RevModPhys.82

Berry phase effects on electronic properties

Di Xiao

Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge,Tennessee 37831, USA

Ming-Che Chang

Department of Physics, National Taiwan Normal University, Taipei 11677, Taiwan

Qian Niu

Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA

Published 6 July 2010

Ever since its discovery the notion of Berry phase has permeated through all branches of physics.Over the past three decades it was gradually realized that the Berry phase of the electronic wavefunction can have a profound effect on material properties and is responsible for a spectrum ofphenomena, such as polarization, orbital magnetism, various quantum, anomalous, or spin Halleffects, and quantum charge pumping. This progress is summarized in a pedagogical manner in thisreview. A brief summary of necessary background is given and a detailed discussion of the Berryphase effect in a variety of solid-state applications. A common thread of the review is the semiclassicalformulation of electron dynamics, which is a versatile tool in the study of electron dynamics in thepresence of electromagnetic fields and more general perturbations. Finally, a requantization method isdemonstrated that converts a semiclassical theory to an effective quantum theory. It is clear that theBerry phase should be added as an essential ingredient to our understanding of basic materialproperties.

DOI: 10.1103/RevModPhys.82.1959 PACS numbers: 71.70.Ej, 72.10.Bg, 73.43.f, 77.84.s

CONTENTS

I. Introduction 1960A. Topical overview 1960B. Organization of the review 1961C. Basic concepts of the Berry phase 1962

1. Cyclic adiabatic evolution 19622. Berry curvature 19633. Example: The two-level system 1964

D. Berry phase in Bloch bands 1965II. Adiabatic Transport and Electric Polarization 1966

A. Adiabatic current 1966B. Quantized adiabatic particle transport 1967

1. Conditions for nonzero particle transport incyclic motion 1967

2. Many-body interactions and disorder 19683. Adiabatic pumping 1969

C. Electric polarization of crystalline solids 19691. The Rice-Mele model 1970

III. Electron Dynamics in an Electric Field 1971A. Anomalous velocity 1971B. Berry curvature: Symmetry considerations 1972C. The quantum Hall effect 1973D. The anomalous Hall effect 1974

1. Intrinsic versus extrinsic contributions 19742. Anomalous Hall conductivity as a Fermi

surface property 1975E. The valley Hall effect 1976

IV. Wave Packet: Construction and Properties 1976A. Construction of the wave packet and its orbital

moment 1977

B. Orbital magnetization 1978

C. Dipole moment 1979

D. Anomalous thermoelectric transport 1980

V. Electron Dynamics in Electromagnetic Fields 1980

A. Equations of motion 1981

B. Modified density of states 1981

1. Fermi volume 1982

2. Streda formula 1982

C. Orbital magnetization: Revisit 1982

D. Magnetotransport 1983

1. Cyclotron period 1983

2. The high-field limit 1984

3. The low-field limit 1984

VI. Electron Dynamics Under General Perturbations 1985

A. Equations of motion 1985

B. Modified density of states 1986

C. Deformed crystal 1986

D. Polarization induced by inhomogeneity 1987

1. Magnetic-field-induced polarization 1988

E. Spin texture 1989

VII. Quantization of Electron Dynamics 1989

A. Bohr-Sommerfeld quantization 1989

B. Wannier-Stark ladder 1990

C. de Haas–van Alphen oscillation 1990

D. Canonical quantization Abelian case 1991

VIII. Magnetic Bloch Bands 1992

A. Magnetic translational symmetry 1993

B. Basics of magnetic Bloch band 1994C. Semiclassical picture: Hyperorbits 1995D. Hall conductivity of hyperorbit 1996

IX. Non-Abelian Formulation 1997

REVIEWS OF MODERN PHYSICS, VOLUME 82, JULY–SEPTEMBER 2010

0034-6861/2010/823/195949 ©2010 The American Physical Society1959

http://dx.doi.org/10.1103/RevModPhys.82.1959

A. Non-Abelian electron wave packet 1997B. Spin Hall effect 1998C. Quantization of electron dynamics 1998D. Dirac electron 1999E. Semiconductor electron 2000F. Incompleteness of effective Hamiltonian 2001G. Hierarchy structure of effective theories 2001

X. Outlook 2002Acknowledgments 2003Appendix: Adiabatic Evolution 2003References 2003

I. INTRODUCTION

A. Topical overview

In 1984, Michael Berry wrote a paper that has gener-ated immense interest throughout the different fields ofphysics including quantum chemistry Berry, 1984. Thispaper is about the adiabatic evolution of an eigenenergystate when the external parameters of a quantum systemchange slowly and make up a loop in the parameterspace. In the absence of degeneracy, the eigenstate willsurely come back to itself when finishing the loop, butthere will be a phase difference equal to the time inte-gral of the energy divided by plus an extra, which isnow commonly known as the Berry phase.

The Berry phase has three key properties that makethe concept important Shapere and Wilczek, 1989;Bohm et al., 2003. First, it is gauge invariant. The eigen-wave function is defined by a homogeneous linear equa-tion the eigenvalue equation, so one has the gaugefreedom of multiplying it with an overall phase factorwhich can be parameter dependent. The Berry phase isunchanged up to integer multiple of 2 by such aphase factor, provided the eigenwave function is kept tobe single valued over the loop. This property makes theBerry phase physical, and the early experimental studieswere focused on measuring it directly through interfer-ence phenomena.

Second, the Berry phase is geometrical. It can be writ-ten as a line integral over the loop in the parameterspace and does not depend on the exact rate of changealong the loop. This property makes it possible to ex-press the Berry phase in terms of local geometricalquantities in the parameter space. Indeed, Berry himselfshowed that one can write the Berry phase as an integralof a field, which we now call the Berry curvature, over asurface suspending the loop. A large class of applica-tions of the Berry phase concept occur when the param-eters themselves are actually dynamical variables of slowdegrees of freedom. The Berry curvature plays an essen-tial role in the effective dynamics of these slow vari-ables. The vast majority of applications considered inthis review are of this nature.

Third, the Berry phase has close analogies to gaugefield theories and differential geometry Simon, 1983.This makes the Berry phase a beautiful, intuitive, andpowerful unifying concept, especially valuable in today’sever specializing physical science. In primitive terms, the

Berry phase is like the Aharonov-Bohm phase of acharged particle traversing a loop including a magneticflux, while the Berry curvature is like the magnetic field.The integral of the Berry curvature over closed surfaces,such as a sphere or torus, is known to be topological andquantized as integers Chern numbers. This is analo-gous to the Dirac monopoles of magnetic charges thatmust be quantized in order to have a consistentquantum-mechanical theory for charged particles inmagnetic fields. Interestingly, while the magnetic mono-poles are yet to be detected in the real world, the topo-logical Chern numbers have already found correspon-dence with the quantized Hall conductance plateaus inthe spectacular quantum Hall phenomenon Thouless etal., 1982.

This review concerns applications of the Berry phaseconcept in solid-state physics. In this field, we are typi-cally interested in macroscopic phenomena which is slowin time and smooth in space in comparison with theatomic scales. Not surprisingly, the vast majority of ap-plications of the Berry phase concept are discussed here.This field of physics is also extremely diverse, and wehave many layers of theoretical frameworks with differ-ent degrees of transparency and validity Ashcroft andMermin, 1976; Marder, 2000. Therefore, a unifying or-ganizing principle such as the Berry phase concept isparticularly valuable.

We focus our attention on electronic properties, whichplay a dominant role in various aspects of material prop-erties. The electrons are the glue of materials and theyare also the agents responding swiftly to external fieldsand giving rise to strong and useful signals. A basic para-digm of the theoretical framework is based on the as-sumption that electrons are in Bloch waves travelingmore or less independently in periodic potentials of thelattice, except that the Pauli exclusion principle has tobe satisfied and electron-electron interactions are takencare of in some self-consistent manner. Much of our in-tuition on electron transport is derived from the semi-classical picture that electrons behave almost as free par-ticles in response to external fields provided one uses theband energy in place of the free-particle dispersion. Be-cause of this, first-principles studies of electronic prop-erties tend to document only the energy band structuresand various density profiles.

There has been overwhelming evidence that such asimple picture cannot give complete account of effectsto first order in electromagnetic fields. A prime exampleis the anomalous velocity, a correction to the usual qua-siparticle group velocity from the band energy disper-sion. This correction can be understood from a linearresponse analysis: the velocity operator has off-diagonalelements and electric field mixes the bands so that theexpectation value of the velocity acquires an additionalterm to first order in the field Karplus and Luttinger,1954; Kohn and Luttinger, 1957. The anomalous veloc-ity can also be understood as due to the Berry curvatureof the Bloch states, which exist in the absence of theexternal fields and manifest in the quasiparticle velocitywhen the crystal momentum is moved by external forces

1960 Xiao, Chang, and Niu: Berry phase effects on electronic properties

Rev. Mod. Phys., Vol. 82, No. 3, July–September 2010

Chang and Niu, 1995, 1996; Sundaram and Niu, 1999.This understanding enabled us to make a direct connec-tion with the topological Chern number formulation ofthe quantum Hall effect Thouless et al., 1982; Kohmoto,1985, providing motivation as well as confidence in ourpursuit of the eventually successful intrinsic explanationof the anomalous Hall effect Jungwirth et al., 2002; Na-gaosa et al., 2010.

Interestingly enough, the traditional view cannot evenexplain some basic effects to zeroth order of the fields.The two basic electromagnetic properties of solids as amedium are the electric polarization and magnetization,which can exist in the absence of electric and magneticfields in ferroelectric and ferromagnetic materials. Theirclassical definitions were based on the picture of boundcharges and currents, but these are clearly inadequatefor the electronic contribution and it was known that thepolarization and orbital magnetization cannot be deter-mined from the charge and current densities in the bulkof a crystal at all. A breakthrough on electric polariza-tion was made in the early 1990s by linking it with thephenomenon of adiabatic charge transport and express-ing it in terms of the Berry phase1 across the entire Bril-louin zone Resta, 1992; King-Smith and Vanderbilt,1993. Based on the Berry phase formula, one can nowroutinely calculate polarization related properties usingfirst-principles methods, with a typical precision of thedensity functional theory. The breakthrough on orbitalmagnetization came only recently, showing that it notonly consists of the orbital moments of quasiparticlesbut also contains a Berry curvature contribution of to-pological origin Thonhauser et al., 2005; Xiao et al.,2005; Shi et al., 2007.

In this article, we follow the traditional semiclassicalformalism of quasiparticle dynamics, only to make itmore rigorous by including the Berry curvatures in thevarious facets of the phase space including the adiabatictime parameter. All of the above-mentioned effects aretransparently revealed with complete precision of thefull quantum theory. A number of new and related ef-fects, such as anomalous thermoelectric, valley Hall, andmagnetotransport, are easily predicted, and other effectsdue to crystal deformation and order parameter inho-mogeneity can also be explored without difficulty. More-over, by including Berry phase induced anomalous trans-port between collisions and “side jumps” duringcollisions which is itself a kind of Berry phase effect,the semiclassical Boltzmann transport theory can givecomplete account of linear response phenomena inweakly disordered systems Sinitsyn, 2008. On a micro-scopic level, although the electron wave-packet dynam-ics is yet to be directly observed in solids, the formalismhas been replicated for light transport in photonic crys-tals, where the associated Berry phase effects are vividlyexhibited in experiments Bliokh et al., 2008. Finally, it

is possible to generalize the semiclassical dynamics in asingle band into one with degenerate or nearly degener-ate bands Culcer et al., 2005; Shindou and Imura, 2005,and one can study transport phenomena where inter-band coherence effects as in spin transport, only to real-ize that the Berry curvatures and quasiparticle magneticmoments become non-Abelian i.e., matrices.

The semiclassical formalism is certainly amendable toinclude quantization effects. For integrable dynamics,such as Bloch oscillations and cyclotron orbits, one canuse the Bohr-Sommerfeld or Einstein-Brillouin-Kellerquantization rule. The Berry phase enters naturally as ashift to the classical action, affecting the energies of thequantized levels, e.g., the Wannier-Stark ladders and theLandau levels. A high point of this kind of application isthe explanation of the intricate fractal-like Hofstadterspectrum Chang and Niu, 1995, 1996. A recent break-through has also enabled us to find the density of quan-tum states in the phase space for the general case in-cluding nonintegrable systems Xiao et al., 2005,revealing Berry curvature corrections which should havebroad impacts on calculations of equilibrium as well astransport properties. Finally, one can execute a general-ized Peierls substitution relating the physical variables tothe canonical variables, turning the semiclassical dynam-ics into a full quantum theory valid to first order in thefields Chang and Niu, 2008. Spin-orbit coupling andanomalous corrections to the velocity and magnetic mo-ment are all found from a simple explanation of thisgeneralized Peierls substitution.

Therefore, it is clear that Berry phase effects in solid-state physics are not something just nice to be foundhere and there; the concept is essential for a coherentunderstanding of all the basic phenomena. It is the pur-pose of this review to summarize a theoretical frame-work which continues the traditional semiclassical pointof view but with a much broader range of validity. It isnecessary and sufficient to include the Berry phases andgradient energy corrections in addition to the energydispersions in order to account for all phenomena up tofirst order in the fields.

