QUANTUM FIELD THEORY IN CONDENSED MATTER PHYSICS · 2014-11-22 · LITERATURE G. Rickayzen: Green’s functions and Condensed Matter H. Bruus and K. Flensberg: Many-Body Quantum Theory

QUANTUM FIELD THEORY IN CONDENSED MATTER PHYSICS

Wolfgang Belzig

FORMAL MATTERS

“Wahlpflichtfach” 4h lecture + 2h exerciseLecture: Mon & Thu, 10-12, P603Tutorial: Mon, 14-16 (P912) or 16-18 (P712)50% of exercises needed for examLanguage: “English” (German questions allowed)No lecture on 21. April, 25. April, 9. May, 2. June, 13. June, 16. June, 23. June, 14. July

EXERCISE GROUPS

• Two groups @ 14h (P912) and 16h (P712)• Distribution: see list• Exercise sheets usually 5-8 days before exercise

(see webpage for preview)• Question on exercise to one of the tutors

(preferably the one who is responsible)• Plan: 18.4. Milena Filipovic; 2.5. Martin Bruderer; 9.5. Fei Xu

16.5. Cecilia Holmqvist; 23.5. Peter Machon

LITERATURE

G. Rickayzen: Green’s functions and Condensed MatterH. Bruus and K. Flensberg: Many-Body Quantum Theory in Condensed Matter PhysicsJ. Rammer: Quantum Field Theory of Non-equilibium StatesG. Mahan: Quantum Field Theoretical MethodsFetter & Walecka: Quantum Theory of Many Particle SystemsYu. V. Nazarov & Ya. Blanter: Quantum TransportJ. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986)

CONTENT

A. Introduction, the ProblemB. Formalities, DefinitionsC. Diagrammatic MethodsD. Disordered ConductorsE. Electrons and PhononsF. SuperconductivityG. Quasiclassical MethodsH. Nonequilibrium and Keldysh formalismI. Quantum Transport and Quantum Noise

PHYSICAL OVERVIEW

Solid state physics: Many electrons and phonons interacting with the lattice potential and among each other (band structure, disorder, electron-electron interaction, electron-phonon interaction)

We need to explain:

Why some materials are conductors, semiconductors, insulators, superconductors or ferromagnets

Thermodynamic quantities, transport coefficients, electric and magnetic susceptibilities

All phenomena follow from the same Hamiltonian, but with different microscopic parameters (number of electrons per atom, lattice structure, nucleus masses...)

THE THEORETICAL PROBLEM

Due to the huge number of degrees of freedom (M states, N particles: MN), a wave function treatment is not feasible.

Quantum fields ( ) are more suitable, since they can represent a large number of identical particles

The central objects are expectation values of quantum field operators - Greens function

Physical observable are obtained directly from the Greens functions, e.g. density

Ψ(x),ck†,...

G(x, x ') = Ψ(x)Ψ†( ′x )

n(x) = G(x, x)

THE TECHNICAL PROBLEM

Often approximations are based on some perturbation expansion up to some finite order (first or second)

In many practical problems we need non-pertubative solutions:

n ~ e− t /τ ≈ 1− t / τ

kBTc ~ ω ce−1/NV not expandable in V

goes negative

Critical temperature

Exponential decay

CONTENT

A. Introduction, the ProblemB. Formalities, DefinitionsC. Diagrammatic MethodsD. Disordered ConductorsE. Electrons and PhononsF. SuperconductivityG. Quasiclassical MethodsH. Nonequilibrium and Keldysh formalismI. Quantum Transport and Quantum Noise

A. INTRODUCTION

1. Greens functions (general definition, Poisson equation, Schrödinger equation, retarded, advanced and causal)Linear response (general theory of response functions, Kubo formula, Greens function)

2. Statement of the problem (physical problem, 2nd quantization, field operators, electron-phonon interaction, single particle potentials)

THE HAMILTONIAN

H =P2j2M j

+Uj (Rj )

⎛

⎝⎜⎞

⎠⎟j∑

+pi2

2m+Ui (

ri )⎛⎝⎜

⎞⎠⎟i

∑

+12

e2ri −rji≠ j

∑ + VijRj ,ri( )

ij∑

Ions in the harmonic potential

Electrons in the periodic potential

Electron-electron and Elektron-phonon interaction

THE HAMILTONIAN

H = ω qλ a†qλa qλ +12

⎛⎝⎜

⎞⎠⎟qλ

∑+ kσc

†kσc kσ

kσ∑

+ c†k + qσc†k '− qσ 'V (

q)c k 'σ 'c kσk′k qσ ′σ∑

+ gqλc†k + qσc kσ a†− qλ + a qλ( )

kqσλ∑

Ions in the harmonic potential

Electrons in the periodic potential

Electron-electron and Electron-phonon interaction

SUMMARY OF A.

