,cl. = e−nSinst./~ — ‘classical’ contribution

An,qu. — quantum fluctuations (imported from single well): Gs.w.(0, 0; t) ∼ 1√sin ωt

An,qu. ∼n∏

i

1√

sin(−iω(τi+1 − τi))∼

n∏

i

e−ω(τi+1−τi)/2 ∼ e−ωτ/2

G(a,±a; τ) ≃∑

n even / odd

Kne−nSinst./~e−ωτ/2

τn/n!︷︸︸︷∫ τ

0

dτ1

∫ τ1

0

dτ2 · · ·

∫ τn−1

0

dτn

=∑

n even / odd

e−ωτ/2 1

n!

(τKe−Sinst./~

)n

Using ex =∑∞

n=0 xn/n!, N.B. non-perturbative in ~!

G(a, a; τ) ≃ Ce−ωτ/2 cosh(τKe−Sinst./~

)

G(a,−a; τ) ≃ Ce−ωτ/2 sinh(τKe−Sinst./~

)


Lecture XIII 39

Consistency check: main contribution from

n = 〈n〉 ≡

∑

n nXn/n!∑

n Xn/n!= X = τKe−Sinst./~

no. per unit time, n/τ exponentially small, and indep. of τ , i.e. dilute gas

hω

S

A

SA

x

Oscillator StatesExact States

Ψ

⊲ Physical interpretation: For infinite barrier — two independent oscillators,coupling splits degeneracy — symmetric/antisymmetric

G(a,±a; τ) ≃ 〈a|S〉e−ǫSτ/~〈S| ± a〉 + 〈a|A〉e−ǫAτ/~〈A| ± a〉

|〈a|S〉|2 = 〈a|S〉〈S| − a〉 =C

2, |〈a|A〉|2 = −〈a|A〉〈A| − a〉 =

C

2

Setting: ǫA/S = ~ω/2± ∆ǫ/2

G(a,±a; τ) ≃C

2

(e−(~ω−∆ǫ)τ/2~ ± e−(~ω+∆ǫ)τ/2~

)= Ce−ωτ/2

cosh(∆ǫτ/~)sinh(∆ǫτ/~)

.

⊲ Remarks:

(i) Legitimacy? How do (neglected) terms O(~2) compare to ∆ǫ?

In fact, such corrections are bigger but act equally on |S〉 and |A〉i.e. ∆ǫ = ~Ke−Sinst./~ is dominant contribution to splitting

V(q)

τ

V(q)

q

−

q qqm

m

(ii) Unstable States and Bounces: survival probability: G(0, 0; t)? No even/odd effect:

G(0, 0; τ) = Ce−ωτ/2 exp[τKe−Sinst/~

] τ=it= Ce−iωt/2 exp

[

−Γ

2t

]

Decay rate: Γ ∼ |K|e−Sinst/~ (i.e. K imaginary) N.B. factor of 2


Lecture XIV 40

Lecture XIV: Coherent States

Discuss element of PI construction which demands generalisation

⊲ Generalisation of FPI to many-body systems problematicdue to particle indistinguishability and statistics

Can second quantisation help? automatically respects particle statistics

Require complete basis on the Fock space to construct PI

Such eigenstates exist and are known as Coherent Statesreference: Negele and Orland

⊲ Coherent States (Bosons)

What are eigenstates of Fock space operators: ai and a†i s.t. [ai, a

†j] = δij?

Being a state of the Fock space, an eigenstate |φ〉 can be expanded as

|φ〉 =∑

n1,n2,···

Cn1,n2,···

(a†1)

n1

√n1

(a†2)

n2

√n2

· · · |0〉

N.B. notation |0〉 for vacuum state!

(i) a†i |φ〉 = φi|φ〉? — in fact, eigenstate of a

†i can not exist:

if the minimum occupation of |φ〉 is n0, the minimum of a†i |φ〉 is n0 + 1

(ii) ai|φ〉 = φi|φ〉? — can exist and given by: |φ〉 ≡ exp[∑

i φia†i ]|0〉 N.B. φ ≡ φi

Proof: since ai commutes with all a†j for j 6= i — focus on one element i

a exp(φa†)|0〉 = [a, exp(φa†)]|0〉 =

=∞

∑

n=0

φn

n![a, (a†)n]|0〉 =

∞∑

n=1

nφn

n!(a†)n−1|0〉 = φ exp(φa†)|0〉

a(a†)n = aa†(a†)n−1 = (1 + a†a)(a†)n−1 = (a†)n−1 + a†a(a†)n−1 = n(a†)n−1 + (a†)na

i.e. |φ〉 is eigenstate of all ai with eigenvalue φi — known as Bosonic coherent state

⊲ Properties of coherent state:

• Hermitian conjugation:

∀i : 〈φ|a†i = 〈φ|φi

φi is complex conjugate of φi


Lecture XIV 41

• By direct application of ∂φi(and operator commutativity):

∀i : a†i |φ〉 = ∂φi

|φ〉

• Overlap: with 〈θ| = (|θ〉)† = 〈0|eP

iθiai

〈θ|φ〉 = 〈0|eP

iθiai |φ〉 = e

P

iθiφi〈0|φ〉 = exp

[

∑

i

θiφi

]

i.e. states are not orthogonal! operators not Hermitian

• Norm: 〈φ|φ〉 = exp

[

∑

i

φiφi

]

• Completeness — resolution of id. (for proof see notes)

∫

∏

i

dφidφi

πe−

P

iφiφi|φ〉〈φ| = 1F

where dφidφi = dRe φidIm φi

⊲ Coherent States (Fermions)

Following bosonic case, seek state |η〉 s.t.

ai|η〉 = ηi|η〉, η = ηi

But anticommutativity [ai, aj ]+ = 0 (i 6= j) implies eigenvalues ηi anticommute!!

ηiηj = −ηjηi

ηi can not be ordinary numbers — in fact, they obey...

⊲ Grassmann Algebra

In addition to anticommutativity, defining properties:

(i) η2i = 0 (cf. fermions) but note: these are not operators, i.e. [ηi, ηi]+ 6= 1

(ii) Elements ηi can be added to and multiplied by ordinary complex numbers

c + ciηi + cjηj, ci, cj ∈ C

(iii) Grassmann numbers anticommute with fermionic creation/annihilation operators[ηi, aj]+ = 0

⊲ Calculus of Grassmann variables:

(iv) Differentiation: ∂ηiηj = δij

N.B. ordering ∂ηiηjηi = −ηj∂ηi

ηi = −ηj for i 6= j


Lecture XIV 42

(v) Integration:∫

dηi = 0,∫

dηiηi = 1i.e. differentiation and integration have the same effect!!

⊲ Gaussian integration:∫

dηdη e−ηaη =

∫

dηdη (1 − ηaη) = a

∫

dηη

∫

dηη = a

∫

dηdη e−ηT Aη = detA (exercise)

cf. ordinary complex variables

⊲ Functions of Grassmann variables:

Taylor expansion terminates at low order since η2 = 0, e.g.

F (η) = F (0) + ηF ′(0)

Using rules∫

dηF (η) =

∫

dη [F (0) + ηF ′(0)] = F ′(0) ≡ ∂ηF [η]

i.e. differentiation and integration have same effect on F [η]!

Usually, one has a function of many variables F [η], say η = η1, · · · ηN

F (η) =∞

∑

n=0

1

n!

∂nF (0)

∂ηi · · ·∂ηj

ηj · · · ηi

but series must terminate at n = N

with these preliminaries we are in a position to introduce the

⊲ Fermionic coherent state: |η〉 = exp[−∑

i ηia†i ]|0〉 i.e. η = ηi

Proof (cf. bosonic case)

a exp(−ηa†)|0〉 = a(1 − ηa†)|0〉 = ηaa†|0〉 = η|0〉 = η exp(−ηa†)|0〉

Other defining properties mirror bosonic CS — problem set

⊲ Differences:

(i) Adjoint: 〈η| = 〈0|e−P

iaiηi ≡ 〈0|e

P

iηiai but N.B. ηi not related to ηi!