B. Organization of the review

This review can be divided into three main parts. InSec. II we consider the simplest example of Berry phasein crystals: the adiabatic transport in a band insulator. Inparticular, we show that induced adiabatic current dueto a time-dependent perturbation can be written as aBerry phase of the electronic wave functions. Based onthis understanding, the modern theory of electric polar-ization is reviewed. In Sec. III the electron dynamics inthe presence of an electric field is discussed as a specificexample of the time-dependent problem, and the basicformula developed in Sec. II can be directly applied. Inthis case, the Berry phase is made manifest as transversevelocity of the electrons, which gives rise to a Hall cur-rent. We then apply this formula to study the quantum,anomalous, and valley Hall effect.

1Also called Zak’s phase, it is independent of the Berry cur-vature which only characterizes Berry phases over loops con-tinuously shrinkable to zero Zak, 1989.

1961Xiao, Chang, and Niu: Berry phase effects on electronic properties


To study the electron dynamics under spatial-dependent perturbations, we turn to the semiclassicalformalism of Bloch electron dynamics, which has provento be a powerful tool to investigate the influence ofslowly varying perturbations on the electron dynamics.In Sec. IV we discuss the construction of the electronwave packet and show that the wave packet carries anorbital magnetic moment. Two applications of the wave-packet approach, the orbital magnetization and anoma-lous thermoelectric transport in ferromagnet, are dis-cussed. In Sec. V the wave-packet dynamics in thepresence of electromagnetic fields is studied. We showthat the Berry phase not only affects the equations ofmotion but also modifies the electron density of states inthe phase space, which can be changed by applying amagnetic field. The formula for orbital magnetization isrederived using the modified density of states. We alsopresent a comprehensive study of the magnetotransportin the presence of the Berry phase. The electron dynam-ics under more general perturbations is discussed in Sec.VI. We also present two applications: electron dynamicsin deformed crystals and polarization induced by inho-mogeneity.

In the remaining part of the review we focus on therequantization of the semiclassical formulation. In Sec.VII the Bohr-Sommerfeld quantization is reviewed indetail. With its help, one can incorporate the Berryphase effect into the Wannier-Stark ladders and the Lan-dau levels very easily. In Sec. VIII we show that thesame semiclassical approach can be applied to systemssubject to a very strong magnetic field. One only has toseparate the field into a quantization part and a pertur-bation. The former should be treated quantum mechani-cally to obtain the magnetic Bloch band spectrum whilethe latter is treated perturbatively. Using this formalism,the cyclotron motion, the splitting into magnetic sub-bands, and the quantum Hall effect are discussed. InSec. IX we review recent advances on the non-AbelianBerry phase in degenerate bands. The Berry connectionnow plays an explicit role in spin dynamics and is deeplyrelated to the spin-orbit interaction. The relativisticDirac electrons and the Kane model in semiconductorsare cited as two applied examples. Finally, we discuss therequantization of the semiclassical theory and the hier-archy of effective quantum theories.

We do not attempt to cover all of the Berry phaseeffects in this review. Interested readers can consult thefollowing: Shapere and Wilczek 1989; Nenciu 1991;Resta 1994, 2000; Thouless 1998; Bohm et al. 2003;Teufel 2003; Chang and Niu 2008. In this review, wefocus on a pedagogical and self-contained approach,with the main focus on the semiclassical formalism ofBloch electron dynamics Chang and Niu, 1995, 1996;Sundaram and Niu, 1999. We start with the simplestcase, then gradually expand the formalism as more com-plicated physical situations are considered. Whenever anew ingredient is added, a few applications are providedto demonstrate the basic ideas. The vast number of ap-plications we discuss is a reflection of the universality ofthe Berry phase effect.

C. Basic concepts of the Berry phase

In this section we introduce the basic concepts of theBerry phase. Following Berry’s original paper Berry,1984, we first discuss how the Berry phase arises duringthe adiabatic evolution of a quantum state. We then in-troduce the local description of the Berry phase in termsof the Berry curvature. A two-level model is used todemonstrate these concepts as well as some importantproperties, such as the quantization of the Berry phase.Our aim is to provide a minimal but self-contained in-troduction. For a detailed account of the Berry phase,including its mathematical foundation and applicationsin a wide range of fields in physics, see Shapere andWilczek 1989 and Bohm et al. 2003, and referencestherein.

1. Cyclic adiabatic evolution

Consider a physical system described by a Hamil-tonian that depends on time through a set of param-eters, denoted by R= R1 ,R2 , . . . , i.e.,

H = HR, R = Rt . 1.1

We are interested in the adiabatic evolution of the sys-tem as Rt moves slowly along a path C in the param-eter space. For this purpose, it is useful to introduce aninstantaneous orthonormal basis from the eigenstates ofHR at each value of the parameter R, i.e.,

HRnR = nRnR . 1.2

However, Eq. 1.2 alone does not completely determinethe basis function nR; it still allows an arbitraryR-dependent phase factor of nR. One can make aphase choice, also known as a gauge, to remove thisarbitrariness. Here we require that the phase of the basisfunction is smooth and single valued along the path C inthe parameter space.2

According to the quantum adiabatic theorem Kato,1950; Messiah, 1962, a system initially in one of itseigenstates n„R0… will stay as an instantaneous eigen-state of the Hamiltonian H„Rt… throughout the pro-cess. A derivation can be found in the Appendix.Therefore the only degree of freedom we have is thephase of the quantum state. We write the state at time tas

2Strictly speaking, such a phase choice is guaranteed only infinite neighborhoods of the parameter space. In the generalcase, one can proceed by dividing the path into several suchneighborhoods overlapping with each other, then use the factthat in the overlapping region the wave functions are relatedby a gauge transformation of form 1.7.



nt = eintexp−i

0

t

dtn„Rt…n„Rt… ,

1.3

where the second exponential is known as the dynamicalphase factor. Inserting Eq. 1.3 into the time-dependentSchrödinger equation

i

tnt = H„Rt…nt 1.4

and multiplying it from the left by n„Rt…, one findsthat n can be expressed as a path integral in the param-eter space

n = C

dR · AnR , 1.5

where AnR is a vector-valued function

AnR = inR

RnR . 1.6

This vector AnR is called the Berry connection or theBerry vector potential. Equation 1.5 shows that, in ad-dition to the dynamical phase, the quantum state willacquire an additional phase n during the adiabatic evo-lution.

Obviously, AnR is gauge dependent. If we make agauge transformation

nR → eiRnR , 1.7

with R an arbitrary smooth function and AnR trans-forms according to

AnR → AnR −

RR . 1.8

Consequently, the phase n given by Eq. 1.5 will bechanged by „R0…−„RT… after the transformation,where R0 and RT are the initial and final points ofthe path C. This observation has led Fock 1928 to con-clude that one can always choose a suitable R suchthat n accumulated along the path C is canceled out,leaving Eq. 1.3 with only the dynamical phase. Becauseof this, the phase n has long been deemed unimportantand it was usually neglected in the theoretical treatmentof time-dependent problems.

This conclusion remained unchallenged until Berry1984 reconsidered the cyclic evolution of the systemalong a closed path C with RT=R0. The phase choicewe made earlier on the basis function nR requireseiR in the gauge transformation Eq. 1.7 to be singlevalued, which implies

„R0… − „RT… = 2 integer. 1.9

This shows that n can be only changed by an integermultiple of 2 under the gauge transformation Eq.1.7 and it cannot be removed. Therefore for a closedpath, n becomes a gauge-invariant physical quantity,

now known as the Berry phase or geometric phase ingeneral; it is given by

n = C

dR · AnR . 1.10

From the above definition, we can see that the Berryphase only depends on the geometric aspect of theclosed path and is independent of how Rt varies intime. The explicit time dependence is thus not essentialin the description of the Berry phase and will bedropped in the following discussion.

2. Berry curvature

It is useful to define, in analogy to electrodynamics, agauge-field tensor derived from the Berry vector poten-tial:

n R =

RA

nR −

RA

nR

= i nRR

nRR

− ↔ . 1.11

This field is called the Berry curvature. Then accordingto Stokes’s theorem the Berry phase can be written as asurface integral

n = S

dR ∧ dR 12

n R , 1.12

where S is an arbitrary surface enclosed by the path C. Itcan be verified from Eq. 1.11 that, unlike the Berryvector potential, the Berry curvature is gauge invariantand thus observable.

If the parameter space is three dimensional, Eqs.1.11 and 1.12 can be recast into a vector form

nR = RAnR , 1.11

n = S

dS · nR . 1.12

The Berry curvature tensor n and vector n are re-

lated by n = n with the Levi-Cività anti-

symmetry tensor. The vector form gives us an intuitivepicture of the Berry curvature: it can be viewed as themagnetic field in the parameter space.

Besides the differential formula given in Eq. 1.11,the Berry curvature can be also written as a summationover the eigenstates:

n R = i

nn

nH/RnnH/Rn − ↔ n − n

2 .

1.13

The curvature can be obtained from Eq. 1.11 usingnH /Rn= n /R nn−n for nn. The sum-mation formula has the advantage that no differentia-tion on the wave function is involved, therefore it can beevaluated under any gauge choice. This property is par-



ticularly useful for numerical calculations, in which thecondition of a smooth phase choice of the eigenstates isnot guaranteed in standard diagonalization algorithms.It has been used to evaluate the Berry curvature in crys-tals with the eigenfunctions supplied from first-principles calculations Fang et al., 2003; Yao et al.,2004.

Equation 1.13 offers further insight on the origin ofthe Berry curvature. The adiabatic approximationadopted earlier is essentially a projection operation, i.e.,the dynamics of the system is restricted to the nth en-ergy level. In view of Eq. 1.13, the Berry curvature canbe regarded as the result of the “residual” interaction ofthose projected-out energy levels. In fact, if all energylevels are included, it follows from Eq. 1.13 that thetotal Berry curvature vanishes for each value of R,

n

n R = 0. 1.14

This is the local conservation law for the Berrycurvature.3 Equation 1.13 also shows that

n R be-comes singular if two energy levels nR and nR arebrought together at certain value of R. This degeneracypoint corresponds to a monopole in the parameterspace; an explicit example is given below.

So far we have discussed the situation where a singleenergy level can be separated out in the adiabatic evo-lution. However, if the energy levels are degenerate,then the dynamics must be projected to a subspacespanned by those degenerate eigenstates. Wilczek andZee 1984 showed that in this situation non-AbelianBerry curvature naturally arises. Culcer et al. 2005 andShindou and Imura 2005 discussed the non-AbelianBerry curvature in the context of degenerate Blochbands. In the following we focus on the Abelian formu-lation and defer the discussion of the non-Abelian Berrycurvature to Sec. IX.

Compared to the Berry phase which is always associ-ated with a closed path, the Berry curvature is truly alocal quantity. It provides a local description of the geo-metric properties of the parameter space. Moreover, sofar we have treated the adiabatic parameters as passivequantities in the adiabatic process, i.e., their time evolu-tion is given from the outset. Later we will show that theparameters can be regarded as dynamical variables andthe Berry curvature will directly participate in the dy-namics of the adiabatic parameters Kuratsuji and Iida,1985. In this sense, the Berry curvature is a more fun-damental quantity than the Berry phase.

3. Example: The two-level system

Consider a concrete example: a two-level system. Thepurpose to study this system is twofold. First, as a simplemodel, it demonstrates the basic concepts as well as sev-eral important properties of the Berry phase. Second, itwill be repeatedly used later in this article, although indifferent physical context. It is therefore useful to gothrough the basis of this model.

The generic Hamiltonian of a two-level system takesthe following form:

H = hR · , 1.15

where are the Pauli matrices. Despite its simple form,the above Hamiltonian describes a number of physicalsystems in condensed-matter physics for which the Berryphase effect has been discussed. Examples include spin-orbit coupled systems Culcer et al., 2003; Liu et al.,2008, linearly conjugated diatomic polymers Su et al.,1979; Rice and Mele, 1982, one-dimensional ferroelec-trics Vanderbilt and King-Smith, 1993; Onoda et al.,2004b, graphene Semenoff, 1984; Haldane, 1988, andBogoliubov quasiparticles Zhang et al., 2006.

Parametrize h by its polar angle and azimuthal angle, h=hsin cos , sin sin , cos , the two eigen-states, with energies ±h, are

u− = sin 2e−i

− cos 2, u+ = cos 2e−i

sin 2

. 1.16

We are, of course, free to add an arbitrary phase to thesewave functions. Consider the lower energy level. TheBerry connection is given by

A = uiu = 0, 1.17a

A = uiu = sin2

2, 1.17b

and the Berry curvature is

= A − A = 12 sin . 1.18

However, the phase of u− is not defined at the southpole =. We can choose another gauge by multiply-ing u− by ei so that the wave function is smooth andsingle valued everywhere except at the north pole. Un-der this gauge we find A=0 and A=−cos2 /2, and thesame expression for the Berry curvature as in Eq. 1.18.This is not surprising because the Berry curvature is agauge-independent quantity and the Berry connection isnot.4

3The conservation law is obtained under the condition thatthe full Hamiltonian is known. However, in practice one usu-ally deals with effective Hamiltonians which are obtainedthrough some projection process of the full Hamiltonian.Therefore there will always be some “residual” Berry curva-ture accompanying the effective Hamiltonian see Chang andNiu 2008 and discussions in Sec. IX.