1. Greens functions obey differential equations with generalized delta-perturbation

2. Analytical properties of GF are related to causality

3. Linear response response of a quantum system (Kubo formula) related to a retarded function

4. Many-body Hamiltonian in second quantization: electrons, phonons, potentials, interaction

CONTENT

A. Introduction, the ProblemB. Formal MattersC. Diagrammatic MethodsD. Disordered ConductorsE. Electrons and PhononsF. SuperconductivityG. Quasiclassical MethodsH. Nonequilibrium and Keldysh formalismI. Quantum Transport and Quantum Noise

B. FORMAL MATTERS

1.Definitions of double-time GF (retarded, advanced, causal, temperature), higher order GF

2.Analytical properties (spectral representation, spectral function, Matsubara frequencies)

3. Single particle Greens functions, spectral function, density of states, quasiparticles

4.Equation of motion for Greens functions5.Wicks theorem

1. DEFINITIONS

Two-time Greens functions for two operators A and BRetarded Greens function

GR (t, ′t ) = −i

AH (t), BH ( ′t )⎡⎣ ⎤⎦ θ(t − ′t )

A, B⎡⎣ ⎤⎦ = AB + BA

= Tr ρ( ) ρ =1Ze−β ( H −µN )

=−1 Bose operators+1 Fermi operators

⎧⎨⎪

⎩⎪Bose: ak , ak

†, ρ(r ) = ψ †(r )ψ (r ),j (r ), H , N

Fermi: ck , ck†, ψ †(r ), ψ (r )

1. DEFINITIONS

Temperature Greens function

G(τ , ′τ ) = −1T A(−iτ )B(−i ′τ )( )

t→ −iτ Wick rotation

AH (t)→ A(−iτ ) = A(τ ) = eHτ Ae−Hτ

Higher-order Greens function

G(τ1,τ 2 ,τ 3,…) ~ T A(τ1) B(τ 2 ) C(τ 3)( )

T A(τ ) B( ′τ )( ) = A(τ ) B( ′τ )θ(τ − ′τ ) − B( ′τ ) A(τ )θ(τ '− τ )

Time-ordering operator

“Later times to the left”

B. FORMAL MATTERS

1.Definitions of double-time GF (retarded, advanced, causal, temperature), higher order GF

2.Analytical properties (spectral representation, spectral function, Matsubara frequencies)

3. Single particle Greens functions, spectral function, density of states, quasiparticles

4.Equation of motion for Greens functions5.Wicks theorem

MATSUBARA GREEN’S FUNCTION

G(τ ) =− A(τ ) B τ > 0

B A(τ ) τ < 0

⎧⎨⎪

⎩⎪A(τ ) = A(−iτ ) = eHτAe−Hτ

Definition of temperature GF

0 < τ ≤ β G(τ − β) = −G(τ )Definition of Matsubara GF for

For we find the symmetry relation

G(ων ) = dτeiωντG(τ )0

β

∫Symmetry implies

ων =πβ

2ν = −1 for bosons 2ν +1( ) = +1 for electrons

⎧⎨⎪

⎩⎪

G(τ ) = 1β

e− iωντ G(ων )ν∑

ν = 0,±1,±2,…

−β < τ ≤ β

SPECTRAL REPRESENTATION

A(x) = 1+ e−βω

Ze−βEn AnmBmnδ (x − Em + En )

nm∑

G(ων ) = dx A(x)iων − x−∞

∞

∫Matsubara GF can be represented as

SAME function as before

G(ων ) = G(iων )Matsubara GF can be found from G as

f (ω )

Inverse question: Can we determine G from MGF?