(ii) Gaussian integration:

∫

dηdη e−ηη = 1 N.B. no π’s

Completeness relation∫

∏

i

dηidηie−

P

iηiηi|η〉〈η| = 1F


Lecture XV 43

Lecture XV: Many-body (Coherent State) Path Integral

Could formulate many-body propagator (Green function), but here, convenient to focus onpartition function.

⊲ Quantum partition function

Z =∑

n∈Fock Space

〈n|e−β(H−µN)|n〉, β = 1/kBT, µ chemical potential

Coherent state representation of Z — insert resolution of id. (fermions/bosons)

∫

d[ψ, ψ]e−P

i ψiψi|ψ〉〈ψ| = 1F , d[ψ, ψ] ≡∏

i

dψidψiπ(1−ζ)/2

each element ψi associated with one basis state, viz. a†i— e.g. i may include position, momentum, spin, lattice site, etc.

Z =

∫

d[ψ, ψ]e−P

i ψiψi∑

n

〈n|ψ〉〈ψ|e−β(H−µN)|n〉

Elimination of |n〉 requires identity: 〈n|ψ〉〈ψ|n〉 = 〈−ζψ|n〉〈n|ψ〉

Proof: E.g. |n〉 = a†1a†2 · · ·a†n|0〉

〈n|ψ〉 = 〈0|an · · ·a2a1|ψ〉 = ψn · · ·ψ2ψ1〈0|ψ〉 = ψn · · ·ψ2ψ1

〈ψ|n〉 = ψ1ψ2 · · · ψn〈n|ψ〉〈ψ|n〉 = ψn · · ·ψ2ψ1ψ1ψ2 · · · ψn = ψ1ψ1ψ2ψ2 · · ·ψnψn

= (−ζψ1ψ1)(−ζψ2ψ2) · · · (−ζψnψn) = 〈−ζψ|n〉〈n|ψ〉

commute through and erase∑

n |n〉〈n|

Z =

∫

d[ψ, ψ]e−P

i ψiψi〈−ζψ|e−β(H−µN)|ψ〉

⊲ Coherent State Path Integral

Applied to general Hamiltonian

H − µN =∑

ij

(hij − µδij)a†iaj +

∑

ijkl

Vijkla†ia

†jakal

N.B. operators are normal ordered

Follow general strategy of Feynman:


Lecture XV 44

(i) Divide ‘time interval’ β into N segments ∆β = β/N

〈−ζψ|e−β(H−µN)|ψ〉 = 〈−ζψ|e−∆β(H−µN)

∧e−∆β(H−µN)

∧ · · · |ψ〉

(ii) At each ∧ insert resolution of id.

1F =

∫

d[ψn, ψn]e−ψn·ψn|ψn〉〈ψn|

i.e. N-independent sets N.B. each ψn is a vector with elements ψin

(iii) Expand exponent in ∆β

〈ψ′|e−∆β(H−µN)|ψ〉 = 〈ψ′|[

1 − ∆β(H − µN)]

|ψ〉 +O(∆β)2

= 〈ψ′|ψ〉 − ∆β〈ψ′|(H − µN)|ψ〉 +O(∆β)2

= 〈ψ′|ψ〉 [1 − ∆β (H(ψ′, ψ) − µN(ψ′, ψ))] +O(∆β)2

≃ eψ′·ψe−∆β(H(ψ′,ψ)−µN(ψ′,ψ))

with H(ψ′, ψ) =〈ψ′|H|ψ〉〈ψ′|ψ〉 =

∑

ij

hijψ′iψj +

∑

ijkl

Vijklψ′iψ

′jψkψl

similarly N(ψ′, ψ) N.B. 〈ψ′|ψ〉 bilinear in ψ, i.e. commutes with everything

Z =

∫ N∏

n=0ψN=−ζψ0,ψN=−ζψ0

d[ψn, ψn]e−

PNn=1[ψn·(ψn−ψn−1)+∆β(H(ψn,ψn−1)−µN(ψn,ψn−1))]

Continuum limit N → ∞

∆βN

∑

n=0

→∫ β

0

dτ,ψn − ψn−1

∆β→ ∂τψ

∣

∣

∣

τ=n∆β,

N∏

n=0

d[ψn, ψn] → D(ψ, ψ)

comment on “small” Grassmann nos.

Z =

∫

ψ(β)=−ζψ(0)ψ(β)=−ζψ(0)

D(ψ, ψ)e−S[ψ,ψ], S[ψ, ψ] =

∫ β

0

dτ(

ψ · ∂τψ +H(ψ, ψ) − µN(ψ, ψ))

With particular example:

S[ψ, ψ] =

∫ β

0

dτ

[

∑

ij

ψi(τ) [(∂τ − µ)δij + hij]ψj(τ) +∑

ijkl

Vijkl ψi(τ)ψj(τ)ψk(τ)ψl(τ)

]

i.e. Quantum partition function expressed as path integral over fields ψj(τ)


Lecture XV 45

⊲ Matsubara frequency representation

Often convenient to express path integral in frequency domain

ψ(τ) =1√β

∑

ωn

ψneiωnτ , ψωn =

1√β

∫ β

0

dτ ψ(τ)e−iωnτ

where, since ψ(τ) = −ζψ(τ + β)

ωn =

2nπ/β, bosons,(2n+ 1)π/β, fermions

, n ∈ Z

ωn are known as Matsubara frequencies

Using1

β

∫ β

0

dτ ei(ωn−ωm)τ = δωnωm

S[ψ, ψ] =∑

ijωn

ψiωn [(iωn − µ) δij + hij ]ψjωn +

+1

β

∑

ijkl

∑

ωn1ωn2ωn3ωn4

Vijklψiωn1ψjωn2

ψkωn3ψlωn4

δωn1+ωn2 ,ωn3+ωn4


Lecture XVI 46

Lecture XVI: Applications and Connections

⊲ Partition Function of ideal (Non-Interacting) Gas of Quantum Particles

Useful for “normalisation” of interacting theories

e.g. Non-interacting fermions: H =∑

α

ǫαa†αaα

As a warm-up exercise, let us first use coherent state representation:

Quantum Partition function

Z0 = tr e−β(H−µN) =∑

n

〈n|e−β(H−µN)|n〉

In coherent state basis:

Z0 =

∫

d[ψ, ψ]e−P

αψαψα〈−ψ|e−β(H−µN)|ψ〉

Using ident. N.B. n2α = nα

e−β(H−µN) = e−βP

α(ǫα−µ)a†αaα =

∏

α

e−β(ǫα−µ)nα =∏

α

[1 +

(e−β(ǫα−µ) − 1

)nα

]

Z0 =

∫

d[ψ, ψ]e−P

αψαψα

∏

α

〈−ψ|ψ〉︷︸︸︷

e−ψαψα

[1 +

(e−β(ǫα−µ) − 1

)(−ψαψα)

]

=∏

α

∫

dψαdψα

1 − 2ψαψα︷︸︸︷

e−2ψαψα

[1 +

(e−β(ǫα−µ) − 1

)(−ψαψα)

]

=∏

α

∫

dψαdψα[1 − 2ψαψα −

(e−β(ǫα−µ) − 1

)ψαψα

]

=∏

α

∫

dψαdψα[−ψαψα(1 + e−β(ǫα−µ))

]

=∏

α

[1 + e−β(ǫα−µ)

]i.e. Fermi − Dirac distribution

Exercise: show (using CS) that in Bosonic case

Z0 =∏

α

∞∑

n=0

e−nβ(ǫα−µ) =∏

α

[1 − e−β(ǫα−µ)

]−1

Bose-Einstein distribution


Lecture XVI 47

⊲ Connection of CSPI with FPI

e.g. Quantum Harmonic oscillator: H =p2

2m+

1

2mω2q2

In second quantised form, H = (a†a + 1/2)~ω, [a, a†] = 1, i.e. bosons!