4One can verify that u= „sin /2e−i ,−cos /2ei−1…

T isalso an eigenstate. The phase accumulated by such a statealong the loop defined by = /2 is =2− 1

2 , which seemsto imply that the Berry phase is gauge dependent. This is be-cause for an arbitrary the basis function u is not singlevalued; one must also trace the phase change in the basis func-tion. For integer value of the function u is single valuedalong the loop and the Berry phase is well defined up to aninteger multiple of 2.



If hR depends on a set of parameters R, then

R1R2=

12

,cos R1,R2

. 1.19

Several important properties of the Berry curvaturecan be revealed by considering the specific case of h= x ,y ,z. Using Eq. 1.19, we find the Berry curvaturein its vector form

=12

hh3 . 1.20

One recognizes that Eq. 1.20 is the field generated by amonopole at the origin h=0 Dirac, 1931; Wu and Yang,1975; Sakurai, 1993, where the two energy levels be-come degenerate. Therefore the degeneracy points actas sources and drains of the Berry curvature flux. Inte-grate the Berry curvature over a sphere containing themonopole, which is the Berry phase on the sphere; wefind

1

2

S2dd = 1. 1.21

In general, the Berry curvature integrated over a closedmanifold is quantized in the units of 2 and equals tothe net number of monopoles inside. This number iscalled the Chern number and is responsible for a num-ber of quantization effects discussed below.

D. Berry phase in Bloch bands

Above we introduced the basic concepts of the Berryphase for a generic system described by a parameter-dependent Hamiltonian. We now consider its realizationin crystalline solids. As we shall see, the band structureof crystals provides a natural platform to investigate theoccurrence of the Berry phase effect.

Within the independent electron approximation, theband structure of a crystal is determined by the follow-ing Hamiltonian for a single electron:

H =p2

2m+ Vr , 1.22

where Vr+a=Vr is the periodic potential with a theBravais lattice vector. According to Bloch’s theorem, theeigenstates of a periodic Hamiltonian satisfy the follow-ing boundary condition:5

nqr + a = eiq·anqr , 1.23

where n is the band index and q is the crystal momen-tum, which resides in the Brillouin zone. Thus the sys-tem is described by a q-independent Hamiltonian with aq-dependent boundary condition Eq. 1.23. To complywith the general formalism of the Berry phase, we make

the following unitary transformation to obtain aq-dependent Hamiltonian:

Hq = e−iq·rHeiq·r =p + q2

2m+ Vr . 1.24

The transformed eigenstate unqr=e−iq·rnqr is just thecell-periodic part of the Bloch function. It satisfies thestrict periodic boundary condition

unqr + a = unqr . 1.25

This boundary condition ensures that all the eigenstateslive in the same Hilbert space. We can thus identify theBrillouin zone as the parameter space of the trans-formed Hamiltonian Hq and unq as the basis func-tion.

Since the q dependence of the basis function is inher-ent to the Bloch problem, various Berry phase effectsare expected in crystals. For example, if q is forced tovary in the momentum space, then the Bloch state willpick up a Berry phase:

n = C

dq · unqiqunq . 1.26

We emphasize that the path C must be closed to make na gauge-invariant quantity with physical significance.

Generally speaking, there are two ways to generate aclosed path in the momentum space. One can apply amagnetic field, which induces a cyclotron motion along aclosed orbit in the q space. This way the Berry phase canmanifest in various magneto-oscillatory effects Mikitikand Sharlai, 1999, 2004, 2007, which have been ob-served in metallic compound LaRhIn5 Goodrich et al.,2002, and most recently graphene systems Novoselovet al., 2005, 2006; Zhang et al., 2005. Such a closed orbitis possible only in two- or three-dimensional 3D sys-tems see Sec. VII.A. Following our discussion in Sec.I.C, we define the Berry curvature of the energy bandsby

nq = q unqiqunq . 1.27

The Berry curvature nq is an intrinsic property of theband structure because it only depends on the wavefunction. It is nonzero in a wide range of materials, inparticular, crystals with broken time-reversal or inver-sion symmetry. In fact, once we have introduced the con-cept of the Berry curvature, a closed loop is not neces-sary because the Berry curvature itself is a local gauge-invariant quantity. It is now well recognized thatinformation on the Berry curvature is essential in aproper description of the dynamics of Bloch electrons,which has various effects on transport and thermody-namic properties of crystals.

One can also apply an electric field to cause a linearvariation in q. In this case, a closed path is realized whenq sweeps the entire Brillouin zone. To see this, we notethat the Brillouin zone has the topology of a torus: thetwo points q and q+G can be identified as the samepoint, where G is the reciprocal lattice vector. This can

5Through out this article, q refers to the canonical momen-tum and k is reserved for mechanical momentum.



be seen by noting that nq and nq+G satisfy thesame boundary condition in Eq. 1.23; therefore, theycan at most differ by a phase factor. The torus topologyis realized by making the phase choice nq= nq+G. Consequently, unq and unq+G satisfy thefollowing phase relation:

unqr = eiG·runq+Gr . 1.28

This gauge choice is called the periodic gauge Resta,2000.

In this case, the Berry phase across the Brillouin zoneis called Zak’s phase Zak, 1989

n = BZ

dq · unqiqunq . 1.29

This phase plays an important role in the formation ofWannier-Stark ladders Wannier, 1962; see Sec. VII.B.We note that this phase is entirely due to the torus to-pology of the Brillouin zone, and it is the only way torealize a closed path in a one-dimensional parameterspace. By analyzing the symmetry properties of Wannierfunctions Kohn, 1959 of a one-dimensional crystal, Zak1989 showed that n is either 0 or in the presence ofinversion symmetry; a simple argument is given in Sec.II.C. If the crystal lacks inversion symmetry, n can as-sume any value. Zak’s phase is also related to macro-scopic polarization of an insulator King-Smith andVanderbilt, 1993; Resta, 1994; Sipe and Zak, 1999; seeSec. II.C.

II. ADIABATIC TRANSPORT AND ELECTRICPOLARIZATION

One of the earlier examples of the Berry phase effectin crystals is the adiabatic transport in a one-dimensional band insulator, first considered by Thouless1983. He found that if the potential varies slowly intime and returns to itself after some time, the particletransport during the time cycle can be expressed as aBerry phase and it is an integer. This idea was later gen-eralized to many-body systems with interactions and dis-order provided that the Fermi energy always lies in abulk energy gap during the cycle Niu and Thouless,1984. Avron and Seiler 1985 studied the adiabatictransport in multiply connected systems. The scheme ofadiabatic transport under one or several controlling pa-rameters has proven very powerful and opened the doorto the field of parametric charge pumping Niu, 1990;Talyanskii et al., 1997; Brouwer, 1998; Switkes et al.,1999; Zhou et al., 1999. It also provides a firm founda-tion to the modern theory of polarization developed inthe early 1990s King-Smith and Vanderbilt, 1993; Ortizand Martin, 1994; Resta, 1994.

A. Adiabatic current

Consider a one-dimensional band insulator under aslowly varying time-dependent perturbation. We assume

the perturbation is periodic in time, i.e., the Hamiltoniansatisfies

Ht + T = Ht . 2.1

Since the time-dependent Hamiltonian still has thetranslational symmetry of the crystal, its instantaneouseigenstates have the Bloch form eiqxunq , t. It is con-venient to work with the q-space representation of theHamiltonian Hq , t see Eq. 1.24 with eigenstatesunq , t. We note that under this parametrization thewave vector q and time t are put on an equal footing asboth are independent coordinates of the parameterspace.

We are interested in the adiabatic current induced bythe variation in external potentials. Apart from an un-important overall phase factor and up to first order inthe rate of the change in the Hamiltonian, the wavefunction is given by see the Appendix

un − i nn

ununun/t

n − n. 2.2

The velocity operator in the q representation has theform vq , t=Hq , t /q.6 Hence, the average veloc-ity in a state of given q is found to first order as

vnq =nqq

− i nn

unH/qununun/t

n − n− c.c. ,

2.3

where c.c. denotes the complex conjugate. Using the factthat unH /qun= n−nun /q un and the iden-tity nunun=1, we find

vnq =nqq

− i un

q un

t − un

t un

q .

2.4

The second term is exactly the Berry curvature qtn de-

fined in the parameter space q , t see Eq. 1.11.Therefore Eq. 2.4 can be recast into a compact form

vnq =nqq

−qtn . 2.5

Upon integration over the Brillouin zone, the zeroth-order term given by the derivative of the band energyvanishes, and only the first-order term survives. The in-duced adiabatic current is given by

6The velocity operator is defined by v r= i /H ,r. Inthe q representation, it becomes vq=e−iq·ri /r ,Heiq·r

=Hq , t /q.



j = − n

BZ

dq

2qt

n , 2.6

where the sum is over filled bands. We have thus derivedthe result that the adiabatic current induced by a time-dependent perturbation in a band is equal to the q inte-gral of the Berry curvature qt

n Thouless, 1983.

B. Quantized adiabatic particle transport

Next we consider the particle transport for the nthband over a time cycle given by

cn = − 0

T

dtBZ

dq

2qt

n . 2.7

Since the Hamiltonian Hq , t is periodic in both t and q,the parameter space of Hq , t is a torus, schematicallyshown in Fig. 1a. By definition 1.12, 2cn is nothingbut the Berry phase over the torus.

In Sec. I.C.3 we showed that the Berry phase over aclosed manifold, the surface of a sphere S2 in that case,is quantized in the unit of 2. Here we prove that it isalso true in the case of a torus. Our strategy is to evalu-ate the surface integral 2.7 using Stokes’s theorem,which requires the surface to be simply connected. To dothat, we cut the torus open and transform it into a rect-angle, as shown in Fig. 1b. The basis function along thecontour of the rectangle is assumed to be single valued.We introduce x= t /T and y=q /2. According to Eq.1.10, the Berry phase in Eq. 2.7 can be written into acontour integral of the Berry vector potential, i.e.,

c =1

2A

B

dxAxx,0 + B

C

dyAy1,y

+ C

D

dxAxx,1 + D

A

dyAy0,y=

1

20

1

dxAxx,0 − Axx,1

− 0

1

dyAy0,y − Ay1,y , 2.8

where the band index n is dropped for simplicity. Nowconsider the integration over x. By definition Axx ,y= ux ,yixux ,y. Recall that ux ,0 and ux ,1describe physically equivalent states, therefore they canonly differ by a phase factor, i.e., eixxux ,1= ux ,0. We thus have

0

1

dxAxx,0 − Axx,1 = x1 − x0 . 2.9

Similarly,

0

1

dyAy0,y − Ay1,y = y1 − y0 , 2.10

where eiyyuy ,1= uy ,0. The total integral is

c =1

2x1 − x0 + y0 − y1 . 2.11

On the other hand, using the phase matching relations atthe four corners A, B, C, and D,

eix0u0,1 = u0,0 ,

eix1u1,1 = u1,0 ,

eiy0u1,0 = u0,0 ,

eiy1u1,1 = u0,1 ,

we obtain

u0,0 = eix1−x0+y0−y1u0,0 . 2.12

The single valuedness of u requires that the exponentmust be an integer multiple of 2. Therefore the trans-ported particle number c, given in Eq. 2.11, must bequantized. This integer is called the first Chern number,which characterizes the topological structure of the map-ping from the parameter space q , t to the Bloch statesuq , t. Note that in our proof we made no reference tothe original physical system; the quantization of theChern number is always true as long as the Hamiltonianis periodic in both parameters.

For a particular case in which the entire periodic po-tential is sliding, an intuitive picture of the quantizedparticle transport is the following. If the periodic poten-tial slides its position without changing its shape, we ex-pect that the electrons simply follow the potential. If thepotential shifts one spatial period in the time cycle, theparticle transport should be equal to the number of filledBloch bands double if the spin degeneracy is counted.This follows from the fact that there is on average onestate per unit cell in each filled band.

1. Conditions for nonzero particle transport in cyclic motion

We have shown that the adiabatic particle transportover a time period takes the form of the Chern number

A=(0,0) B=(1,0)

C=(1,1)D=(0,1)

x

y

(a) (b)

R(t)

q

FIG. 1. Brillouin zone as a torus. a A torus with its surfaceparametrized by q , t. The control parameter Rt is periodicin t. b The equivalence of a torus: a rectangle with periodicboundary conditions: AB=DC and AD=BC. To make use ofStokes’s theorem, we relax the boundary condition and allowthe wave functions on parallel sides to have different phases.



and it is quantized. However, the exact quantizationdoes not guarantee that the electrons will be transportedat the end of the cycle because zero is also an integer.According to the discussion in Sec. I.C.3, the Chernnumber counts the net number of monopoles enclosedby the surface. Therefore the number of transportedelectrons can be related to the number of monopoles,which are degeneracy points in the parameter space.

To formulate this problem, we let the Hamiltoniandepend on time through a set of control parametersRt, i.e.,

Hq,t = H„q,Rt…, Rt + T = Rt . 2.13

The particle transport is now given by, in terms of R,

cn =1

2 dRBZ

dq qRn . 2.14

If Rt is a smooth function of t, as it is usually the casefor physical quantities, then R must have at least twocomponents, say R1 and R2. Otherwise, the trajectory ofRt cannot trace out a circle on the torus see Fig. 1a.To find the monopoles inside the torus, we now relax theconstraint that R1 and R2 can only move on the surfaceand extend their domains inside the torus such that theparameter space of q ,R1 ,R2 becomes a toroid. Thus,the criterion for cn to be nonzero is that a degeneracypoint must occur somewhere inside the torus as one var-ies q, R1, and R2. In the context of quasi-one-dimensional ferroelectrics, Onoda et al. 2004b dis-cussed the situation where R has three components andshowed how the topological structure in the R spaceaffects the particle transport.