No, since has the same MGFG '(ω ) = G(ω ) + (1+ eβω ) f (ω )G(ων ) = G(iων ) = G '(iων ) for an arbitrary analytical function

Unique definition through condition limω →∞

G(ω ) ~ 1 /ω

THE SPECTRAL FUNCTION

A(x) = 1+ e−βω

Ze−βEn AnmBmnδ (x − En + Em )

nm∑ → G(ω ) = dx A(x)

ω − x−∞

∞

∫

G(ων ) = G(iων ) = dx A(x)iων − x−∞

∞

∫

GR (E) = G(E + iδ ) = dx A(x)E − x + iδ−∞

∞

∫ GA (E) = G(E − iδ ) = dx A(x)E − x − iδ−∞

∞

∫

CONSEQUENCES OF ANALYTICAL PROPERTIES

Kramers-Kronig relations (from analycity of G in upper half-plane)

ReGR (E) = 1πP dE ' ImG( ′E )

′E − E−∞

∞

∫ ImGR (E) = −1πP dE ' ReG( ′E )

′E − E−∞

∞

∫Real and imaginary parts of response functions are related

Dissipation (c.f. exercise)

Fluctuation-Dissipation relation (for Bose operators and A=B)

ImGR (E) = π2e−βE −1( ) dteiEt B(t)B†

−∞

∞

∫Fluctuations (spectral density)

B.3 SINGLE PARTICLE GREEN’S FUNCTION

Definition: G(r ,τ; ′r , ′τ ) = − T Ψ(r ,τ ) Ψ(r ',τ ')( )Ψ(r ,τ ) = eHτΨ(r )e−Hτ Ψ(r ,τ ) = eHτΨ†(r )e−Hτ

Spectral representation

A(r , ′r ; x) = 1+ e−βω

Ze−βEmΨnm (

r )Ψmn† (r ')δ (x − Em + En )

nm∑

G(r , ′r ;ων ) = dτeiωντG(r , ′r ;τ )0

β

∫ = dx A(r , ′r , x)

iων − x−∞

∞

∫

MOMENTUM REPRESENTATION

G(k ,τ ) = − T ck (τ )ck( )

A(k , x) = 1+ e

−βx

Ze−βEm cknm

2δ (x − Em + En )

nm∑

G(k ,ων ) = dτeiωντG(

k ,τ )

0

β

∫ = dx A(k , x)

iων − x−∞

∞

∫

Density of states

N(E,k ) = −

1πImGR (

k ,E) = A(

k ,E)

N(E) = −1πIm GR (

k ,E)

k∑ = A(

k ,E)

k∑

G(k ,ων ) =

1iων − k

Free particles

N(E,k ) = δ (k − E)

WICK’S THEOREM

G0(n) (1,2,…,n;1',2 ',…,n ') =

−( )i−1G0 (1,i ')G0(n−1)(1,2,,n;1',, i ' ,,n ')

i=1

n

∑G0(2)(1,2;1',2 ') = G0 (1,1')G0 (2,2 ')

−G0 (1,2 ')G0 (1',2)

Recursion relation for GF of noninteracting particles

Example:

WICK’S THEOREM II

For Fermions

G0(n) (1,2,…,n;1',2 ',…,n ') =

G0 (1,1') G0 (1,2 ') ... G0 (1,n ')G0 (2,1') G0 (2,2 ')

G0 (n,1') ... ... G0 (n,n ')

B. FORMAL MATTERS - SUMMARY

1. Definitions of different GFs: retarded, advanced, causal, temperature (Wick rotation, imaginary time)

2. Analytical properties, all GF determined by the same spectral function A(x), Matsubara GF and frequencies

3. Single particle Green’s function, density of states, relation to observables, quasiparticles as poles of the 1P-GF

4. Equation of motion: 1P-GF obeys differential equation like ordinary GF, interaction couples 1P-GF, 2P-GF,..., NP-GF

5. Wicks theorem: NP-GF for non-interacting particles can be decomposed into products of 2P-GF

C. DIAGRAMMATIC METHODS

Systematic pertubation theory in external

potential, two-particle interaction or/and electron-

phonon interaction

Symbolic language in terms of Feynman diagrams

enables efficient algebraic manipulations

C. DIAGRAMMATIC METHODS

1.Single-particle potential

2. Interacting particles

3.Translational invariant problems

4. Selfenergy and Correlations

5. Screening and random phase approximation (RPA)

C. 1 SINGLE-PARTICLE POTENTIAL

x= +

xDyson equation:

G(1,2) G0 (1,2) U(1)

G(1,2) = G0 (1,2) + d∫ 3G0 (1, 3)U(3)G(3,2)Rules for order-n contribution:

• Draw all topologically distinct and connected diagrams with 2 external and n internal vertices

• Associate a lines with G0 and each internal vertex with U and integrate over all internal coordinates

Dictionary:

C. 2 TWO-PARTICLE INTERACTION

H = H0 +VS0 (τ ) = T exp − H0 (τ ')0

τ

∫⎡⎣⎢

⎤⎦⎥

VI (τ ) = S0−1(τ )VS0 (τ )

SI (τ ) = T exp − VI (τ ')0

τ

∫⎡⎣⎢

⎤⎦⎥

G(1,2) =T SI (β) ψ I (1) ψ I (2)⎡⎣ ⎤⎦ 0

T SI (β)[ ] 0

Expression for 1P-GF in interaction picture

ready for expansion....