Z = tr(e−βH) =

∫ ψ(β)=ψ(0)

ψ(β)=ψ(0)

D[ψ, ψ] exp

[

−

∫ β

0

(ψ∂τψ + ~ωψψ

)]

e−β~ω/2 in D[ψ, ψ], ψ(τ) — complex scalar field

Parameterise complex field in terms of two real scalar fields

ψ(τ) =(mω

2~

)1/2[

q(τ) +ip(τ)

mω

]

Substituting (e.g. ~ωψψ = mω2

2(q2 + p2

(mω)2)) and noting

∫ β

0

dτ qp = −

∫ β

0

dτ pq

Z =

∫ ψ(β)=ψ(0)

ψ(β)=ψ(0)

D[p, q] exp

[

−

∫ β

0

dτ

(p2

2m+

1

2mω2q2 −

ipq

~

)]

cf. (Euclidean time) FPI β = it/~, τ = it′/~, i~

∂q∂τ

= ∂q∂t′

Z =

∫

D[p, q] exp

[i

~

∫ t

0

dt′ (pq −H(p, q))

]

Partition Function of Harmonic Oscillator from CSPI

(i) Bosonic oscillator:

ZB =

J det(∂τ + ~ω)−1

︷︸︸︷∫

D[ψ, ψ] exp

[

−

∫ β

0

dτψ (∂τ + ~ω)ψ

]

=

∫

(∏

n

dψωndψωn

)e−P

nψωn

(iωn+~ω)ψωn

= J∏

ωn

[iωn + ~ω]−1 =J

~ω

∞∏

n=1

[

(~ω)2 +

(2nπ

β

)2]−1

J ′

~ω

∞∏

n=1

[

1 +

(~ωβ

2πn

)2]−1

=J ′

2β sinh(~ωβ/2)

∏∞

n=1[1 + (x/πn)2] = (sinh x)/x

Normalisation: T → 0, Z dominated by g.s. limβ→∞ZB = e−β~ω/2(= ZF )

i.e. J ′ = β ZB =1

2 sinh(~βω/2)


Lecture XVI 48

(ii) Fermionic oscillator: Gaussian Grassmann integration

ZF = J det(∂τ + ~ω) = J∏

ωn

[iωn + ~ω] = J

∞∏

n=0

[

(~ω)2 +

((2n+ 1)π

β

)2]

= J ′

∞∏

n=1

[

1 +

(~ωβ

(2n+ 1)π

)2]

= J ′ cosh(~ωβ/2)

∏∞

n=1[1 + (x/π(2n+ 1))2] = cosh(x/2)

Using normalisation: limβ→∞ZF = e−β~ω/2

J ′ = 2e−β~ω ZF = 2e−β~ω cosh(~βω/2).

cf. direct computation:

ZB = e−β~ω/2∞∑

n=0

e−nβ~ω, ZF = e−β~ω/21∑

n=0

e−nβ~ω.

Note that normalising prefactor J ′ involves only a constant offset of free energy,

F = −kBT lnZ

statistical correlations encoded in content of functional integral

⊲ In notes, two case studies:

(i) Plasma Theory of the weakly interacting electron gas

(ii) BCS theory of superconductivity — a prototype for gauge theories

We will deal with project (ii)


Lecture XVII 49

Lecture XVII: Weakly Interacting Electron Gas: Plasma Theory

⊲ How are the properties of an electron gas influenced by weak Coulomb interaction?

⊲ Qualitative considerations:

When is the intereaction weak? Defining r0 = 1n1/3 as the average electron separation,

the typical p.e. e2

r0and k.e. ~2

mr0lead to the dimensionless ratio, rs =

e2

r0

mr20

~2≡r0a0

, where

a0 is electron Bohr radius, from which one can infer that Coulomb effects dominate atlow density

At rs ∼ 35 there is (believed to be) a transition to an electron solid phaseknown as a Wigner crystal (cf. Mott-Hubbard insulator)

For most metals (2 < rs < 6), k.e. and p.e. comparable; fortunately (thanks toadiabatic continuity) “weak coupling” theory valid even for intermediate rs

⊲ Motivates consideration of weak coupling theory rs ≪ 1: Σ-convention on spin

H =

∫

ddr c†σ(r)p2

2mcσ(r) +

1

2

∫

ddr

∫

ddr′c†σ(r)c†σ′(r

′)e2

|r − r′|cσ′(r

′)cσ(r)

Aim: to explore dielectric properties and ground state energy of electron gas through...

⊲ Quantum partition function: using CSPI formulation

Z ≡ tr e−β(H−µN) =

∫

ψσ(0)=−ψσ(β)ψσ(0)=−ψσ(β)

D(ψσ, ψσ)e−S[ψσ,ψσ ]

S[ψσ, ψσ] =

∫ β

0

dτ

[∫

ddr ψσ(r, τ)

(

∂τ +p2

2m− µ

)

ψσ(r, τ)

+1

2

∫

ddr

∫

ddr′ψσ(r, τ)ψσ′(r′, τ)

e2

|r − r′|ψσ′(r

′, τ)ψσ(r, τ)

]

Expressed in Fourier basis: ψσ(r, τ) =1

√

L3β

∑

k,ωn

ei(k·r−ωnτ)ψk,ωn,σ

S =

∫ β

0

dτ

[∑

k

ψkσ(τ) (∂τ + ǫk − µ)ψkσ(τ) +1

2Ld

∑

q 6=0

4πe2

q2ρq(τ)ρ−q(τ)

]

where ǫk = ~2k2

2mand ρq(τ) =

∫ddr e−iq·rρ(r, τ) ≡

∑

k ψkσ(τ)ψk+q,σ(τ)(N.B. neutralising background exclusion of q = 0 from sum)

With the action quartic in fermionic fields ψ, Z can not be evaluated exactly

For weak interaction, rs ≪ 1, we could expand in Coulomb interaction: Feynman diagram expansion (cf. Gell-Mann—Bruckner theory)


Lecture XVII 50

Alternative — use field integral to isolate leading diagrammatic series expansion— known as the Random Phase Approximation (RPA)

⊲ General principle:

When confronted with interacting field theory, seek decomposition of interactionthrough introduction of auxiliary field which captures low-energy content of theory

In some cases, these fields are identified with the elementary particles that mediatethe interaction (see below); in others, these fields encode the low-energy collective modesof the system (e.g. superfluid, superconductor)

⊲ Decoupling facilitated using the Hubbard-Stratonovich transformation:

e−

R β0 dτ

P

q6=02πe2

Ldq2 ρq(τ)ρ−q(τ)=

∫

Dφ e−

R β0 dτ

P

q6=0

»

q2

8πφq(τ)φ−q(τ)+ ie

2Ld/2(φq(τ)ρ−q(τ)+ρq(τ)φ−q(τ))

–

⊲ Physically, φ represents (scalar) photon field which mediates Coulomb interactionN.B. φ real and periodic φ(τ + β) = φ(τ)

Z =

∫

D(ψσ, ψσ)

∫

Dφ exp

−

∫ β

0

dτ

∫

ddr

[1

8π(∂φ)2 + ψσ

(

∂τ +p2

2m− µ+ ieφ

)

ψσ

]

Gaussian in Grassmann fields, field integral may be performed:using identity

∫D[ψ, ψ] exp[−ψMψ] = detM = exp[ln detM ]

Z =

∫

Dφ exp[

−

∫ β

0

dτ

∫

ddr1

8π(∂φ)2 +

spin︷︸︸︷

2 ln det

(

∂τ +p2

2m− µ+ ieφ

) ]

Setting e = 0, photon field decouples from determinant;recovers partition function of non-interacting electron gas

⊲ Perturbation Theory in e:

Define free particle Green function: G0 = [∂τ + p2

2m− µ]−1 and expand:

ln(1 + x) = −∑

n=1(−x)n/n

ln det

(

∂τ +p2

2m− µ+ ieφ

)

≡ tr ln(

G−10 + ieφ

)

= tr ln G−10 + tr ln

[

1 + ieG0φ]

= tr ln G−10 − tr

[

−ieG0φ+1

2

(

ieG0φ)2

+ · · ·

]

• First order term: for convenience, set k ≡ (k, ωn), etc.