2. Many-body interactions and disorder

Above we considered only band insulators of nonin-teracting electrons. However, in real materials bothmany-body interactions and disorder are ubiquitous. Niuand Thouless 1984 studied this problem and showedthat in the general case the quantization of particletransport is still valid as long as the system remains aninsulator during the whole process.

Consider a time-dependent Hamiltonian of anN-particle system

Ht = i

N pi2

2m+ Uxi,t +

ij

N

Vxi − xj , 2.15

where the one-particle potential Uxi , t varies slowly intime and repeats itself in period T. Note that we havenot assumed any specific periodicity of the potentialUxi , t. The trick is to use the so-called twisted bound-ary condition by requiring that the many-body wavefunction satisfies

x1, . . . ,xi + L, . . . ,xN = eiLx1, . . . ,xi, . . . ,xN ,

2.16

where L is the size of the system. This is equivalent tosolving a -dependent Hamiltonian

H,t = expi xiHtexp− i xi 2.17

with the strict periodic boundary condition

;x1, . . . ,xi + L, . . . ,xN = ;x1, . . . ,xi, . . . ,xN .

2.18

The Hamiltonian H , t together with the boundarycondition 2.18 describes a one-dimensional systemplaced on a ring of length L and threaded by a magneticflux of /eL Kohn, 1964. We note that the abovetransformation 2.17 with the boundary condition 2.18is similar to that of the one-particle case, given by Eqs.1.24 and 1.25.

One can verify that the current operator is given byH , t /. For each , we can repeat the same stepsin Sec. II.A by identifying un in Eq. 2.2 as the many-

body ground-state 0 and un as the excited state. Wehave

j =

− i 0

0

t − 0

t 0

=

− t. 2.19

So far the derivation is formal and we still cannot seewhy the particle transport should be quantized. The keystep is achieved by realizing that if the Fermi energy liesin a gap, then the current j should be insensitive tothe boundary condition specified by Thouless, 1981;Niu and Thouless, 1984. Consequently, we can take thethermodynamic limit and average j over differentboundary conditions. Note that and +2 /L describethe same boundary condition in Eq. 2.16. Thereforethe parameter space for and t is a torus T2 : 02 /L ,0 tT. The particle transport is given by

c = −1

2

0

T

dt0

2/L

d t, 2.20

which, according to the previous discussion, is quan-tized.

We emphasize that the quantization of the particletransport only depends on two conditions:

1 The ground state is separated from the excitedstates in the bulk by a finite energy gap.

2 The ground state is nondegenerate.

The exact quantization of the Chern number in thepresence of many-body interactions and disorder is re-markable. Usually, small perturbations to the Hamil-tonian result in small changes of physical quantities.However, the fact that the Chern number must be aninteger means that it can only be changed in a discon-tinuous way and does not change at all if the perturba-tion is small. This is a general outcome of the topologicalinvariance.



Later we show that the same quantity also appears inthe quantum Hall effect. Equation 2.19, the inducedcurrent, also provides a many-body formulation foradiabatic transport.

3. Adiabatic pumping

The phenomenon of adiabatic transport is sometimescalled adiabatic pumping because it can generate a dccurrent I via periodic variations of some parameters ofthe system, i.e.,

I = ec , 2.21

where c is the Chern number and is the frequency ofthe variation. Niu 1990 suggested that the exact quan-tization of the adiabatic transport can be used as a stan-dard for charge current and proposed an experimentalrealization in nanodevices, which could serve as a chargepump. Later a similar device was realized in the experi-mental study of acoustoelectric current induced by a sur-face acoustic wave in a one-dimensional channel in aGaAs-AlxGa1−x heterostructure Talyanskii et al., 1997.The same idea has led to the proposal of a quantum spinpump in an antiferromagnetic chain Shindou, 2005.

Recently much effort has focused on adiabatic pump-ing in mesoscopic systems Brouwer, 1998; Zhou et al.,1999; Avron et al., 2001, 2004; Sharma and Chamon,2001; Mucciolo et al., 2002; Zheng et al., 2003. Experi-mentally both charge and spin pumping have been ob-served in a quantum dot system Switkes et al., 1999;Watson et al., 2003. Instead of the wave function, thecentral quantity in a mesoscopic system is the scatteringmatrix. Brouwer 1998 showed that the pumped chargeover a time period is given by

Qm =e

AdX1dX2

m

IS

X1

SX2

, 2.22

where m labels the contact, X1 and X2 are two externalparameters whose trace encloses the area A in the pa-rameter space, and label the conducting channels,and S is the scattering matrix. Although the physicaldescriptions of these open systems are dramatically dif-ferent from the closed ones, the concepts of gauge fieldand geometric phase can still be applied. The integrandin Eq. 2.22 can be thought as the Berry curvatureX1X2

=−2IX1u X2

u if we identify the inner productof the state vector with the matrix product. Zhou et al.2003 showed the pumped charge spin is essentiallythe Abelian non-Abelian geometric phase associatedwith scattering matrix S.

C. Electric polarization of crystalline solids

Electric polarization is one of the fundamental quan-tities in condensed-matter physics, essential to anyproper description of dielectric phenomena of matter.Despite its importance, the theory of polarization incrystals had been plagued by the lack of a proper micro-scopic understanding. The main difficulty lies in the fact

that in crystals the charge distribution is periodic inspace, for which the electric dipole operator is not welldefined. This difficulty is most exemplified in covalentsolids, where the electron charges are continuously dis-tributed between atoms. In this case, simple integrationover charge density would give arbitrary values depend-ing on the choice of the unit cell Martin, 1972, 1974. Ithas prompted the question whether the electric polariza-tion can be defined as a bulk property. These problemsare eventually solved by the modern theory of polariza-tion King-Smith and Vanderbilt, 1993; Resta, 1994,where it is shown that only the change in polarizationhas physical meaning and it can be quantified using theBerry phase of the electronic wave function. The result-ing Berry phase formula has been very successful infirst-principles studies of dielectric and ferroelectric ma-terials. Resta and Vanderbilt 2007 reviewed recentprogress in this field.

Here we discuss the theory of polarization based onthe concept of adiabatic transport. Their relation is re-vealed by elementary arguments from macroscopic elec-trostatics Ortiz and Martin, 1994. We begin with

· Pr = − r , 2.23

where Pr is the polarization density and r is thecharge density. Coupled with the continuity equation

rt

+ · j = 0, 2.24

Eq. 2.23 leads to

· Pt

− j = 0. 2.25

Therefore up to a divergence-free part,7 the change inthe polarization density is given by

P = 0

T

dt j. 2.26

Equation 2.26 can be interpreted in the following way:The polarization difference between two states is givenby the integrated bulk current as the system adiabati-cally evolves from the initial state at t=0 to the finalstate at t=T. This description implies a time-dependentHamiltonian Ht, and the electric polarization can beregarded as “unquantized” adiabatic particle transport.The above interpretation is also consistent with experi-ments, as it is always the change in the polarization thathas been measured Resta and Vanderbilt, 2007.

Obviously, the time t in the Hamiltonian can be re-placed by any scalar that describes the adiabatic process.For example, if the process corresponds to a deforma-tion of the crystal, then it makes sense to use the param-

7The divergence-free part of the current is usually given bythe magnetization current. In a uniform system, such currentvanishes identically in the bulk. Hirst 1997 gave an in-depthdiscussion on the separation between polarization and magne-tization current.



eter that characterizes the atomic displacement within aunit cell. For general purpose, we assume the adiabatictransformation is parametrized by a scalar t with0=0 and T=1. It follows from Eqs. 2.6 and 2.26that

P = en

0

1

dBZ

dq2dq

n , 2.27

where d is the dimensionality of the system. This is theBerry phase formula obtained by King-Smith andVanderbilt 1993.

In numerical calculations, a two-point version of Eq.2.27 that involves only the initial and final states of thesystem is commonly used to reduce the computationalload. This version is obtained under the periodic gaugesee Eq. 1.28.8 The Berry curvature q

is written asq

A−Aq. Under the periodic gauge, A is periodic

in q, and integration of qA over q vanishes. Hence

P = e BZ

dq2dAq

n =0

1

. 2.28

In view of Eq. 2.28, both the adiabatic transport andthe electric polarization can be regarded as the manifes-tation of Zak’s phase, given in Eq. 1.29.

A price must be paid, however, to use the two-pointformula, namely, the polarization in Eq. 2.28 is deter-mined up to an uncertainty quantum. Since the integral2.28 does not track the history of , there is no infor-mation on how many cycles has gone through. Accord-ing to our discussion on quantized particle transport inSec. II.B, for each cycle an integer number of electronsare transported across the sample, hence the polariza-tion is changed by multiple of the quantum

eaV0

, 2.29

where a is the Bravais lattice vector and V0 is the volumeof the unit cell.

Because of this uncertainty quantum, the polarizationmay be regarded as a multivalued quantity with eachvalue differed by the quantum. With this in mind, con-sider Zak’s phase in a one-dimensional system with in-version symmetry. Now we know that Zak’s phase is just2 /e times the polarization density P. Under spatial in-version, P transforms to −P. On the other hand, inver-sion symmetry requires that P and −P describes thesame state, which is only possible if P and −P differ bymultiple of the polarization quantum ea. Therefore P iseither 0 or ea /2 modulo ea. Any other value of P willbreak the inversion symmetry. Consequently, Zak’sphase can only take the value 0 or modulo 2.

King-Smith and Vanderbilt 1993 further showedthat, based on Eq. 2.28, the polarization per unit cellcan be defined as the dipole moment of the Wanniercharge density,

P = − en dr rWnr2, 2.30

where Wnr is the Wannier function of the nth band,

Wnr − R = NV0BZ

dq23eiq·r−Runkr . 2.31

In this definition, one effectively maps a band insulatorinto a periodic array of localized distributions with trulyquantized charges. This resembles an ideal ionic crystalwhere the polarization can be understood in the classicalpicture of localized charges. The quantum uncertaintyfound in Eq. 2.29 is reflected by the fact that the Wan-nier center position is defined only up to a lattice vector.

Before concluding, we point out that the polarizationdefined above is clearly a bulk quantity as it is given bythe Berry phase of the ground-state wave function. Amany-body formulation was developed by Ortiz andMartin 1994 based on the work of Niu and Thouless1984.

Recent development in this field falls into two catego-ries. On the computational side, calculating polarizationin finite electric fields has been addressed, which has astrong influence on density functional theory in ex-tended systems Nunes and Vanderbilt, 1994; Nunes andGonze, 2001; Souza et al., 2002. On the theory side,Resta 1998 proposed a quantum-mechanical positionoperator for extended systems. It was shown that theexpectation value of such an operator can be used tocharacterize the phase transition between the metallicand insulating states Resta and Sorella, 1999; Souza etal., 2000 and is closely related to the phenomenon ofelectron localization Kohn, 1964.

1. The Rice-Mele model

So far our discussion of the adiabatic transport andelectric polarization has been rather abstract. We nowconsider a concrete example: a one-dimensional dimer-ized lattice model described by the following Hamil-tonian:

H = j t

2+ − 1j

2cj

†cj+1 + H.c. + − 1jcj†cj,

2.32

where t is the uniform hopping amplitude, is thedimerization order, and is a staggered sublattice po-tential. It is the prototype Hamiltonian for a class ofone-dimensional ferroelectrics. At half filling, the systemis a metal for ==0, and an insulator otherwise. Riceand Mele 1982 considered this model in the study ofsolitons in polyenes. It was later used to study ferroelec-tricity Vanderbilt and King-Smith, 1993; Onoda et al.,

8A more general phase choice is given by the path-independent gauge unq ,=eiq+G·runq+G ,, whereq is an arbitrary phase Ortiz and Martin, 1994.



2004b. If =0, it reduces to the celebrated Su-Shrieffer-Heeger model Su et al., 1979.

Assuming periodic boundary conditions, the Blochrepresentation of the above Hamiltonian is given byHq=hq ·, where

h = t cosqa

2,− sin

qa

2, . 2.33

This is the two-level model discussed in Sec. I.C.3. Itsenergy spectrum consists of two bands with eigenener-gies ±= ± 2+2 sin2 qa /2+ t2 cos2 qa /21/2. The degen-eracy point occurs at

= 0, = 0, q = /a . 2.34

The polarization is calculated using the two-point for-mula 2.28 with the Berry connection given by

Aq = qA + qA = sin2

2q , 2.35

where and are the spherical angles of h.Consider the case of =0. In the parameter space of

h, it lies in the x-y plane with = /2. As q varies from 0to 2 /a, changes from 0 to if 0 and 0 to − if0. Therefore the polarization difference betweenP and P− is ea /2. This is consistent with the obser-vation that the state with P− can be obtained by shift-ing the state with P by half of the unit cell length a.