C. 2 TWO-PARTICLE INTERACTION

G(1,2) G0 (1,2)

Rules for order-n contribution:• Draw all topologically distinct and connected diagrams

with 2 external and 2n internal vertices• Connect all vertices with direct sold lines and all internat

vertices with dashed lines• Integrate over all internal coordinates• ...

Dictionary: −V (1,2)

++=

C. 3 TRANSLATIONAL INVARIANCE

Feynman rules for order-n contribution im momentum and frequency space:

• Draw the same diagrams as in real space• Associate lines with Fourier-transformed GF and V• Momentum conservation at vertices• Sum over internal momenta and frequencies• ...

++=

G(k,ων ) G0 (k,ων )−V (q)

C. 4 SELFENERGY AND CORRELATIONS

G = G0 +G0ΣG

Summing all diagram which can be cut by cutting a single line

G =1

G0−1 − ΣPhysical implications

• real part of selfenergy implies energy shifts• imaginary part leads to finite lifetimeHartree-Fock approximations• leads to (diverging) real self energy• neglects correlations in 2P-GF

= +

Dyson equation

C. 5 SCREENING AND RANDOM PHASE APPROXIMATION (RPA)

Dressing the interaction

Physical implications• screening of the interaction by creation of electron-hole pairs• equivalent diagram occurs in charge response to external potential -> relation to dielectric functionConsequences: Finite life time & Thomas-Fermi screening

= +

Polarization bubble

Π(k,ων )V (q,ων ) =

V0 (q)1− Π(q,ων )V0 (q)

C. DIAGRAMMATIC METHODS

Systematic pertubation theory for Greens functions in external potential, two-particle interaction or/and electron-phonon interactionSymbolic language in terms of Feynman diagrams enables efficient algebraic manipulationsSelfenergy leads to an improved approximation by summing infinite series of certain diagram classesHartree-Fock diagrams provide a first rough approximationScreening of the interaction and finite lifetime in the RPA

CONTENT

A. Introduction, the ProblemB. Formalities, DefinitionsC. Diagrammatic MethodsD. Disordered ConductorsE. Electrons and PhononsF. SuperconductivityG. Quasiclassical MethodsH. Nonequilibrium and Keldysh formalismI. Quantum Transport and Quantum Noise

D. DISORDERED CONDUCTORS

1.Greens function in a random potential

2.Conductivity and polarization bubble

3.Vertex corrections and transport life time

4.Diffuson, Cooperon and weak localization

5.Localization (?)

D.1 GREENS FUNCTION IN A RANDOM POTENTIAL

Electron motion in a random potential:

• randomly distributed impurities • semiclassical picture neglects wave nature• equivalent to Boltzmann-equation treatment (Drude

conductivity), mean free path l• systematic expansion in wave nature: ~1/kFl

D.1 GREENS FUNCTION IN A RANDOM POTENTIAL

x= +G(1,2) = G0 (1,2) + d∫ 3G0 (1, 3)U(3)G(3,2)

Field theory method, Dyson equation:

• The impurities are randomly distributed and we have to average over the positions

• This introduces correlations between scattering events, since scattering at different impurities averages out, but not at the same impurity!

AVERAGING OVER RANDOM IMPURITY POSITIONS

G(1,2) =

= + + +

+ + +

Averaging leads to correlations between scattering events at the same impurity

crossed lines-> self energy

SELF ENERGY FOR ELASTIC IMPURITY SCATTERING

Σ(1,2)= + +crossed lines

Only diagrams, which cannot be “cut”, c.f. double counting

G(k,ωµ ) =1

G0−1(k,ωµ ) − Σ(k,ωµ )

Solving the Dyson equation

Σ(k,ωµ ) = k,ωµ k,ωµ′k ,ωµk −

′k ′k − k

SELF ENERGY IN BORN APPROXIMATION

Σ(k,ωµ ) = −i 12τ k

sgn(ωµ )