2tr[G0φ] = 2∑

k

G0(k)︷︸︸︷

〈k|G0|k〉

1√

L3βφk=0

︷︸︸︷

〈k|φ|k〉 =2

√

L3β

∑

k

1

−iωn + ǫk − µφ0 = 0


Lecture XVII 51

φ0 = 0 due to neutralising background• Second order term:

2 ×e2

2tr[G0φ]2 = e2

∑

k,q

G0(k)︷︸︸︷

〈k|G0|k〉

1√

βL3φq

︷︸︸︷

〈k|φ|k + q〉

G0(k + q)︷︸︸︷

〈k + q|G0|k + q〉

1√

βL3φ−q

︷︸︸︷

〈k + q|φ|k〉=e2

2

∑

q

Π(q)φ−qφq

where “density-density” response function,

Π(q) =2

βL3

∑

k

1

−iωn + ǫk − µ

1

−iωn − iωm + ǫk+q − µ

Combined with bare term, to leading order in e2 (Random Phase Approximation),

Z = Z0

∫

Dφ e−S[φ], S[φ] =1

2

∑

q

D−1(q)︷︸︸︷(

q2

4π− e2Π(q)

)

|φq|2 +O(e4)

Z0 denotes partition function of non-interacting gas

⊲ Physically, D−1(q) denotes dynamically screened Coulomb interaction

D−1(q) = ǫ(q)q2

4π, ǫ(q) = 1 −

4πe2

q2Π(q)

where ǫ(q) is the energy and momentum dependent effective dielectric function

mq,

4πe 2

q2

k,ωn

ωmωω

n

+

+k+q,

ωmq,χ( )

=

-1

ωmq,=

=

++ ...

+

Diagrammatic interpretation:

D(q) =4π

q2

1

1 − 4πe2

q2 Π(q)=

4π

q2

∞∑

n=0

(

e2Π(q)4π

q2

)n


Lecture XVIII 52

Lecture XVIII: Random Phase Approximation

⊲ Previously, we have seen that the quantum partition function of theweakly interacting electron gas can be written as field integral

Z = Z0

∫

Dφ e−S[φ], S[φ] =1

2

∑

q=(ωm,q)

D−1(q)︷︸︸︷(

q2

4π− e2Π(q)

)

|φq|2 + O(e4)

where dielectric properties found to be controlled by density-density response function

Π(q) =2

βL3

∑

k

1

iωn − ǫk + µ

1

iωn + iωm − ǫk+q + µ

To understand form of χ(q), we have to digress and discuss

⊲ Matsubara Summations

Basic idea: by introducing auxiliary function g(z) that has simple polesof strength unity at z = iωn, Cauchy’s theorem implies

∑

ωn

f(iωn) =1

2πi

∮

C

dz g(z)f(z)

where contour C encloses only poles of g(z)

e.g. g(z) =

β

exp(βz) − 1, bosons

−β

exp(βz) + 1, fermions

Then, moving contour to infinity

Poles at Matsubarafrequencies

z

c

z

Pole of f

z

∑

ωn

f(iωn) =

→ 0︷︸︸︷

limR→∞

R

2πi

∫ 2π

0

dθg(Reiθ)f(Reiθ) −

if simple∑

P

g(zP )f(zP )

︷︸︸︷

1

2πi

∑

P :f(zP )=0

∮

dzg(z)f(z)


Lecture XVIII 53

Applied to χ(q),

χ(q) = −2

βL3

∑

k

[

g(ǫk − µ)

iωm + ǫk − ǫk+q

+g(ǫk+q − µ − i2πm

β)

−iωm − ǫk + ǫk+q

]

=2

L3

∑

k

nF (ǫk) − nF (ǫk+q)

iωm + ǫk − ǫk+q

,

where nF(ǫ) =1

eβ(ǫ−µ) + 1is the Fermi distribution function

Finally, for |q| ≪ kF ≡ (2mµ)1/2 and kBT ≪ µ, k summation Lindhard function

χ(q) ≃ −2ν(µ)

(

1 −ωm

vF |q|tan−1

[vF |q|

ωm

])

where ν(µ) is density of states at Fermi level

• Static Limit: For |ωm| ≪ kF |q|/m, χ(0,q) ≃ −2ν(µ), i.e.

D(0,q) ≃4πe2

q2

1

1 + 24πe2

q2 ν(µ)

Fourier transformed, static screened Coulomb interactione2

|r|e−|r|/λTF

where λTF = 2 × 4πe2ν(µ) — Thomas-Fermi screening length

i.e. At long time scales (low frequencies), bare Coulomb interaction isrenormalised (screened) by collective charge fluctuations

Physically, focusing a single electron, because it is negatively charged, other electrons

will be repelled. As a result, a positively charged cloud of radius λTF will form

balancing the negative charge of the electron. When viewed from a distance larger

than λTF, the electron+cloud behaves as a neutral particle.

• High Frequency Limit: For |ωn| ≫ kF |q|/m, χ(ωm,q) ≃ −q2

mω2m

n,

where n = N/L3 is the total number density (including spin)

D(q) =4πe2

q2

1

1 + 4πe2nmω2

m

i.e. real time response (iωm → ω + i0) singular whenωp = 4πe2n/m — Plasma frequency

In this case, there is a resonance which couples to the excitation mode where the pos-

itively charged background and the negatively charged electrons are moving uniformly

against each other.


Lecture XVIII 54

• Ground State Energy

limβ→∞

Z ∼ e−βEg.s. .

In the RPA approximation, Z = Z0 ×const.

det D−1/2= Z0 × const.

∏

q

D(q)1/2

i.e. Eg.s. = Eg.s.(e = 0) −1

2β

∑

q

ln D(q)


Lecture XIX 55

Lecture XIX: Bose-Einstein Condensation

Previously, we have seen how the functional field integral technique can be developed to

explore the impact of the electronic degrees of freedom on the effective Coulomb interaction

in a metal. However, our considerations did not engage any non-trivial mean-fields: the

platform of the non-interacting electron system remains adiabatically connected to that of

the weakly interacting system. In the following we will explore a problem in which the de-

velopment of a non-trivial ground state — the Bose-Einstein condensate — is accompanied

by the appearance of collective modes absent in the non-interacting system.

⊲ Following our earlier considerations, we begin with a Hamiltonian describing a Bosesystem of size L subject to a weak short-ranged repulsive contact interaction:

H =

∫

ddr a†(r)H0a(r) +g

2

∫

ddr a†(r)a†(r)a(r)a(r)

⊲ CSPI: Z = tre−β(H−µN) =∫

ψ(β)=ψ(0)D(ψ, ψ) e−S[ψ,ψ],

S =

∫ β

0

dτ

∫

ddr[

ψ(r, τ)(∂τ + H0 − µ)ψ(r, τ) +g

2(ψ(r, τ)ψ(r, τ))2

]

,

⊲ Bose-Einstein Condensation

As a warm-up, consider non-interacting Bose gas with spectrum ǫa

Z0 ≡ Z∣

∣

∣

g=0=

∫

p.b.c.