Figure 2 shows the calculated polarization for arbi-trary and . If the system adiabatically evolves along aloop enclosing the degeneracy point 0,0 in the ,space, then the polarization will be changed by ea, whichmeans that if we allow , to change in time along thisloop, for example, t=0 sint and t=0 cost, aquantized charge of e is pumped out of the system afterone cycle. On the other hand, if the loop does not con-tain the degeneracy point, then the pumped charge iszero.

III. ELECTRON DYNAMICS IN AN ELECTRIC FIELD

The dynamics of Bloch electrons under the perturba-tion of an electric field is one of the oldest problems insolid-state physics. It is usually understood that whilethe electric field can drive electron motion in momen-tum space, it does not appear in the electron velocity;the latter is simply given by see, for example, Chap. 12of Ashcroft and Mermin 1976

vnq =nqq

. 3.1

Through recent progress on the semiclassical dynamicsof Bloch electrons it has been made increasingly clearthat this description is incomplete. In the presence of anelectric field, an electron can acquire an anomalous ve-locity proportional to the Berry curvature of the bandChang and Niu, 1995, 1996; Sundaram and Niu, 1999.This anomalous velocity is responsible for a number oftransport phenomena, in particular various Hall effects,which we study in this section.

A. Anomalous velocity

Consider a crystal under the perturbation of a weakelectric field E, which enters into the Hamiltonianthrough the coupling to the electrostatic potential r.A uniform E means that r varies linearly in space andbreaks the translational symmetry of the crystal so thatBloch’s theorem cannot be applied. To avoid this diffi-culty, one can let the electric field enter through a uni-form vector potential At that changes in time. Usingthe Peierls substitution, the Hamiltonian is written as

Ht =p + eAt2

2m+ Vr . 3.2

This is the time-dependent problem studied in last sec-tion. Transforming to the q-space representation, wehave

Hq,t = Hq +e

At . 3.3

Introduce the gauge-invariant crystal momentum

k = q +e

At . 3.4

The parameter-dependent Hamiltonian can be simplywritten as H„kq , t…. Hence the eigenstates of the time-dependent Hamiltonian can be labeled by a single pa-rameter k. Moreover, because At preserves the trans-lational symmetry, q is still a good quantum number andis a constant of motion q=0. It then follows from Eq.3.4 that k satisfies the following equation of motion:

k = −e

E . 3.5

-0.5

0.5

0

FIG. 2. Color online Polarization as a function of and inthe Rice-Mele model. The unit is ea with a the lattice constant.The line of discontinuity can be chosen anywhere dependingon the particular phase choice of the eigenstate.



Using /q= /k and /t=−e /E /k, the gen-eral formula 2.5 for the velocity in a given state k be-comes

vnk =nkk

−e

Enk , 3.6

where nk is the Berry curvature of the nth band:

nk = ikunk kunk . 3.7

We can see that, in addition to the usual band dispersioncontribution, an extra term previously known as ananomalous velocity also contributes to vnk. This veloc-ity is always transverse to the electric field, which willgive rise to a Hall current. Historically the anomalousvelocity was obtained by Karplus and Luttinger 1954,Kohn and Luttinger 1957, and Adams and Blount1959; its relation to the Berry phase was realized muchlater. In Sec. V we rederive this term using a wave-packet approach.

B. Berry curvature: Symmetry considerations

The velocity formula 3.6 reveals that, in addition tothe band energy, the Berry curvature of the Bloch bandsis also required for a complete description of the elec-tron dynamics. However, the conventional formula Eq.3.1 has much success in describing various electronicproperties in the past. It is thus important to know underwhat conditions the Berry curvature term cannot be ne-glected.

The general form of the Berry curvature nk can beobtained via symmetry analysis. The velocity formula3.6 should be invariant under time-reversal and spatialinversion operations if the unperturbed system has thesesymmetries. Under time reversal, vn and k change signwhile E is fixed. Under spatial inversion, vn, k, and Echange sign. If the system has time-reversal symmetry,the symmetry condition on Eq. 3.6 requires that

n− k = − nk . 3.8

If the system has spatial inversion symmetry, then

n− k = nk . 3.9

Therefore, for crystals with simultaneous time-reversaland spatial inversion symmetry the Berry curvature van-ishes identically throughout the Brillouin zone. In thiscase Eq. 3.6 reduces to the simple expression 3.1.However, in systems with broken either time-reversal orinversion symmetries, their proper description requiresthe use of the full velocity formula 3.6.

There are many important physical systems whereboth symmetries are not simultaneously present. For ex-ample, in the presence of ferromagnetic or antiferro-magnetic ordering the crystal breaks the time-reversalsymmetry. Figure 3 shows the Berry curvature on theFermi surface of fcc Fe. As shown the Berry curvature isnegligible in most areas in the momentum space anddisplays sharp and pronounced peaks in regions wherethe Fermi lines intersection of the Fermi surface with

010 plane have avoided crossings due to spin-orbitcoupling. Such a structure has been identified in othermaterials as well Fang et al., 2003. Another example isprovided by single-layered graphene sheet with stag-gered sublattice potential, which breaks inversion sym-metry Zhou et al., 2007. Figure 4 shows the energyband and Berry curvature of this system. The Berry cur-vature at valley K1 and K2 have opposite signs due totime-reversal symmetry. We note that as the gap ap-proaches zero, the Berry phase acquired by an electronduring one circle around the valley becomes exactly ±.This Berry phase of has been observed in intrinsic

-103

-102

-101

0

101

102

103

104

105

-3

-2

-1

0

1

2

3

4

5

H(100)(000)

(101)H(001)

FIG. 3. Color online Fermi surface in 010 plane solid linesand the integrated Berry curvature −zk in atomic unitscolor map of fcc Fe. From Yao et al., 2004.

(eV)

(a

2)

1312

1612

1912

1312

1612

1912

( / )xk a

−80

−40

0

40

80

~ ~~ ~−1

− 0.5

0

0.5

1

(a)

(b)

FIG. 4. Color online Energy bands top panel and Berrycurvature of the conduction band bottom panel of agraphene sheet with broken inversion symmetry. The first Bril-louin zone is outlined by the dashed lines, and two inequiva-lent valleys are labeled as K1 and K2. Details are presented inXiao, Yao, and Niu 2007.



graphene sheet Novoselov et al., 2005; Zhang et al.,2005.

C. The quantum Hall effect

The quantum Hall effect was discovered by Klitzing etal. 1980. They found that in a strong magnetic field theHall conductivity of a two-dimensional 2D electron gasis exactly quantized in the units of e2 /h. The exact quan-tization was subsequently explained by Laughlin 1981based on gauge invariance and was later related to atopological invariance of the energy bands Thouless etal., 1982; Avron et al., 1983; Niu et al., 1985. Since thenit has blossomed into an important research field incondensed-matter physics. In this section we focus onlyon the quantization aspect of the quantum Hall effectusing the formulation developed so far.

Consider a two-dimensional band insulator. It followsfrom Eq. 3.6 that the Hall conductivity of the system isgiven by

xy =e2

BZ

d2k

22kxky, 3.10

where the integration is over the entire Brillouin. Onceagain we encounter the situation where the Berry curva-ture is integrated over a closed manifold. Here xy is theChern number in the units of e2 /h, i.e.,

xy = ne2

h. 3.11

Therefore the Hall conductivity is quantized for a two-dimensional band insulator of noninteracting electrons.

Historically the quantization of the Hall conductivityin a crystal was first shown by Thouless et al. 1982 formagnetic Bloch bands see also Sec. VIII. It was shownthat, due to the magnetic translational symmetry, thephase of the wave function in the magnetic Brillouinzone carries a vortex and leads to a nonzero quantizedHall conductivity Kohmoto, 1985. However, it is clearfrom the above derivation that for the quantum Halleffect to occur the only condition is that the Chern num-ber of the bands must be nonzero. It is possible that insome materials the Chern number can be nonzero evenin the absence of an external magnetic field. Haldane1988 constructed a tight-binding model on a honey-comb lattice which displays the quantum Hall effect withzero net flux per unit cell. Another model is proposedfor semiconductor quantum well where the spin-orbitinteraction plays the role of the magnetic field Qi et al.,2006; Liu et al., 2008 and leads to a quantized Hall con-ductance. The possibility of realizing the quantum Halleffect without a magnetic field is attractive in device de-sign.

Niu et al. 1985 further showed that the quantizedHall conductivity in two-dimensions is robust againstmany-body interactions and disorder see also Avronand Seiler 1985. Their derivation involves the sametechnique discussed in Sec. II.B.2. A two-dimensionalmany-body system is placed on a torus by assuming pe-

riodic boundary conditions in both directions. One canthen thread the torus with magnetic flux through itsholes Fig. 5 and make the Hamiltonian H1 ,2 de-pend on the flux 1 and 2. The Hall conductivity iscalculated using the Kubo formula

H = ie2n0

0v1nnv20 − 1 ↔ 20 − n2 , 3.12

where n is the many-body wave function with 0 theground state. In the presence of flux, the velocity opera-tor is given by vi=H1 ,2 /i with i= e /i /Liand Li the dimensions of the system. We recognize thatEq. 3.12 is the summation formula 1.13 for the Berrycurvature 12

of the state 0. The existence of a bulkenergy gap guarantees that the Hall conductivity re-mains unchanged after thermodynamic averaging, whichis given by

H =e2

0

2/L1

d10

2/L2

d212. 3.13

Note that the Hamiltonian H1 ,2 is periodic in iwith period 2 /Li because the system returns to itsoriginal state after the flux is changed by a flux quantumh /e and i changed by 2 /Li. Therefore the Hall con-ductivity is quantized even in the presence of many-body interaction and disorder. Due to the high precisionof the measurement and the robustness of the quantiza-tion, the quantum Hall resistance is now used as theprimary standard of resistance.

The geometric and topological ideas developed in thestudy of the quantum Hall effect has a far-reaching im-pact on modern condensed-matter physics. The robust-ness of the Hall conductivity suggests that it can be usedas a topological invariance to classify many-body phasesof electronic states with a bulk energy gap Avron et al.,1983: states with different topological orders Hall con-ductivities in the quantum Hall effect cannot be adia-batically transformed into each other; if that happens, aphase transition must occur. The Hall conductivity hasimportant applications in strongly correlated electronsystems, such as the fractional quantum Hall effect Wenand Niu, 1990, and most recently the topological quan-tum computing for a review, see Nayak et al. 2008.

ϕ1

ϕ2

FIG. 5. Magnetic flux going through the holes of the torus.



D. The anomalous Hall effect

Next we discuss the anomalous Hall effect, which re-fers to the appearance of a large spontaneous Hall cur-rent in a ferromagnet in response to an electric fieldalone for early works in this field see Chien and West-gate 1980. Despite its century-long history and impor-tance in sample characterization, the microscopicmechanism of the anomalous Hall effect has been a con-troversial subject and it comes to light only recently fora recent review see Nagaosa et al. 2010. In the past,three mechanisms have been identified: the intrinsiccontribution Karplus and Luttinger, 1954; Luttinger,1958, the extrinsic contributions from the skew Smit,1958, and side-jump scattering Berger, 1970. The lattertwo describe the asymmetric scattering amplitudes forspin-up and spin-down electrons. It was later realizedthat the scattering-independent intrinsic contributioncomes from the Berry phase supported anomalous ve-locity. This will be our primary interest here.

The intrinsic contribution to the anomalous Hall ef-fect can be regarded as an “unquantized” version of thequantum Hall effect. The Hall conductivity is given by

xy =e2

dk

2dfkkxky, 3.14

where fk is the Fermi-Dirac distribution function.However, unlike the quantum Hall effect, the anoma-lous Hall effect does not require a nonzero Chern num-ber of the band; for a band with zero Chern number, thelocal Berry curvature can be nonzero and give rise to anonzero anomalous Hall conductivity.

Consider a generic Hamiltonian with spin-orbit SOsplit bands Onoda, Sugimoto, and Nagaosa, 2006,

H =2k2

2m+ k · ez − z, 3.15

where 2 is the SO split gap in the energy spectrum ±

=2k2 /2m±2k2+2 and gives a linear dispersion inthe absence of . This model also describes spin-polarized two-dimensional electron gas with Rashba SOcoupling, with the SO coupling strength and theexchange field Culcer et al., 2003. Obviously the termbreaks time-reversal symmetry and the system is ferro-magnetic. However, the term alone will not lead to aHall current as it only breaks the time-reversal symme-try in the spin space. The SO interaction is needed tocouple the spin and orbital part together. The Berry cur-vature is given by, using Eq. 1.19,

± = 2

22k2 + 23/2 . 3.16

The Berry curvatures of the two energy bands have op-posite sign and are highly concentrated around the gap.In fact, the Berry curvature has the same form of theBerry curvature in one valley of the graphene, shown inFig. 4. One can verify that the integration of the Berry

curvature of a full band, 20qdq±, is ± for the upper

and lower bands, respectively.9

Figure 6 shows the band dispersion, and the intrinsicHall conductivity Eq. 3.14 as the Fermi energysweeps across the SO split gap. As shown when theFermi energy F is in the gap region, the Hall conduc-tivity reaches its maximum value about −e2 /2h. If F−, the states with energies just below −, which con-tribute most to the Hall conductivity, are empty. If F, contributions from upper and lower bands canceleach other, and the Hall conductivity decreases quicklyas F moves away from the band gap. It is only when−F, the Hall conductivity is resonantly en-hanced Onoda, Sugimoto, and Nagaosa, 2006.