GR(A) (k,E) = 1

E − k + (−)i2τ k

1τ k

= 2πNiN(0) V (k − k 'F )2

vF′

Self energy

Momentum life time (->mean free path)

Impurity averagedGreens function

l = vFτ

SOME CONSEQUENCES

Spectral density

Fourier trafo

Density of states N(E) = N(0)

GR (r,E) = −πN(0)kFr

eikF r− r /2l

A(k,E) = 12πτ

1E − k( )2 + (1 / 2τ )2

SUMMARY D1: IMPURITY AVERAGED GREENS FUNCTION

G(k,ωµ ) =1

iωµ − k +i2τ k

sgn(ωµ )

= +

= +SelfconsistentBorn approximation

T-matrixapproximation

THE FUTURE

Mo 13.6. (pentecost) no lecture, no exerciseTh 16.6. exercise (Martin)Mo 20.6. 2x lectureTh 23.6. (corpus christi), no lectureMo 27.6. lecture, exercise (Cecilia)Th 30.6. lectureMo 4.7. lecture, exercise (Peter)Th 7.7. lectureMo 11.7. lecture + exam preparationTh 14.7. no lecture

D.2 CONDUCTIVITY AND TRANSPORT LIFE TIME

• Current as response to external electric field

• Linearization in electric field - linear response

• Use the Kubo formula to relate the conductivity to

the two-particle Green’s function

CURRENT AS LINEAR RESPONSE

Conductivity Jα = σαβEβE = −

∂A∂t

↔E = iω

A

Current density operatorj1 =

−ie2m

Ψ† ∇Ψ( ) − ∇Ψ†( )Ψ⎡⎣ ⎤⎦ ↔ j1(q) =−e2m

(2k + q)ck†ck+q

k∑

Electric current J(r) = j1(r) −emΨ†(r)Ψ(r)A(r)

Jα (r,t) = −e2

mn(r,t)Aα (r,t) − dt 'dr 'G jj

αβ (r,t;r ',t ')Aβ (r ',t ')−∞

t

∫Jα (q,ω ) = −

e2

mnAα (q,ω ) −G

jjαβ (q,ω )Aβ (q,ω ) = −Kαβ (q,ω )Aβ (q,ω )

CURRENT-CURRENT RESPONSE FUNCTION

G jjαβ (1,1') =

e2

4m∇ ′2 − ∇1'( ) ∇2 − ∇1( )G (2) (2,2 ';1,1') ′2 → ′1 −

2→1−

Two-particle Greens function

G (2) (2,2 ';1,1') = − T Ψ(2) Ψ(2 ') Ψ(1') Ψ(1)

=G (2) (2,2 ';1,1') + ≈

EVALUATION OF CONDUCTIVITY

Approximations (k ± q / 2) = (k) ± vFq / 2

Exchange of the order of frequency and momentum sum

G jjαβ (q,ων ) = −

2e2

mT kαkβG(k + q / 2,ωµ )G(k − q / 2,ωµ +ων )

k∑

µ∑

= −2e2

mT N(0) d∫

ν∑ ×

kFαkFβ

iωµ − +vFq2

+ i2τsgn(ωµ )

1

i ωµ +ων( ) − −vFq2

+ i2τsgn(ωµ +ων ) vF

Current-current response function

BOLTZMANN CONDUCTIVITY

Jα (q,ω ) = i 3ne

2

mvF2

vFαvFβω − vF

q + i / τσαβ (

q,ω )

Eβ (q,ω )

Result coincides with Boltzmann equation treatment

ω EF

q kFValidity

For slowly varying fields: vFq1τ

ql 1( )

σ =σ 0

1− iωτ σ 0 =

ne2τm

Drude conductivity

D.3 VERTEX CORRECTION AND TRANSPORT LIFE-TIME

= +

So far neglegted: Captured in Vertex correction:

Integral equation forVertex function:

Γ(k,q)2k + q

INTEGRAL EQUATION

= +

Γα (k,q) = 2k + q( )α + Ni V (k − k ') 2G(k '+ q2,ωµ +ων )G(k '−

q2,ωµ )Γα (k ',q)

k '∑

• Integral equation for the vertex function• Vertex function is a vector• Vertex function depends parametrically on frequencies• Equivalent to Boltzmann equation

Γ(k,q)

Γ(k,q)