D(ψ, ψ) e−P

aωnψa,ωn

(−iωn+ǫa−µ)ψaωn = const.∏

a,ωn

1

−iωn + ǫa − µ

where, w.l.o.g., we assume ǫa ≥ 0 and ǫ0 = 0

While stability of integral requires µ ≤ 0, precise value fixed by

N(µ) = −∂µF =1

β∂µ lnZ0 = −

1

β

∑

a,ωn

1

iωn − ǫa + µ=

∑

a

nB(ǫa) ,

where, using Matsubara summation, nB(ǫ) =1

eβ(ǫ−µ) − 1(Bose-Einstein distribution)

T

µ

0

TC

As T reduced, µ increased until, at T = TBEC, µ = 0. For T < TBEC, µ re-mains zero and a macroscopic number of particles N − N1 condense into ground state:Bose-Einstein condensation

i.e. for T < TBEC,∑

a>0

nB(ǫa)∣

∣

∣

µ=0≡ N1 < N


Lecture XIX 56

⊲ How can this phenomenon be incorporated into the path integral?

Although condensate characterised by ground state component ψ0(τ)for T < TBEC, fluctuations are unbound (µ = 0 = ǫ0 and action for ψ0,0 vanishes!)

Here we must treat field ψ0(τ) as a (time-independent)Lagrange multiplier to be used to fix the number of particles below TBEC:

S0|µ=0− ≃ −βψ0µψ0 +∑

a6=0,ωn

ψaωn(−iωn + ǫa − µ)ψaωn

i.e. N = −∂µF |µ=0− =1

β∂µ lnZ0|µ=0− = ψ0ψ0 −

1

β

∑

a6=0,ωn

1

iωn − ǫa= ψ0ψ0 +N1

i.e. ψ0ψ0 = N0 translates to number of particles in condensate

⊲ Weakly Interacting Bose Gas

As in electron gas Presence of interaction prevents exact evaluation of field integraltherefore, turn to saddle-point + fluctuations analysis

⊲ Landau theory

In saddle-point (mean-field) approximation dominant contribution to Z =∫

D(ψ, ψ)e−S

controlled by ψ0(ωn = 0) (i.e. time-independent) sector of theory

1

βS[ψ0, ψ0] = −µψ0ψ0 +

g

2Ld(ψ0ψ0)

2

SS1

Re ψ0

Im ψ0Re ψ0

Im ψ0

Minimum action obtained from saddle-point equation: ψ0(−µ+g

Ldψ0ψ0) = 0

⊲ For µ < 0 (i.e. above the condensation threshold of the non-interacting system),s.p.e. exhibits only trivial solution ψ0 = 0 — no stable condensate

⊲ Below condensation threshold (i.e. for µ ≥ 0), equation solved by any

configuration with |ψ0| = γ ≡√

µLd/g

N.B. ψ0ψ0 ∝ Ld, reflecting the macroscopic population of the ground state


Lecture XIX 57

⊲ The equation couples only to the modulus of ψ0, i.e. the solution of the stationaryphase equation is “continuously degenerate”: Each configuration ψ0 = γ exp(iφ), φ ∈[0, 2π] is a solution. One ground state chosen spontaneous symmetry breaking.

Self-consistent calculation of µ = µ(N) demands considerationof low-energy fluctuations around the mean-field solution

By taking into account such fluctuations, we will be also ableto address the phenomenon of superfluidity


Lecture XX 58

Lecture XX: Superfluidity

Previously, we have seen that, when treated in a mean-field or saddle-point approximation,

the field theory of the weakly interacting Bose gas shows a transition to a Bose-Einstein

condensed phase when µ = 0 where the order parameter, the complex condensate wave-

function ψ0 acquires a non-zero expectation value, |ψ0| = γ ≡√

µLd/g. The spontaneous

breaking of the continuous symmetry associated with the phase of the order parameter is

accompanied by the appearance of massless collective phase fluctuations. In the following,

we will explore the properties of these fluctuations and their role in the phenomenon of

superfluidity.

⊲ Starting with model action for the Bose system, (~ = 1)

S[ψ, ψ] =

∫ β

0

dτ

∫

ddr

[

ψ(r, τ)

(

∂τ −∂2

2m− µ

)

ψ(r, τ) +g

2(ψ(r, τ)ψ(r, τ))2

]

a saddle-point analysis of the action revealed that, for µ > 0, the field ψacquires a constant non-zero expectation value: ψ0 = ψ0 = (µLd/g)1/2 ≡ γ

SS1

Re ψ0

Im ψ0Re ψ0

Im ψ0

In the following, we will explore the effect of fluctuations around the mean-field

To do so, it is convenient to effect the reparameterisation ψ(r, τ) = [ρ(r, τ)]1/2eiφ(r,τ)

Using 1.

∫ β

0

dτψ∂τψ =

1

2

∫ β

0

dτ∂τ (ρ1/2ρ1/2) = −

ρ

2

∣∣∣

β

0= 0

︷︸︸︷∫ β

0

dτρ1/2∂τρ1/2 +

∫ β

0

dτiρ∂τφ

2. ∂(ρ1/2eiφ) =

(1

2ρ1/2∂ρ+ iρ1/2∂φ

)

eiφ

3.

∫ β

0

dτψ∂2ψ = −

∫ β

0

dτ∂ψ · ∂ψ = −

∫ β

0

(1

4ρ(∂ρ)2 + ρ(∂φ)2

)

the action takes the form

S[ρ, φ] =

∫ β

0

dτ

∫

ddr

iρ∂τφ+1

2m

[1

4ρ(∂ρ)2 + ρ(∂φ)2

]

− µρ+gρ2

2


Lecture XX 59

Then, discarding gradient terms involving massive fluctuations δρ,an expansion in δρ ≡ ρ− ρ0 at µ = µBEC = 0

S[δρ, φ] ≃ S0[ρ0] +

∫ β

0

dτ

∫

ddr

[

iδρ∂τφ+gδρ2

2+

ρ0

2m(∂φ)2

]

• First term has canonical structure ‘momentum × ∂τ (coordinate)’cf. canonically conjugate pair

• Second term records energy cost of “massive” fluctuations fromMexican hat potential minimum

• Third term measures energy cost of spatially varying massless phase flucutations:i.e. φ is a Goldstone mode

Gaussian integration over δρ:

∫

D(δρ) exp[

−

∫ β

0

dτ

∫

ddr

g

2

(

δρ+i

g∂τφ

)2

+(∂τφ)2

2g︷︸︸︷(

iδρ∂τφ+gδρ2

2

) ]

= const.× exp

[

−

∫ β

0

dτ

∫

ddr(∂τφ)2

2g

]

effective low energy action

S[φ] ≃ S0 +1

2

∫ β

0

dτ

∫

ddr

[1

g(∂τφ)2 +

ρ0

m(∂φ)2

]

.

cf. Lagrangian formulation of harmonic medium (or massless Klein-Gordon field)

S =

∫

dt

∫

ddr

[m

2φ2 −

1

2ksa

2(∂φ)2

]

=

∫

dx ∂µφ∂µφ

i.e. low-energy excitations involve collective phase fluctuations with a spectrum ωk =gρ0

m|k|

⊲ Physical ramifications: consider quantum mechanical current density operator

j(r, τ) =1

2

[

a†(r, τ)p

ma(r, τ) −

(p

ma†(r, τ)

)

a(r, τ)

]

fun. int−→

i

2m

[(∂ψ(r, τ))ψ(r, τ) − ψ(r, τ)∂ψ(r, τ)

]≃ρ0

m∂φ(r, τ)

i.e. ∂φ is measure of (super)current flowVariation of action S[δ, φ]

i∂τφ = −gδρ, i∂τδρ =ρ0

m∂2φ = ∂ · j

• First equation: system adjusts to spatial fluctuations of densityby dynamical phase fluctuation


Lecture XX 60

• Second equation continuity equation (conservation of mass)

Crucially, stationary equations possess steady state solution with non-vanishingcurrent flow: setting ∂τφ = ∂τδρ = 0, obtain δρ = 0 and ∂ · j = 0

i.e. for T < TBEC, a configuration with a uniformdensity profile can support a steady state divergenceless (super)flow

Notice that a ‘mass term’ in the phase φ action would spoil this property,i.e. the phenomenon of superflow is intimately linked to the Goldstone mode

⊲ Steady state current flow in normal environments is prevented by the mechanism ofenergy dissipation, i.e. particles scatter off imperfections inside the system and therebyconverting part of their energy into the creation of elementary excitations

How can dissipative loss of energy be avoided?