1. Intrinsic versus extrinsic contributions

The above discussion does not take into account thefact that, unlike insulators, in metallic systems electronscattering can be important in transport phenomena.Two contributions to the anomalous Hall effect arisesdue to scattering: i the skew scattering that refers tothe asymmetric scattering amplitude with respect to thescattering angle between the incoming and outgoingelectron waves Smit, 1958 and ii the side jump whichis a sudden shift of the electron coordinates during scat-tering Berger, 1970. In a more careful analysis, a sys-

9Since the integration is performed in an infinite momentumspace, the result is not quantized in the unit of 2.

k

Energy

(a)

-

0.1

0.2

0.3

0.4

Hallconductivity

Fermi energy

-

(b)

FIG. 6. Anomalous Hall effect in a simple two-band model. aEnergy dispersion of spin-split bands. b The Hall conductiv-ity −xy in the units of e2 /h as a function of Fermi energy.



tematic study of the anomalous Hall effect based on thesemiclassical Boltzmann transport theory has been car-ried out Sinitsyn, 2008. The basic idea is to solve thefollowing Boltzmann equation:

gk

t− eE ·

kf

=

k

kkgk − gk −f

eE · rkk ,

3.17

where g is the nonequilibrium distribution function, kkrepresents the asymmetric skew scattering, and rkk de-scribes the side-jump of the scattered electrons. TheHall conductivity is the sum of different contributions

H = Hin + H

sk + Hsj , 3.18

where Hin is the intrinsic contribution given by Eq.

3.14, Hsk is the skew scattering contribution, which is

proportional to the relaxation time , and Hsj is the side

jump contribution, which is independent of . Note that,in addition to Berger’s original proposal, H

sj also in-cludes two other contributions: the intrinsic skew-scattering and anomalous distribution function Sinitsyn,2008.

An important question is how to identify the domi-nant contribution to the anomalous Hall effect AHEamong these mechanisms. If the sample is clean and thetemperature is low, the relaxation time can be ex-tremely large, and the skew scattering is expected todominate. On the other hand, in dirty samples and athigh temperatures H

sk becomes small compared to bothH

in and Hsj . Because the Berry phase contribution H

in isindependent of scattering, it can be readily evaluatedusing first-principles methods or effective Hamiltonians.Excellent agreement with experiments has been demon-strated in ferromagnetic transition metals and semicon-ductors Jungwirth et al., 2002; Fang et al., 2003; Yao etal., 2004, 2007; Xiao, Yao, et al., 2006, which leaves littleroom for the side-jump contribution.

In addition, a number of experimental results alsogave favorable evidence for the dominance of the intrin-sic contribution Lee et al., 2004b; Mathieu et al., 2004;Sales et al., 2006; Zeng et al., 2006; Chun et al., 2007. Inparticular, Tian et al. 2009 recently measured theanomalous Hall conductivity in Fe thin films. By varyingthe film thickness and the temperature, they are able tocontrol various scattering process such as the impurityscattering and the phonon scattering. Figure 7 showstheir measured ah as a function of xxT2. One can seethat although ah in different thin films and at differenttemperatures shows a large variation at finite xx, theyconverge to a single point as xx approaches zero, wherethe impurity-scattering-induced contribution should bewashed out by the phonon scattering and only the intrin-sic contribution survives. It turns out that this convergedvalue is very close to the bulk ah of Fe, which confirmsthe dominance of the intrinsic contribution in Fe.

In addition to the semiclassical approach Sinitsyn etal., 2005; Sinitsyn, 2008, there are a number of worksbased on a full quantum-mechanical approach Nozières

and Lewiner, 1973; Onoda and Nagaosa, 2002; Dugaev etal., 2005; Inoue et al., 2006; Onoda, Sugimoto, and Na-gaosa, 2006; Kato et al., 2007; Sinitsyn et al., 2007; Onodaet al., 2008. In both approaches, the Berry phase sup-ported intrinsic contribution to the anomalous Hall ef-fect has been firmly established.

2. Anomalous Hall conductivity as a Fermi surface property

An interesting aspect of the intrinsic contribution tothe anomalous Hall effect is that the Hall conductivityEq. 3.14 is given as an integration over all occupiedstates below the Fermi energy. It seems to be against thespirit of the Landau Fermi-liquid theory, which statesthat the transport property of an electron system is de-termined by quasiparticles at the Fermi energy. This is-sue was first raised by Haldane 2004, and he showedthat the Hall conductivity can be written, in the units ofe2 /2h, as the Berry phase of quasiparticles on theFermi surface, up to a multiple of 2. Therefore theintrinsic Hall conductivity is also a Fermi surface prop-erty. This observation suggests that the Berry phase onthe Fermi surface should be added as a topological in-gredient to the Landau Fermi-liquid theory.

For simplicity, consider a two-dimensional system. Weassume there is only one partially filled band. If we writethe Berry curvature in terms of the Berry vector poten-tial and integrate Eq. 3.14 by part, one finds

xy2D =

e2

d2k

2 f

kyAkx

−f

kxAky

. 3.19

Note that the Fermi distribution function f is a step func-tion at the Fermi energy. If we assume the Fermi surfaceis a closed loop in the Brillouin zone, then

xy2D =

e2

2h dk · Ak. 3.20

The integral is nothing but the Berry phase along theFermi circle in the Brillouin zone. The three-dimensional case is more complicated; Haldane 2004showed that the same conclusion can be reached.

Wang et al. 2007 implemented Haldane’s idea infirst-principles calculations of the anomalous Hall con-ductivity. From a computational point of view, the ad-

0 2 4 6

0.8

1.2

1.6

Theory

AH(103

-1cm

-1)

93 nm 10 nm37 nm 6.5 nm23 nm 5 nm20 nm 4 nm15 nm

2

xx(T)(1010 -2cm -2)

BulkFe

FIG. 7. Color online ah vs xxT2 in Fe thin films withdifferent film thickness over the temperature range of5–300 K. From Tian et al., 2009.



vantage lies in that the integral over the Fermi sea isconverted to a more efficient integral on the Fermi sur-face. On the theory side, Shindou and Balents 2006,2008 derived an effective Boltzmann equation for qua-siparticles on the Fermi surface using the Keldysh for-malism, where the Berry phase of the Fermi surface isdefined in terms of the quasiparticle Green’s functions,which nicely fits into the Landau Fermi-liquid theory.

E. The valley Hall effect

A necessary condition for the charge Hall effect tomanifest is the broken time-reversal symmetry of thesystem. In this section we discuss another type of Halleffect which relies on inversion symmetry breaking andsurvives in time-reversal invariant systems.

We use graphene as our prototype system. The bandstructure of intrinsic graphene has two degenerate andinequivalent Dirac points at the corners of the Brillouinzone, where the conduction and valance bands toucheach other, forming a valley structure. Because of theirlarge separation in momentum space, the intervalleyscattering is strongly suppressed Morozov et al., 2006;Morpurgo and Guinea, 2006; Gorbachev et al., 2007,which makes the valley index a good quantum number.Interesting valley-dependent phenomena, which con-cerns about the detection and generation of valley po-larization, are currently being explored Akhmerov andBeenakker, 2007; Rycerz et al., 2007; Xiao, Yao, and Niu,2007; Yao et al., 2008.

The system we are interested in is graphene with bro-ken inversion symmetry. Zhou et al. 2007 recently re-ported the observation of a band-gap opening in epitax-ial graphene, attributed to the inversion symmetrybreaking by the substrate potential. In addition, in bi-ased graphene bilayer inversion symmetry can be explic-itly broken by the applied interlayer voltage McCannand Fal’ko, 2006; Ohta et al., 2006; Min et al., 2007. It isthis broken inversion symmetry that allows a valley Halleffect. Introducing the valley index z= ±1 which labelsthe two valleys, we can write the valley Hall effect as

jv = Hv z E , 3.21

where Hv is the valley Hall conductivity, and the valley

current jv= zv is defined as the average of the valleyindex z times the velocity operator v. Under time rever-sal, both the valley current and electric field are invari-ant z changes sign because the two valleys switch whenthe crystal momentum changes sign. Under spatial in-version, the valley current is still invariant but the elec-tric field changes sign. Therefore, the valley Hall con-ductivity can be nonzero when the inversion symmetry isbroken even if time-reversal symmetry remains.

In the tight-binding approximation, the Hamiltonianof a single graphene sheet can be modeled with anearest-neighbor hopping energy t and a site energy dif-ference between sublattices. For relatively low doping,we can resort to the low-energy description near the

Dirac points. The Hamiltonian is given by Semenoff,1984

H =32

atqxzx + qyy +

2z, 3.22

where is the Pauli matrix accounting for the sublatticeindex and q is measured from the valley center K1,2

= ±4 /3ax with a the lattice constant. The Berry cur-vature of the conduction band is given by

q = z3a2t2

22 + 3q2a2t23/2 . 3.23

Note that the Berry curvatures in two valleys have op-posite sign as required by time-reversal symmetry. Theenergy spectrum and the Berry curvature are shown inFig. 4. Once the structure of the Berry curvature is re-vealed, the valley Hall effect becomes transparent: uponthe application of an electric field, electrons in differentvalleys will flow to opposite directions transverse to theelectric field, giving rise to a net valley Hall current inthe bulk.

We remark that as goes to zero, the Berry curvaturevanishes everywhere except at the Dirac points where itdiverges. Meanwhile, the Berry phase around the Diracpoints becomes exactly ± see also Sec. VII.C.

As shown the valley transport in systems with brokeninversion symmetry is a very general phenomenon. Itprovides a new and standard pathway to potential appli-cations of valleytronics or valley-based electronic appli-cations in a broad class of semiconductors Gunawan etal., 2006; Xiao, Yao, and Niu, 2007; Yao et al., 2008.

IV. WAVE PACKET: CONSTRUCTION ANDPROPERTIES

Our discussion so far has focused on crystals undertime-dependent perturbations, and we have shown thatthe Berry phase will manifest itself as an anomalousterm in the electron velocity. However, in general situa-tions the electron dynamics can be also driven by pertur-bations that vary in space. In this case, the most usefuldescription is provided by the semiclassical theory ofBloch electron dynamics, which describes the motion ofa narrow wave packet obtained by superposing theBloch states of a band see, for example, Chap. 12 ofAshcroft and Mermin 1976. The current and next sec-tions are devoted to the study of the wave-packet dy-namics, where the Berry curvature naturally appears inthe equations of motion.

In this section we discuss the construction and thegeneral properties of the wave packet. Two quantitiesemerge in the wave-packet approach, i.e., the orbitalmagnetic moment of the wave packet and the dipolemoment of a physical observable. For their applications,we consider the problems of orbital magnetization andanomalous thermoelectric transport in ferromagnets.



A. Construction of the wave packet and its orbital moment

We assume the perturbations are sufficiently weakthat transitions between different bands can be ne-glected; i.e., the electron dynamics is confined within asingle band. Under this assumption, we construct a wavepacket using the Bloch functions nq from the nthband,

W0 = dqwq,tnq . 4.1

There are two requirements on the envelope functionwq , t. First, wq , t must have a sharp distribution inthe Brillouin zone such that it makes sense to speak ofthe wave vector qc of the wave packet given by

qc = dqqwq,t2. 4.2

To first order, wq , t2 can be approximated by q−qc and one has

dqfqwq,t2 fqc , 4.3

where fq is an arbitrary function of q. Equation 4.3 isuseful in evaluating various quantities related to thewave packet. Second, the wave packet has to be nar-rowly localized around its center of mass, denoted by rc,in the real space, i.e.,

rc = W0rW0 . 4.4

Using Eq. 4.3 we obtain

rc = −

qcarg wqc,t + Aq

nqc , 4.5

where Aqn= iunqqunq is the Berry connection of

the nth band defined using unq=e−ik·rnq. A morerigorous consideration of the wave-packet constructionis given by Hagedorn 1980.

In principle, more dynamical variables, such as thewidth of the wave packet in both the real space andmomentum space, can be added to allow a more elabo-rate description of the time evolution of the wavepacket. However, in short period the dynamics is domi-nated by the motion of the wave-packet center, and Eqs.4.2 and 4.5 are sufficient requirements.

When more than one band come close to each otheror even become degenerate, the single-band approxima-tion breaks down and the wave packet must be con-structed using Bloch functions from multiple bands. Cul-cer et al. 2005 and Shindou and Imura 2005developed the multiband formalism for electron dynam-ics, which will be presented in Sec. IX. For now, we fo-cus on the single-band formulation and drop the bandindex n for simplicity.

The wave packet, unlike a classical point particle, hasa finite spread in real space. In fact, since it is con-structed using an incomplete basis of the Bloch func-tions, the size of the wave packet has a nonzero lower

bound Marzari and Vanderbilt, 1997. Therefore, awave packet may possess a self-rotation around its cen-ter of mass, which will in turn give rise to an orbitalmagnetic moment. By calculating the angular momen-tum of a wave packet directly, one finds Chang and Niu,1996

mq = −e

2W0r − rc jW0

= − ie

2qu Hq − qqu , 4.6

where Hq=e−iq·rHeiq·r is the q-dependent Hamiltonian.Equation 4.3 is used to obtain this result. This showsthat the wave packet of a Bloch electron generally ro-tates around its mass center and carries an orbital mag-netic moment in addition to its spin moment.