HOMOGENEOUS PART

q = 0Γα (k) = 2kα + Ni V (k − k ') 2G(k ',ωµ +ων )G(k ',ωµ )Γα (k ')

k '∑

depends weakly on k

We make the Ansatz Γα (k) = γ kα γ =independent of k

depend only on ′k

Multiplying with k

γ = 2 + γ Ni V (k − k ') 2G(k ',ωµ +ων )G(k ',ωµ )k ⋅′k

k 2k '∑

Close to Fermienergy

γ = 2 + γ NiN(0) V (k − k 'F )2k ⋅′kF

k 2 vF′

1/2π ′τ

d ′k G(k ',ωµ +ων )G(k ',ωµ )∫

SOLVING THE EQUATION

d ′k G(k ',ωµ +ων )G(k ',ωµ )∫ = 2πiθ(−ωµ )θ(ωµ +ων )

iων + i / τ

γ = 2 + γ i′τθ(−ωµ )θ(ωµ +ων )

iων + i / τ

c.f. D.2

γ − 2 = i′τθ(−ωµ )θ(ωµ +ων )

iων + i1τ− 1

′τ⎛⎝⎜

⎞⎠⎟

1τtr

1τ tr

= 2πNiN(0) V (k −k 'F )

21−k ⋅′kF

k 2⎛⎝⎜

⎞⎠⎟ vF′

Weighted average of scattering processes (backscattering contributes stronger)

Transport lifetime:

CORRECTION TO DRUDE CONDUCTIVITY

σ (ω ) = 2e2

mN(0) kF

3

3γ / 2

ω + i / τ= σ tr

11− iωτ tr

• Vertex correction leads to a different scattering time in the conductivity than the momentum life-time in the single-particle Greens function

• For s-wave scattering

Vertex correction important for anisotropic problems

V (k − k ') = V0 = const.k ⋅′k vF′

= 0 τ tr = τ

D.4 DIFFUSON, COOPERON AND WEAK LOCALIZATION

How does diffusive motion appear in the quantum theory? Diffusion ladder (Diffuson)

How does quantum interference change the diffusion picture? Maximally crossed diagrams (Cooperon)

What is the effect on the conductivity?Reduced conductivity (weak localization)

SUMMARY:D. DISORDERED CONDUCTORS

Scattering at randomly distributed leads to correlations between scattering events

G(1,2) =

= + + +

+ + +crossed lines-> self energy

SUMMARY:D. DISORDERED CONDUCTORS

1P-GF contains momentum lifetime defined through t-matrix

= +SelfconsistentBorn approximation

T-matrixapproximation

G(k,ωµ ) =1

iωµ − k +i2τ k

sgn(ωµ )

= +

D. CONDUCTIVITY AND VERTEX CORRECTION

= +

Conductivity:

Integral equation for vertex correction:

Quasiclassical (Drude)

σ (ω ) = σ tr

1− iωτ tr

Γα (k)

Small for isotropic scattering

D. QUANTUM COOPERON

Ladder diagrams for Diffuson and Cooperon

=D

C=

D. QUANTUM CORRECTIONS

Diffuson describes classical diffusion

Cooperon describes weak localization to conductivity

∂∂t

− D0∂2

∂x2⎛⎝⎜

⎞⎠⎟D(x,t) = δ (x − ′x )δ (t − ′t )

Quantum correction• 3D: small correction• 2D: universal ~e^2/h• 1D: strong localisationgives access to dephasing time

CONTENT

A. Introduction, the ProblemB. Formalities, DefinitionsC. Diagrammatic MethodsD. Disordered ConductorsE. Electrons and PhononsF. SuperconductivityG. Quasiclassical MethodsH. Nonequilibrium and Keldysh formalismI. Quantum Transport and Quantum Noise

E. ELECTRONS AND PHONONSGREENS FUNCTION FOR FREE PHONONS

Field operators (displacement) uj =1NM

Qλ qeλ (q)ei

qRj

qλ∑

Green’s function

Dλ ′λ (q′q ;τ , ′τ ) = 2 ωλ qωλ ' q ' T Qλ q (τ ) Qλ ' q ' (τ ')⎡⎣ ⎤⎦

Hamiltonian Hph = ωλ qa†λ qaλ q

λ q∑ Qλ q =

12ωλ q

a†λ q + aλ q( )

Free Phonons Dλ ′λ (q′q ;τ , ′τ ) = δλ ′λ δ q ′q Dλ (

q,τ )