Trivially, no energy can be exchanged if there are no elementary excitations to create

In reality, this means that the excitations of the system should beenergetically inaccessible (k.e. of carriers too small to create excitations)

But this is not the case here! there is no energy gap (ωk ∝ |k|)

However, there is an ingenuous argument due to Landau (see notes) showingthat a linear excitation spectrum can stabilize dissipationless transport


Lecture XXI 61

Lecture XXI: Cooper instability

In the final section of the course, we will explore a pairing instability of the electron gas

which leads to condensate formation and the phenomenon of superconductivity.

⊲ History:

• 1911 discovery of superconductivity (Onnes)

• 1951 “isotope effect” — clue to (conventional) mechanism

• 1956 Development of (correct) phenomenology (Ginzburg-Landau)

• 1957 BCS theory of conventional superconductivity (Bardeen-Cooper-Schrieffer)

• 1976 Discovery of unconventional superconductivity in heavy fermions (Steglich)

• 1986 Discovery of high temperature superconductivity in cuprates (Bednorz-Muller)

• ???? awaiting theory?

⊲ (Conventional) mechanism: exchange of phonons can induce (space-nonlocal)attractive pairwise interaction between electrons

H ′ = H0 − |M |2∑

kk′q

~ωq

~2ω2q − (ǫk − ǫk−q)2

c†k−qσc

†k′+qσ′ck′σ′ckσ

Physically electrons can lower their energy by sharing lattice polarisation of another

By exploiting interaction, electron pairs can condense into macroscopic phasecoherent state with energy gap to quasi-particle excitations

To understand why, let us consider the argument marshalled by Cooper which lead to

the development of a consistent many-body theory.

⊲ Cooper instability

Consider two electrons propagating above a filled Fermi sea:Is a weak pairwise interaction V (r1 − r2) sufficient to create a bound state?

Consider variational state

ψ(r1, r2) =

spin singlet︷︸︸︷

1√2(| ↑1〉 ⊗ | ↓2〉 − | ↑2〉 ⊗ | ↓1〉)

spatial symm. gk = g−k︷︸︸︷∑

|k|≥kF

gkeik·(r1−r2)

Applied to Schrodinger equation: Hψ = Eψ

∑

k

gk [2ǫk + V (r1 − r2)] eik·(r1−r2) = E

∑

k

gkeik·(r1−r2)


Lecture XXI 62

Fourier transforming equation: ×L−d∫d(r1 − r2)e

−ik′·(r1−r2)

∑

k′

Vk−k′gk′ = (E − 2ǫk)gk, Vkk′ =1

Ld

∫

dr V (r)ei(k−k′)·r

If we assume Vk−k′ =

− VLd |ǫk − ǫF |, |ǫk′ − ǫF | < ωD

0 otherwise

− V

Ld

∑

k′

gk′ = (E − 2ǫk)gk 7→ − V

Ld

∑

k

1

E − 2ǫk

∑

k′

gk′ =∑

k

gk 7→ − V

Ld

∑

k

1

E − 2ǫk= 1

Using1

Ld

∑

k

=

∫ddk

(2π)d=

∫

ν(ǫ) dǫ ∼ ν(ǫF )

∫

dǫ, where ν(ǫ) =1

|∂kǫk|is DoS

V

Ld

∑

k

1

2ǫk − E≃ ν(ǫF )V

∫ ǫF +ωD

ǫF

dǫ

2ǫ− E=ν(ǫF )V

2ln

(2ǫF + 2ωD − E

2ǫF − E

)

= 1

In limit of weak coupling, i.e. ν(ǫF )V ≪ 1

E ≃ 2ǫF − 2ωDe− 2

ν(ǫF )V

• i.e. pair forms a bound state (no matter how small interaction!)

• energy of bound state is non-perturbative in ν(ǫF )V

⊲ Radius of pair wavefunction: g(r) =∑

k gkeik·r, gk = 1

2ǫk−E× const., ∂k = ∂ǫk

∂k

∂∂ǫ

〈r2〉 =

∫ddr r2|g(r)|2∫ddr|g(r)|2 =

∑

k |∂kgk|2∑

k |gk|2≃v2

F

∫ ǫF +ωD

ǫF

4dǫ(2ǫ−E)4

∫ ǫF +ωD

ǫF

dǫ(2ǫ−E)2

=4

3

v2F

(2ǫF − E)2

if binding energy 2ǫF − E ∼ kBTc, Tc ∼ 10K, vF ∼ 108cm/s, ξ0 = 〈r2〉1/2 ∼ 104A,i.e. other electrons must be important

⊲ BCS wavefunction

Two electrons in a paired state has wavefunction

φ(r1 − r2) = (| ↑1〉 ⊗ | ↓2〉 − | ↓1〉 ⊗ | ↑2〉)g(r1 − r2)

with zero centre of mass momentum

Drawing analogy with Bose condensate, let us examine variational state

ψ(r1 · · · r2N) = NN∏

n=1

φ(r2n−1 − r2n)


Lecture XXI 63

Is state compatible with Pauli principle? Using g(r1 − r2) =∑

k gkeik·(r1−r2)

or, in Fourier representation

∫ddr1

Lde−ik1·r1

∫ddr2

Lde−ik2·r2g(r1 − r2) =

∑

k

gkδk1,kδk2,−k

or in second quantised form,

FT[

g(r1 − r2)c†↑(r1)c

†↓(r2)|Ω〉

]

=∑

k

gk c†k↑c

†−k↓|Ω〉

Then, of the terms in the expansion of

|ψ〉 =

N∏

n=1

[∑

kn

gknc†kn↑c†−kn↓

]

|Ω〉

those with all kns different survive

Generally, more convenient to work in grand canonical ensemblewhere one allows for (small) fluctuations in the total particle density

|ψ〉 =∏

k

(uk + vkc†k↑c

†−k↓)|Ω〉 ∼

cf. coherent state︷︸︸︷

exp

[∑

k

gkc†k↑c

†−k↓

]

|Ω〉

i.e. statistical independence of pair occupation

In non-interacting electron gas vk =

1 |k| < kF

0 |k| > kF

In interacting system, to determine the variational parameters vk,one can use a variational principle, i.e. to minimise

〈ψ|H − ǫF N |ψ〉

⊲ BCS Hamiltonian

However, since we are interested in both the ground state energy, and the spectrum ofquasi-particle excitations, we will follow a different route and explore a simplified modelHamiltonian

H =∑

k

ǫkc†kσckσ − V

∑

kk′

c†k′↑c

†−k′↓c−k↓ck↑


Lecture XXII 64

Lecture XXII: BCS Superconductivity

⊲ Recall: We have seen that phonon exchange a pairing interaction which rendersa single pair of electrons unstable towards the formation of a bound state (Cooper)

Motivated by this consideration, we have proposed a many-body generalisationof the pair state in the form of the variational BCS state

|ψ〉 =∏

k

(uk + vkc†k↑c

†−k↓)|Ω〉

and within which one may show that the “anomalous average” bk = 〈ψ|c†k↑c

†−k↓|ψ〉

acquires a non-zero expectation

In principle, we could now proceed with the Ansatz for |ψ〉 and employ a variational

analysis. However, instead, we will make use of this Ansatz to develop an approximation

scheme to expand the second quantised BCS Hamiltonian. Indeed, such an approach will

lead to the same phenomenology.