We emphasize that the orbital moment is an intrinsicproperty of the band. Its final expression Eq. 4.6 doesnot depend on the actual shape and size of the wavepacket and only depends on the Bloch functions. Undersymmetry operations, the orbital moment transforms ex-actly like the Berry curvature. Therefore unless the sys-tem has both time-reversal and inversion symmetry,mq is in general nonzero. Information of the orbitalmoment can be obtained by measuring magnetic circulardichroism MCD spectrum of a crystal Souza andVanderbilt, 2008; Yao et al., 2008. In the single-bandcase, MCD directly measures the magnetic moment.

This orbital moment behaves exactly like the electronspin. For example, upon the application of a magneticfield, the orbital moment will couple to the field througha Zeeman-like term −mq ·B. If one can construct awave packet using only the positive energy states theelectron branch of the Dirac Hamiltonian, its orbitalmoment in the nonrelativistic limit is exactly the Bohrmagneton Sec. IX. For Bloch electrons, the orbital mo-ment can be related to the electron g factor Yafet,1963. Consider a specific example. For the graphenemodel with broken-inversion symmetry, discussed inSec. III.E, the orbital moment of the conduction band isgiven by Xiao, Yao, and Niu, 2007

mz,q = z3ea2t2

42 + 3q2a2t2. 4.7

So orbital moments in different valleys have oppositesigns, as required by time-reversal symmetry. Interest-ingly, the orbital moment at exactly the band bottomtakes the following form:

mz = zB , B

=e

2m, 4.8

where m is the effective mass at the band bottom. Theclose analogy with the Bohr magneton for the electronspin is transparent. In realistic situations, the momentcan be 30 times larger than the spin moment and can beused as an effective way to detect and generate the val-ley polarization Xiao, Yao, and Niu, 2007; Yao et al.,2008.



B. Orbital magnetization

A closely related quantity to the orbital magnetic mo-ment is the orbital magnetization in a crystal. Althoughthis phenomenon has been known for a long time, ourunderstanding of orbital magnetization in crystals hasremained in a primitive stage. In fact, there was noproper way to calculate this quantity until recently whenthe Berry phase theory of orbital magnetization is devel-oped Thonhauser et al., 2005; Xiao et al., 2005; Shi et al.,2007. Here we provide a pictorial derivation of the or-bital magnetization based on the wave-packet approach.This derivation gives an intuitive picture of differentcontributions to the total orbital magnetization.

The main difficulty of calculating the orbital magneti-zation is exactly the same as when calculating the elec-tric polarization: the magnetic dipole erp is not de-fined in a periodic system. For a wave packet this is nota problem because it is localized in space. As shown inthe previous section, each wave packet carries an orbitalmoment. Thus, it is tempting to conclude that the orbitalmagnetization is simply the thermodynamic average ofthe orbital moment. As it turns out, this is only part ofthe contribution. There is another contribution due tothe center-of-mass motion of the wave packet.

For simplicity, consider a finite system of electrons ina two-dimensional lattice with a confining potential Vr.We further assume that the potential Vr varies slowlyat atomic length scale such that the wave-packet descrip-tion of the electron is still valid on the boundary. In thebulk where Vr vanishes, the electron energy is just thebulk band energy; near the boundary, it will be tilted updue to the increase in Vr. Thus to a good approxima-tion, we can write the electron energy as

r,q = q + Vr . 4.9

The energy spectrum in real space is shown in Fig. 8.Before proceeding further, we need to generalize the

velocity formula 3.6, which is derived in the presenceof an electric field. In our derivation the electric field

enters through a time-dependent vector potential Atso that we can avoid the technical difficulty of calculat-ing the matrix element of the position operator. How-ever, the electric field may be also given by the gradientof the electrostatic potential. In both cases, the velocityformula should stay the same because it should be gaugeinvariant. Therefore, in general a scalar potential Vrwill induce a transverse velocity of the following form:

1

Vrq . 4.10

This generalization will be justified in Sec. VI.Now consider a wave packet in the boundary region

of the finite system Fig. 8. It will feel a force Vr dueto the presence of the confining potential. Consequently,according to Eq. 4.10 the electron acquires a transversevelocity, whose direction is parallel with the boundaryFig. 8. This transverse velocity will lead to a boundarycurrent of the dimension current densitywidth in twodimensions given by

I =e

dx dq

22

dV

dxfq + Vzq , 4.11

where x is in the direction perpendicular to the bound-ary, and the integration range is taken from deep intothe bulk to outside the sample. Recall that for a currentI flowing in a closed circuit enclosing a sufficiently smallarea A, the circuit carries a magnetic moment given byIA. Therefore the magnetization magnetic moment perunit area has the magnitude of the current I. IntegratingEq. 4.11 by part, we obtain

Mf =1

e dfxy , 4.12

where xy is the zero-temperature Hall conductivityfor a system with Fermi energy :

xy =e2

dq

2d „ − q…zq . 4.13

Since the boundary current corresponds to the globalmovement of the wave-packet center, we call this contri-bution the “free current” contribution, whereas the or-bital moment are due to “localized” current. Thus thetotal magnetization is

Mz = dq2dfqmzq +

1

e dfxy . 4.14

The orbital magnetization has two different contribu-tions: one is from the self-rotation of the wave packet,and the other is due to the center-of-mass motion. Gatand Avron obtained an equivalent result for the specialcase of the Hofstadter model Gat and Avron, 2003a,2003b.

The above derivation relies on the existence of a con-fining potential, which seems to contradict the fact thatthe orbital magnetization is a bulk property. This is awrong assertion as the final expression Eq. 4.14 isgiven in terms of the bulk Bloch functions and does not

ε(k)

ε'

ε'+dε'

∇V

Ωz

I

x

ε~

FIG. 8. Electron energy in a slowly varying confining poten-tial Vr. In addition to the self-rotation, wave packets near theboundary will also move along the boundary due to the poten-tial V. Level spacings between different bulk q states are ex-aggerated; they are continuous in the semiclassical limit. Theinset shows directions of the Berry curvature, the effectiveforce, and the current carried by a wave packet on the leftboundary.



depend on the boundary condition. Here the boundaryis merely a tool to expose the “free current” contribu-tion because in a uniform system, the magnetization cur-rent always vanishes in the bulk. Finally, in more rigor-ous approaches Xiao et al., 2005; Shi et al., 2007 theboundary is not needed and the derivation is based on apure bulk picture. It is similar to the quantum Hall ef-fect, which can be understood in terms of either the bulkstates Thouless et al., 1982 or the edge states Halperin,1982.

C. Dipole moment

The finite size of the wave packet not only allows anorbital magnetic moment but also leads to the conceptof the dipole moment associated with an operator.

The dipole moment appears naturally when we con-sider the thermodynamic average of a physical quantity,

with its operator denoted by O. In the wave-packet ap-proach, the operator is given by

Or = drcdqc

23 grc,qcWOr − rW , 4.15

where gr ,q is the distribution function, W¯ W de-notes the expectation in the wave-packet state, and r− r plays the role as a sampling function, as shown inFig. 9. An intuitive way to view Eq. 4.15 is to think ofthe wave packets as small molecules, then Eq. 4.15 isthe quantum-mechanical version of the familiar coarsegraining process which averages over the length scalelarger than the size of the wave packet. A multipoleexpansion can be carried out. But for most purposes thedipole term is enough. Expand the function to firstorder of r−rc:

r − r = r − rc − r − rc · r − rc . 4.16

Inserting the function into Eq. 4.15 yields

Or = dq23gr,qWOWrc=r

− · dq23gr,qWOr − rcWrc=r.

4.17

The first term is obtained if the wave packet is treated asa point particle. The second term is due to the finite sizeof the wave packet. We can see that the bracket in thesecond integral has the form of a dipole of the operatorO defined by

PO = WOr − rcW . 4.18

The dipole moment of an observable is a general conse-quence of the wave-packet approach and must be in-cluded in the semiclassical theory. Its contribution ap-pears only when the system is inhomogeneous.

In particular, we find the following:

1 If O=e, then Pe=0. This is consistent with the factthat the charge center coincides with the mass cen-ter of the electron.

2 If O= v, one finds the expression for the local cur-rent,

jL = dq23gr,qr + dq

23gr,qmq .

4.19

We explain the meaning of local later. Interestingly,this is the second time we encounter the quantitymq but in an entirely different context. The physi-cal meaning of the second term becomes transparentif we make reference to the self-rotation of the wavepacket. The self-rotation can be thought as localizedcircuit. Therefore if the distribution is not uniform,the localized circuit will contribute to the local cur-rent jL see Fig. 10.

3 If O is the spin operator s, then Eq. 4.18 gives thespin dipole

Ps = usi

q− Aqu . 4.20

The spin dipole shows that in general the spin centerand the mass center do not coincide, which is usually

L

rrc

l

FIG. 9. Sampling function and a wave packet at rc. The widthL of the sampling function is sufficiently small so that it can betreated as a function at the macroscopic level and is suffi-ciently large so that it contains a large number of wave packetsof width l inside its range. Equation 4.15 is indeed a micro-scopic average over the distance L around the point r. See Sec.6.6 in Jackson 1998 for an analogy in macroscopic electro-magnetism.

wave packet

W(r;rc ,kc)

rc

FIG. 10. The wave-packet description of a charge carrierwhose center is rc ,qc. A wave packet generally possesses twokinds of motion: the center-of-mass motion and the self-rotation around its center. From Xiao, Yao, et al., 2006.



due to the spin-orbit interaction. The time derivative ofthe spin dipole contributes to the total spin current Cul-cer et al., 2004.

D. Anomalous thermoelectric transport

In applying the above concepts, we consider the prob-lem of anomalous thermoelectric transport in ferromag-nets, which refers to the appearance of a Hall currentdriven by statistical forces, such as the gradient of tem-perature and chemical potential Chien and Westgate,1980. Similar to the anomalous Hall effect, there arealso intrinsic and extrinsic contributions, and we focuson the former.

A question immediately arises when one tries to for-mulate this problem. Recall that in the presence of anelectric field the electron acquires an anomalous velocityproportional to the Berry curvature, which gives rise toa Hall current. In this case, the driving force is of me-chanical nature: it exists on the microscopic level andcan be described by a perturbation to the Hamiltonianfor the carriers. On the other hand, transport can be alsodriven by the statistical force. However, the statisticalforce manifests on the macroscopic level and makessense only through the statistical distribution of the car-riers. Since there is no force acting directly on individualparticles, the obvious cause for the Berry phase assistedtransport is eliminated. This conclusion would introducea number of basic contradictions to the standard trans-port theory. First, a chemical potential gradient wouldbe distinct from the electric force, violating the basis forthe Einstein relation. Second, a temperature gradientwould not induce an intrinsic charge Hall current, vio-lating the Mott relation. Finally, it is also unclearwhether the Onsager relation is satisfied or not.

It turns out the correct description of anomalous ther-moelectric transport in ferromagnets requires knowl-edge of both the magnetic moment and orbital magneti-zation. First, as shown in Eq. 4.19, the local current isgiven by

jL = dq2dgr,qr + dq

2dfr,qmq ,

4.21

where in the second term we have replaced the distribu-tion function gr ,q with the local Fermi-Dirac functionfr ,q, which is sufficient for a first order calculation.Second, in ferromagnetic systems, it is important to dis-count the contribution from the magnetization current.It was argued that the magnetization current cannotbe measured by conventional transport experimentsCooper et al., 1997. Therefore the transport current isgiven by

j = jL − Mr . 4.22

Using Eq. 4.14, one finds

j = dq2dgr,qr −

1

e dfAH . 4.23

Equation 4.23 is the most general expression for thetransport current. We notice that the contribution fromthe orbital magnetic moment mq cancels out. Thisagrees with the intuitive picture developed in Sec. IV.B,i.e., the orbital moment is due to the self-rotation of thewave packet, therefore it is localized and cannot contrib-ute to transport see Fig. 10.

In the presence of a statistical force, there are twoways for a Hall current to occur. The asymmetric scat-tering will have an effect on the distribution gr ,q,which is obtained from the Boltzmann equation Berger,1972. This results in a transverse current in the firstterm of Eq. 4.23. In addition, there is an intrinsic con-tribution comes from the orbital magnetization, which isthe second term of Eq. 4.23. Note that the spatial de-pendence enters through Tr and r in the distribu-tion function. It is straightforward to verify that for theintrinsic contribution to the anomalous thermoelectrictransport both the Einstein relation and Mott relationstill hold Xiao, Yao, et al., 2006; Onoda et al., 2008.Hence, the measurement of this type of transport, suchas the anomalous Nernst effect, can give further insightinto the intrinsic mechanism of the anomalous Hall ef-fect. Much experimental efforts have been put along thisline. The intrinsic contribution has been verified inCuCr2Se4−xBrx Lee et al., 2004a, 2004b, La1−xSrxCoO3Miyasato et al., 2007, Nd2Mo2O7 and Sm2Mo2O7 Ha-nasaki et al., 2008, and Ga1−xMnx Pu et al., 2008.