Dλ (q,ων ) = dτeiωντDλ (

q,τ )0

β

∫ Dλ(0) (q,ων ) =

2ωλ q

ων2 +ωλ q

2

E.1 ELECTRON-PHONON INTERACTION HAMILTONIAN

H = ω qλa†qλa qλ

qλ∑ + kσc

†kσc kσ

kσ∑

+ c†k + qσc†k '− qσ 'V (

q)c k 'σ 'c kσk′k qσ ′σ∑

+ gqλc†k + qσc kσ a†− qλ + a qλ( )

kqσλ∑

gqλ = iqeλ (q)

2Mωλ q

ρ0Ve− ph (q)

V (q) = Ze2q2 + qTF

2Screened Coulomb interaction (Thomas-Fermi)

Interaction constant

k,ωµ q,ων

k + q,ωµ +ων

E.3 PERTURBATION THEORY FOR ELECTRONS

Selfenergy ΣR (E) = d dω0

∞

∫ α 2F(ω ) 1E + +ω + iδ

+1

E + +ω − iδ⎡⎣⎢

⎤⎦⎥0

∞

∫Combined coupling and density of states of the phonons

α 2F(ω ) = N(0) dΩk

4πdΩk '

4πgkk 'λ

2δ (ω −ωλ

k −′k )∫∫

λ∑

Effective massm*

m= 1+ λ

Combined coupling and density of states of the phonons

λ =∂∂E

ΣR (E)E=0

= 2 dω α 2F(ω ) 1ω0

∞

∫Finite life-time

1τ e− ph (E)

= −2

1+ λℑ ΣR (E)⎡⎣ ⎤⎦ =

2π1+ λ

dω α 2F(ω )0

E

∫ ~ E 3

F. SUPERCONDUCTIVITY

1.Cooper instability2.Bardeen-Cooper-Schrieffer (BCS) theory3.Quasiclassical approximation4. Impurity scattering and dirty limit 5.Applications

Phenomenology (discovery by H. Kamerlingh Onnes, 1911)• perfect conductor

persistent current with decay-time >100000y• perfectly diamagnetic (for small external fields)

• occurs below a critical temperature in some metals and ceramics

Al (1.2K), Hg (4K), Nb (9K), MgB2 (40K), YBCO (92K),HBCCO (164K)

MS = −H → B = µ0 H + M( ) = 0 → χ = −1

σ (ω = 0) = ∞

F.1 COOPER INSTABILITY

Attractive interaction leads to instability of the Fermi sea

bq = ck+q↓ck↑k∑Response to amplitude of a pair

q,ω = 0 q,ω = 0

k,ωµ

−k + q,−ωµ

′k ,ωµ

− ′k + q,−ωµ

F.1 COOPER INSTABILITY

Divergent building block for a certain critical temperature

1 = VTc G(k,ωµ )G(−k,−ωµ )k ,ωµ <ωD

∑ = VTc1

ωµ2 + k

2k ,ωµ <ωD

∑

Tc = 1.13ωDe−

1N (0)VSolution:

• The Fermi sea becomes unstable at the critical temperature!• Finite Tc for arbitrarily small interaction!

Simplified interaction:

g 2 D ≈ V for ωµ , ′ωµ <ωD

0 else

⎧⎨⎪

⎩⎪

F.2 BCS THEORY

Mean field approximation for the interaction termc↑†c†↓c↓c↑

= c↑†c†↓ c↓c↑ + c↑

†c†↓ − c↑†c†↓( )

δ

c↓c↑ + c↑†c†↓ c↓c↑ − c↓c↑( )

δ

+ c↑†c†↓ − c↑

†c†↓( ) c↓c↑ − c↓c↑( )~δ 2 neglected

≈ c↑†c†↓ c↓c↑ + c↑

†c†↓ c↓c↑ − c↑†c†↓ c↓c↑

Mean field hamiltonian c−k+q↓ck↑ = bkδq,0

HMF = kckσ†

kσ∑ ckσ − ck↑

† c−k↓† Δ + c−k↓ck↑ Δ*( ) + Δ* bk

k∑

k∑

Δ = V bkk∑

Quadratic hamiltonian, can be diagonalized by the method of “equation of motion”

F.2 BCS THEORY

Nambu formalism: ψ k =ck↑c†−k↓

⎛

⎝⎜⎜

⎞

⎠⎟⎟

ψ k† = ck↑

† c−k↓( )Gorkov-Greens function:

G(k,τ ) = − Tck↑(τ )

c−k↓(τ )

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗ ck↑(0) c−k↓(0)( ) =

G(k,τ ) F(k,τ )−F*(k,τ ) G(k,τ )