Since we expect quantum fluctuations in bk to be small, we may set

c†k↑c

†−k↓ = bk+

small︷︸︸︷

c†k↑c

†−k↓ − bk

(cf. our approach to BEC where a†0 was replaced by a C-number) so that

H − µN =∑

kσ

ζkc†kσckσ − V

∑

kk′

c†k′↑c

†−k′↓c−k↓ck↑, ζk = ǫk − µ

≃∑

k

ζkc†kσckσ − V

∑

kk′

(

bkc−k′↓ck′↑ + bk′c†k↑c

†−k↓ − bkbk′

)

Then, if we set V∑

kbk ≡ ∆, we obtain Bogoliubov-de Gennes or Gor’kov Hamiltonian

H − µN =∑

k

ζkc†kσckσ −

∑

k

(

∆c−k′↓ck′↑ + ∆c†k↑c

†−k↓

)

+|∆|2

V

=∑

k

(

c†k↑ c−k↓

)(ζk −∆−∆ −ζk

) (ck↑c†−k↓

)

+|∆|2

V

For simplicity, let us for now assume that ∆ is real

Bilinear in fermion operators, H − µN is diagonalised by canonical transformation

(ck↑c†−k↓

)

=

OT

︷︸︸︷(uk −vkvk uk

) (γk↑

γ†−k↓

)

where anticommutation relation requires OTO = 1,i.e. u2

k+ v2

k= 1 (orthogonal transformations)


Lecture XXII 65

Substituting, one finds that the Hamiltonian diagonalised if

2ζkukvk + ∆(v2k − u2

k) = 0

i.e. setting uk = sin θk and vk = cos θk,

tan 2θk = −∆

ζ, sin 2θk =

∆√

ζ2k

+ ∆2, cos 2θk = −

ζk√

ζ2k

+ ∆2

(N.B. for complex ∆ = |∆|eiφ, vk = eiφ cos θk)

As a result

H − µN =∑

k

(ζk − (ζ2k

+ ∆2)1/2) −∆2

V+

∑

kσ

(ζ2k

+ ∆2)1/2 γ†kσγkσ

Quasi-particle excitations, created by γ†kσ, have minimum energy ∆

Energy gap rigidity of ground state

Ground state wavefunction identified as vacuum state of algebra γkσ, γ†kσ,

i.e state which is annihilated by all the quasi-particle operators γkσ.

Condition met uniquely by the state

|ψ〉 ≡∏

k

γ−k↓γk↑|Ω〉 =∏

k

(ukc−k↓ − vkc†k↑)(ukck↑ + vkc

†−k↓)|Ω〉

=∏

k

(vk)(ukc−k↓c†k↓ − vkc

†k↑c

†−k↓)|Ω〉 = const.×

∏

k

(

uk + vkc†k↑c

†−k↓

)

|Ω〉

cf. variational analysis in fact, const. = 1

Note that phase of ∆ is arbitrary,i.e. ground state is continuously degenerate (cf. BEC)

⊲ Self-consistency condition: BCS gap equation

∆ = V∑

k

bk = V∑

k

〈ψ|c†k↑c

†−k↓|ψ〉 = V

∑

k

ukvk =V

2

∑

k

sin 2θk =V

2

∑

k

∆√

ζ2k

+ ∆2

i.e. 1 =V

2

∑

k

1√

ζ2k

+ ∆2=V Ldν(ǫF )

2

∫ ωD

−ωD

dζ1

√

ζ2 + ∆2

if ωD ≫ ∆, ∆ ≃ 2ωDe− 1

ν(ǫF )V Ld

Ground state: In limit ∆ → 0, v2k7→ θ(ǫF − ǫk), and the ground state collapses

to the filled Fermi sea with chemical potential ǫF

As ∆ becomes non-zero, states in the vicinity of the Fermi surface rearrange into acondensate of paired states


Lecture XXII 66

Excitations: Spectrum of quasi-particle excitations√

ζ2k

+ ∆2 shows rigid energy gap ∆.An excitation can be either the creation of a quasi-particle at positive energy or theelimination of a quasi-particle (the creation of a quasi-hole) at negative energy. Inthe ground state, all negative-energy quasi-particle states are filled.

Density of quasi-particle excitations near Fermi surface

ρ(ǫ) =1

Ld

∑

kσ

δ(ǫ−√

ζk + ∆2) =

∫

dζ1

Ld

∑

kσ

δ(ζ − ζk)

︸︷︷︸

ν(ζ)

δ(ǫ−√

ζ + ∆2)

≈ ν(ǫF )∑

s=±1

∫ ∞

0

dζδ(ζ − s[ǫ2 − ∆2]1/2

)

∣∣∣∂[ζ2+∆2]1/2

∂ζ

∣∣∣

= 2ν(ǫF )Θ(ǫ− ∆)ǫ

(ǫ2 − ∆2)1/2,

Spectral weight transferred from Fermi surface to interval [∆,∞]


Lecture XXIII 67

Lecture XXIII: Field Theory of Superconductivity

Following our discussion of the field theory of BEC and superfluidity in the weakly

interacting Bose gas, we turn now to condensation phenomena in Fermi systems

⊲ Starting point is BCS Hamiltonian for local pairing interaction: g ≡ VLd

H =

∫

ddr

[∑

σ

c†σ(r)p2

2mcσ(r) − g c†↑(r)c

†↓(r)c↓(r)c↑(r)

]

⊲ Quantum partition function: Z = tr e−β(H−µN)

Z =

∫

ψ(β)=−ψ(0)

D(ψ, ψ) exp

−

x≡(τ,r)R

dx︷︸︸︷∫ β

0

dτddr[ ∑

σ

ψσ

[G(p)0 ]−1

︷︸︸︷(

∂τ +p2

2m− µ

)

ψσ − gψ↑ψ↓ψ↓ψ↑

]

where ψσ(r, τ) denote Grassmann (anticommuting) fields

Analysis of interacting QFT?

• Perturbative expansion in g?Transition to condensate non-perturbative in g

• “Mean-field” analysis:condensation of pair wavefunction signalled by “anomalous average” 〈c†↑c

†↓〉

⊲ Hubbard-Stratonovich” decoupling: introducing complex commuting field ∆(r, τ)

egR

dx ψ↑ψ↓ψ↓ψ↑ =

∫

D(∆,∆) exp

−∫

dx

1

g(∆+gψ↑ψ↓)(∆+gψ↓ψ↑)−gψ↑ψ↓ψ↓ψ↑

︷︸︸︷[1

g|∆(r, τ)|2 + (∆ψ↓ψ↑ + ∆ψ↑ψ↓)

]

Z =

∫

D(ψ, ψ)

∫

D(∆,∆)e−R

dx|∆|2

g exp[

−∫

dx

Nambu spinor︷︸︸︷

( ψ↑ ψ↓ )

Gor′kov Ham. G−1

︷︸︸︷(

[G(p)0 ]−1 ∆

∆ [G(h)0 ]−1

) (ψ↑

ψ↓

) ]

‘free particle/hole’ Hamiltonian: [G(p/h)0 ]−1 = ∂τ

+/−( p2

2m− µ)

N.B.∫dx ψ↓∂τψ↓ = −

∫dx (∂τ ψ↓)ψ↓ =

∫dxψ↓∂τ ψ↓

Using Gaussian field integral:∫D(ψ, ψ) exp[−

∫ψAψ] = det A = exp[ln det A]

Z =

∫

D(∆,∆) exp

[

−∫

dx1

g|∆|2 + lndet G−1[∆]

]

i.e. Z expressed as functional field integral over single complex scalar field ∆(x)