⎛

⎝⎜⎜

⎞

⎠⎟⎟

Hamiltonian: H =k −Δ

Δ* −k

⎛

⎝⎜⎜

⎞

⎠⎟⎟

G(k,ωµ ) = (iωµ − H )−1 =

−iωµ − kτ 3 + Δωµ2 + k

2 + Δ2

Δ = 0 −ΔΔ* 0

⎛

⎝⎜⎞

⎠⎟

F.2 BCS THEORY

Consequences: • Pair amplitude/anomalous GF F(k,ω) is finite for finite Δ• Normal GF G is different from the free fermion form• Δ has to be determined selfconsistently, i.e. it depends on F

G(k,ωµ ) = −iωµ + k

ωµ2 + k

2 + Δ2 ; F(k,ωµ ) =Δ

ωµ2 + k

2 + Δ2

Unusual spectral properties (viz. quasiparticles):Excitation energies determined from poles of retarded GF

G(k,E) = E + kE2 − k

2 − Δ2Energy gapEk = ± k

2 + Δ 2

k =k2

2m− EF

Ek =k =0

Δ > 0

F.2 BCS THEORY

Finite pairing amplitude (determined self-consistently)

Δ = λT Δωµ2 + Δ2

µ∑ Δ(T = 0) ~ωDe

−1/λ

Tc ~ωDe−1/λ

Δ = 3.53kBTcUniversal BCS ratio

Excitation gap

Ek = ± k2 + Δ2

Density of states

N(E) =E

E2 − Δ2E > Δ

0 E < Δ

⎧

⎨⎪

⎩⎪

F.3 QUASICLASSICAL APPROXIMATION

G(r , p,ωµ ) = d 3ρG(r + ρ / 2, r − ρ / 2,ωµ )∫

g(r , vF ,ωµ ) =iπ

dpτ 3G(r , p,ωµ )∫

• Integrating out fast oscillation• Equation for the envelope function

Wigner representation (strongly peaked at Fermi momentum)

Quasiclassical Greens function

EILENBERGER EQUATION

−ivF∇g(r , vF ,ωµ ) =

iωµτ 3 − iΔ + Σ '(r ,ωµ )τ 3, g(r , vF ,ωµ )⎡⎣ ⎤⎦

• transport like equation• homogenous (additional condition necessary• impurities, phonon scattering are in self-energy• occurence of coherence lengths ξ0 or ξT obvious

ξ0 =vF2Δ

ξT =vF2πT

limp = vFτ imp

EILENBERGER EQUATION

Homogenous Differential equation supplemented by normalization g2 (r , vF ,ωµ ) = 1

Vector potential: ∇g(r )→ ∇− ieAτ 3, g(

r )⎡⎣ ⎤⎦ = ∇, g(r )⎡⎣ ⎤⎦

Selfconsistency: Δ(r ) = λT go.d .(r , vF ,ωµ ) vFµ

∑Current density:

j (r ) = −ieN(0)πT vFTrτ 3g(

r , vF ,ωµ ) vFµ∑

IMPURITY SCATTERING

Impurity selfenergy in selfconsistent Born approximation

Σimp (r ,ωµ ) = Ni V

2 τ 3G(r , p,ωµ )τ 3

p∑

=i

2τ impg(r , vF ,ωµ ) vF τ 3

g(ωµ ) vF = g(ωµ )

Homogenous solution

g(ωµ ) =1Ωµ

ωµτ 3 + Δτ1( )0 = ωµτ 3 + Δτ1 +

12τ imp

g(vF ,ωµ ) vF , g(vF ,ωµ )

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Anderson theorem:Thermodynamics not affected(Tc, Delta, density of states)

g1(r ,ωµ ) = −τ impg0 (

r ,ωµ )∇g0 (r ,ωµ )

THE DIRTY LIMIT

g(r , vF ,ωµ ) = g0 (r ,ωµ ) +

vFg1(r ,ωµ ) +…Expansion in anisotropy

Using Eilenberger and normalization

−∇ g0 (r ,ωµ )∇g0 (

r ,ωµ )( ) = ωµτ 3 + Δ, g0 (r ,ωµ )⎡⎣ ⎤⎦

Usadel equation

Current density:j (r ) = i πσ

2eT Trτ 3g0 (

r ,ωµ )∇g0 (r ,ωµ )

µ∑

σ = 2e2N(0)D D =13vF2τ imp