⊲ To proceed, we must invoke some approximation:


Lecture XXIII 68

• Mean-field theory: far below transition (T ≪ Tc, β ≫ βc) fluctuations small saddle-point approximation in ∆ constant — Gap equation

• Ginzburg-Landau theory: since transition is continuous, close to Tc,we may develop perturbative expansion in (small) ∆

Noting : G−1 = G−10

[

1 + G0

(0 ∆∆ 0

)]

, G0 ≡ G(∆ = 0)

ln det G−1 = tr ln G−1 = tr ln G−10 − 1

2tr

[

G0

(0 ∆∆ 0

)]2

+ · · ·

N.B. ln(1 + z) = −∑∞

n=1(−z)n/n

• Zeroth order term ‘free particle’ contribution, viz. Z0 = det G−10

Using id. =∑

k≡k,ωn

|k〉〈k|, ∆k =1

√

βLd

∫

dx

eiωnτ−ik·r

︷︸︸︷

eik·x ∆(x)

• Second order term

tr G(p)0 ∆G

(h)0 ∆ =

∑

kk′

G(p)0 (k)

∆k′−k/√βLd

︷︸︸︷

〈k|∆|k′〉 G(h)0 (k′)〈k′|∆|k〉

q=k′−k=

∑

q

∆q∆−q

pairing susceptibility Π(q)︷︸︸︷

1

βLd

∑

k

G(p)0 (k)G

(h)0 (k + q)

Combined with bare term, one obtains

Z =

∫

D[∆,∆]e−S[∆,∆], S =∑

ωn,q

[1

g+ Π(ωn,q)

]

|∆ωn,q|2 +O(∆4)

In principle, one can evaluate Π(q, ωn) explicitly;however we can proceed more simply by considering...

⊲ ‘Gradient expansion’: Π(q, ωn) = Π(0, 0) +q2

2∂2|q|Π(0, 0) +O(iωn,q

4)

Ginzburg-Landau theory

S[∆] = β

∫

ddr

[t

2|∆|2 +

K

2|∂∆|2 + u|∆|4 + · · ·

]

t2

= 1g

+ Π(0, 0), K = ∂2|q|Π(0, 0) > 0 and u > 0


Lecture XXIII 69

Note structural similarity to weakly interacting Bose gas

⊲ Landau Theory: If we assume that dominant contribution to Z = e−βF arises fromminumum action, i.e. spatially homogeneous ∆ that minimises

S[∆]

βLd=t

2|∆|2 + u|∆|4

i.e. |∆|(t+ 4u|∆|2

)= 0, |∆| =

0 t > 0√

t/4u t < 0

i.e. for t < 0, spontaneous breaking of continuous U(1) symmetry associatedwith phase gapless fluctuations — Goldstone modes

⊲ Transition Temperature: Using identity 1Ld

∑

k =∫

ddk(2π)d =

∫ν(ǫ)dǫ

Π(0, 0) = − 1

βLd

∑

ωn,k

1

ω2n + (k2/2m− µ)2

≃ − 1

β

∑

ωn

∫ ∞

−∞

dζν(ζ + µ)

ω2n + ζ2

≃ −πν(µ)

β

∑

ωn

1

|ωn|

Introducing energy cut-off at Debye frequency ωD = (2nmax + 1)π/β

Π(0, 0) ≃ −ν(µ)nmax∑

n=−nmax

1

2n+ 1≃ −2ν(µ)

∫ nmax

0

dn

2n+ 1≃ −ν(µ) ln

(βωDπ

)

Transition when t/2 ≡ 1/g + Π(0, 0) = 0, i.e. kBT < kBTc = πωD exp[

− 1ν(µ)g

]

NearTct

2=

g

2− ν(µ) ln

(βωDπ

)

−

= 0︷︸︸︷

(g − ν(µ) ln

(βcωDπ

)

)

= ν(µ) ln(T/Tc) = ν(µ) ln(1 + (T − Tc)/Tc) ≃ ν(µ)

(T − TcTc

)

i.e. physically t is ‘reduced temperature’


Lecture XXIV 70

Lecture XXIV: †Superconductivity and Gauge Invariance

To establish origin of perfect diamagnetism and zero resistance,

one must accommodate electromagnetic field in Ginzburg-Landau Action

⊲ Inclusion of electromagnetic field into BCS action: p → p− eA (c = 1)

LEM = −FµνFµν/4, Fµν = ∂µAν − ∂νAµ

Repitition of field theory in presence of vector field obtains

generalised Ginzburg-Landau theory: Z =

∫

DA

∫

D[∆, ∆]e−S

S = β

∫

dr[ t

2|∆|2 +

K

2|(∂ + i2eA)∆|2 + u|∆|4+

LEM

︷︸︸︷

1

2(∂ × A)2

]

focusing only on spatial fluctuations of A

⊲ Gauge Invariance: Action invariant under local gauge transformation

A 7→ A′ = A − ∂φ(r), ∆ 7→ ∆′ = e−2ieφ(r)∆

(∂ + i2eA)∆ 7→ (∂ + i2e(A − ∂φ))e−2ieφ(r)∆ = e−2ieφ(r)(∂ + i2eA)∆

i.e. |(∂ + i2eA)∆|2 (as well as ∂ × A) invariant

⊲ Anderson-Higgs mechanism:

phase of complex order parameter ∆ = |∆|e−2ieφ(r) absorbed into A 7→ A′ = A − ∂φ(r)

S = β

∫

dr

[t

2|∆|2 +

K

2(∂|∆|)2 −

m2ν

2A2 + u|∆|4 +

1

2(∂ ×A)2

]

where m2ν = 4e2K|∆|2

i.e. massless phase degrees of freedom φ(r) have disappeared!and photon field A has acquired a ‘mass’ !

Example of a general principle:

“Below Tc, Goldstone bosons and the gauge field conspire to create massive excitations,and the massless excitations are unobservable”, cf. electroweak theory

⊲ Meissner effect: minimisation of action w.r.t. A

∂×

B︷︸︸︷

(∂ × A) +m2νA = 0 7→ (∂2 − m2

ν)B = 0

B = 0 is the only constant uniform solution perfect diamagnetism


Lecture XXIV 71

Free energy of superconductor first proposed on phenomenological grounds — how?...& why is crude gradient expansion so successful?

⊲ Statistical Field Theory: Ferromagnetism Revisited

Superconducting phase transition is an example of a critical phenomena

Close to the critical point, the thermodynamic properties of a systemare dictated by ‘universal’ characteristics

To understand why, consider simpler prototype system:the classical Ising ferromagnet:

βH = −J∑

〈ij〉

Szi S

zj + H

∑

i

Szi , Sz

i = ±1

Equilibrium Phase diagram?What happens in the vicinity of critical point?

(1) First order transition — order parameter (magnetisation) changes discontinuouslycorrelation length remains finite

(2) Second order transition — order parameter changes continuouslycorrelation length diverges

...motivates consideration of “hydrodynamic” theory of classical partition function

Z = e−βF =

∫

DS(r) e−βH[S(r)]

βH constrained (only) by symmetry (translation, rotation, etc.)

βH [S(r)] =

∫

dr

[t

2S2 +

K

2(∂S)2 + uS4 + · · ·+ BS

]

cf. Ginzburg-Landau Theory of superconductor

Landau theory: S(r) = S const.

F (S)

Ld=

t

2S2 + uS4

⊲ Generally second order phase transitions divide into Universality classeswith the same characteristic critical behaviour

E.g. (1) Ising model — liquid/gas: S → density ρ, H → pressure P

E.g. (2) Superconductivity — classical XY ferromagnet

subject of statistical field theory...


Quantum Condensed Matter Field Theory - Internet Archive€¦ · [11] C. Kittel, Quantum Theory of Solids, Wiley (1963). Not to be confused with the other famous text by the same

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