Equation 4.23 is not limited to transport driven bystatistical forces. As shown later, at the microscopic levelthe mechanical force generally has two effects: it candrive the electron motion directly and appears in theexpression for r; it can also make the electron energyand the Berry curvature spatially dependent, hence alsomanifest in the second term in Eq. 4.23. The latter pro-vides another route for the Berry phase to enter thetransport problems in inhomogeneous situations, whichcan be caused by a nonuniform distribution function or aspatially dependent perturbation, or both. One exampleis the electrochemical potential −er+r /e, whichcan induce an anomalous velocity term in the equationof motion through −er and also affect the distributionfunction through r.

V. ELECTRON DYNAMICS IN ELECTROMAGNETICFIELDS

In the previous section we discussed the constructionand general properties of a wave packet. Now we are setto study its dynamics under external perturbations. Themost common perturbations to a crystal are the electro-magnetic fields. The study of the electron dynamics un-der such perturbations dates back to Bloch, Peierls,Jones, and Zener in the early 1930s and is continued bySlater 1949, Luttinger 1951, Adams 1952, Karplusand Luttinger 1954, Kohn and Luttinger 1957, Adamsand Blount 1959, Blount 1962a, Brown 1968, Zak1977, Rammal and Bellissard 1990, and Wilkinsonand Kay 1996. In this section we present the semiclas-



sical theory based on the wave-packet approach Changand Niu, 1995, 1996.

A. Equations of motion

In the presence of electromagnetic fields, the Hamil-tonian is given by

H =p + eAr2

2m+ Vr − er , 5.1

where Vr is the periodic lattice potential and Ar andr are the electromagnetic potentials. If the lengthscale of the perturbations is much larger than the spatialspread of the wave packet, the approximate Hamil-tonian that the wave packet “feels” may be obtained bylinearizing the perturbations about the wave-packet cen-ter rc as

H Hc + H , 5.2

Hc =p + eArc2

2m+ Vr − erc , 5.3

H =e

2mAr − Arc,p − eE · r − rc , 5.4

where , is the anticommutator. Naturally, we can thenconstruct the wave packet using the eigenstates of thelocal Hamiltonian Hc. The effect of a uniform Arc is toadd a phase to the eigenstates of the unperturbedHamiltonian. Therefore the wave packet can be writtenas

Wkc,rc = e−ie/Arc·rW0kc,rc , 5.5

where W0 is the wave packet constructed using the un-perturbed Bloch functions.

The wave-packet dynamics can be obtained from thetime-dependent variational principles Kramer and Sa-raceno, 1981. The basic recipe is to first obtain the La-grangian from the following equation:

L = Wi

t− HW , 5.6

then obtain the equations of motion using the Eulerequations. Using Eq. 4.5 we find that Wi /tW=eA ·Rc− /targ wkc , t. For the wave-packet en-ergy, we have WHW=−mk ·B. This is expected aswe already showed that the wave packet carries an or-bital magnetic moment mk that will couple to the mag-netic field. Using Eq. 4.5, we find that the Lagrangianis given by, up to some unimportant total time-derivativeterms dropping the subscript c on rc and kc,

L = k · r − Mk + er − er · Ar,t + k · Ank ,

5.7

where Mk=0k−B ·mk with 0k the unperturbedband energy. The equations of motion are given by

r =Mkk

− kk , 5.8a

k = − eE − er B . 5.8b

Compared to the conventional equations of motion forBloch electrons Ashcroft and Mermin, 1976, there aretwo differences: 1 the electron energy is modified bythe orbital magnetic moment and 2 the electron veloc-ity gains an extra velocity term proportional to the Berrycurvature. As shown, in the case of only an electric field,Eq. 5.8a reduces to the anomalous velocity formula3.6 we derived before.

B. Modified density of states

The Berry curvature not only modifies the electrondynamics but also has a profound effect on the electrondensity of states in the phase space Xiao et al., 2005.10

Recall that in solid-state physics the expectation valueof an observable, in the Bloch representation, is given by

nk

fnknkOnk , 5.9

where fnk is the distribution function. In the semiclassi-cal limit, the sum is converted to an integral in k space,

k

→1

V dk

2d , 5.10

where V is the volume and 2d is the density of states,i.e., number of states per unit k volume. From a classicalpoint of view, the constant density of states is guaran-teed by the Liouville theorem, which states that the vol-ume element is a conserved quantity during the timeevolution of the system.11 However, as shown below, thisis no longer the case for the Berry phase modified dy-namics.

The time evolution of a volume element V=rk isgiven by

1

V

V

t= r · r + k · k . 5.11

Insert the equations of motion 5.8 into Eq. 5.11. Af-ter some algebra, we find

10The phase space is spanned by k and r. Although the La-grangian depends on both k ,r and their velocities, the depen-dence on the latter is only to linear order. This means that themomenta, defined as the derivative of the Lagrangian with re-spect to these velocities, are functions of k ,r only and areindependent of their velocities. Therefore, k ,r also span thephase space of the Hamiltonian dynamics.

11The actual value of this constant volume for a quantumstate, however, can be determined only from the quantizationconditions in quantum mechanics.



V =V0

1 + e/B · . 5.12

The fact that the Berry curvature is generally k depen-dent and the magnetic field is r dependent implies thatthe phase-space volume V changes during time evolu-tion of the state variables r ,k.

Although the phase-space volume is no longer con-served, it is a local function of the state variables and hasnothing to do with the history of time evolution. We canthus introduce a modified density of states

Dr,k =1

2d1 +e

B · 5.13

such that the number of states in the volume elementDnr ,kV remains constant in time. Therefore, the cor-rect semiclassical limit of the sum in Eq. 5.9 is

OR = dkDr,kOr − RW, 5.14

where ¯ W is the expectation value in a wave packet,which could includes the dipole contribution due to thefinite size of the wave packet see Eq. 4.18. In a uni-form system it is given by

O = dkDkfkOk . 5.15

We emphasize that although the density of states is nolonger a constant, the dynamics itself is still Hamil-tonian. The modification comes from the fact that thedynamical variables r and k are no longer canonical vari-ables, and the density of states can be regarded as thephase-space measure Bliokh, 2006b; Duval et al., 2006a,2006b; Xiao, Shi, and Niu, 2006. The phase-space mea-sure dkdr is true only when k and r form a canonical set.However, the phase-space variables obtained from thewave packet are generally not canonical as testified bytheir equations of motion. A more profound reason forthis modification has its quantum-mechanical origin innoncommutative quantum mechanics, discussed in Sec.VII.

In the following we discuss two direct applications ofthe modified density of states in metals and in insulators.

1. Fermi volume

We show that the Fermi volume can be changed lin-early by a magnetic field when the Berry curvature isnonzero. Assume zero temperature, the electron densityis given by

ne = dk2d1 +

e

B · F − . 5.16

We work in the canonical ensemble by requiring theelectron number fixed, therefore, to first order of B, theFermi volume must be changed by

VF = − dk2d

e

B · . 5.17

It is particularly interesting to look at insulators,where the integration is limited to the Brillouin zone.Then the electron must populate a higher band ifBZdkB · is negative. When this quantity is positive,holes must appear at the top of the valance bands. Dis-continuous behavior of physical properties in a magneticfield is therefore expected for band insulators with anonzero integral of the Berry curvatures Chern num-bers.

2. Streda formula

In the context of the quantum Hall effect, Streda1982 derived a formula relating the Hall conductivityto the field derivative of the electron density at a fixedchemical potential

xy = − e ne

Bz

. 5.18

There is a simple justification for this relation given by athermodynamic argument by considering the followingadiabatic process in two dimensions. A time-dependentmagnetic flux generates an electric field with an emfaround the boundary of some region, and the Hall cur-rent leads to a net flow of electrons across the boundaryand thus a change in electron density inside. Note thatthis argument is valid only for insulators because in met-als the adiabaticity would break down. Using Eq. 5.16for an insulator, we obtain, in two dimensions,

xy = −e2

BZ

dk22xy. 5.19

This is what Thouless et al. 1982 obtained using theKubo formula. The fact that the quantum Hall conduc-tivity can be derived using the modified density of statesfurther confirms the necessity to introduce this concept.

C. Orbital magnetization: Revisit

We have discussed the orbital magnetization using arather pictorial derivation in Sec. IV.B. Here we derivethe formula again using the field-dependent density ofstates Eq. 5.13.

The equilibrium magnetization density can be ob-tained from the grand canonical potential, which, withinfirst order in the magnetic field, may be written as

F = −1

k

ln1 + e−M−

= −1

dk

2d1 +e

B · ln1 + e−M− ,

5.20

where the electron energy M=k−mk ·B includes acorrection due to the orbital magnetic moment mk.The magnetization is then the field derivative at fixedtemperature and chemical potential, M=−F /B,T,with the result



Mr = dk2dfkmk

+1

dk

2d

e

kln1 + e−− . 5.21

Integration by parts of the second term will give us theexact formula obtained in Eq. 4.14. We have thus de-rived a general expression for the equilibrium orbitalmagnetization density, valid at zero magnetic field but atarbitrary temperatures. From this derivation we canclearly see that the orbital magnetization is indeed abulk property. The center-of-mass contribution identi-fied before comes from the Berry phase correction tothe electron density of states.

Following the discussions on band insulators in ourfirst example in Sec. V.B.1, there will be a discontinuityof the orbital magnetization if the integral of the Berrycurvature over the Brillouin zone or the anomalous Hallconductivity is nonzero and quantized. Depending onthe direction of the field, the chemical potential 0 inEq. 5.21 should be taken at the top of the valencebands or the bottom of the conduction bands. The sizeof the discontinuity is given by the quantized anomalousHall conductivity times Eg /e, where Eg is the energygap.

A similar formula for insulators with zero Chern num-ber has been obtained by Thonhauser et al. 2005 andCeresoli et al. 2006 using the Wannier function ap-proach and by Gat and Avron 2003a, 2003b for thespecial case of the Hofstadter model. Recently Shi et al.2007 provided a full quantum-mechanical derivation ofthe formula, and showed that it is valid in the presenceof electron-electron interaction, provided the one-electron energies and wave functions are calculated self-consistently within the framework of the exact currentand spin-density functional theory Vignale and Rasolt,1988.

The appearance of the Hall conductivity is not a coin-cidence. Consider an insulator. The free energy is givenby

dF = − MdB − nd − SdT . 5.22

Using the Maxwell relation, we have

H = − e n

B,T

= − e M

B,T. 5.23

On the other hand, the zero-temperature formula of themagnetization for an insulator is given by

M = BZ

dk23mk +

e

− . 5.24

Inserting this formula into Eq. 5.23 gives us once againthe quantized Hall conductivity.

D. Magnetotransport

The equations of motion 5.8 and the density of statesEq. 5.13 gives us a complete description of the elec-

tron dynamics in the presence of electromagnetic fields.In this section we apply these results to the problem ofmagnetotransport. For simplicity, we set e==1 and in-troduce the shorthand dk=dk / 2d.

1. Cyclotron period

Semiclassical motion of a Bloch electron in a uniformmagnetic field is important to understand various mag-netoeffects in solids. In this case, the equations of mo-tion reduce to

Dkr = v + v · B , 5.25a

Dkk = − v B , 5.25b

where Dk=Dk / 2d=1+ e /B ·.We assume the field is along the z axis. From Eq.

5.25b we can see that motion in k space is confined inthe x-y plane and is completely determined once theenergy and the z component of the wave vector kz isgiven. We now calculate the period of the cyclotron mo-tion. The time for the wave vector to move from k1 to k2is

t2 − t1 = t1

t2dt =

k1

k2 dk

k. 5.26

From the equations of motion 5.25 we have

k =Bv

Dk=

B/k

Dk. 5.27

On the other hand, the quantity /k can be writtenas /k, where k denotes the vector in the planeconnecting points on neighboring orbits of energy and+. Then

t2 − t1 =

B

k1

k2 Dkkdk

. 5.28

Introducing the 2D electron density for given and kz,

n2,kz = kz,k

Dkdkxdky

22 , 5.29

the period of a cyclotron motion can be written as

T = 22

B

n2,kz

. 5.30

We thus recovered the usual expression for the cyclotronperiod, with the 2D electron density Eq. 5.29 definedwith the modified density of states.

In addition, we note that there is a velocity term pro-portional to B in Eq. 5.25, which seems to suggestthere will be a current along the field direction. We showthat after averaging over the distribution function, thiscurrent is actually zero. The current along B is given by



jB = − eB dkfv · = −e

B dkkF ·

= −e

B dkkF − dkFk · , 5.31

where F=−fd and f=F /. The first term

vanishes12 and if there is no magnetic monopole in kspace, the second term also vanishes. In the above cal-culation we did not consider the change in the Fermisurface. Since it always comes in the form f /=−f /, we can use the same technique to provethat the corresponding current also vanishes.

2. The high-field limit

We now consider the magnetotransport at the so-called high-field limit, i.e., c!1, where c=2 /T isthe cyclotron frequency and is the relaxation time. Weconsider a configuration where the electric and magneticfields are perpendicular to each other, i.e., E=Ex, B=Bz, and E ·B=0.

In the high-field limit, c!1, the electron can finishseveral turns between two successive collisions. We canthen assume all orbits are closed. According to the theo-rem of adiabatic drifting Niu and Sundaram, 2001, anoriginally closed orbit remains closed for weak perturba-tions so that

0 = k = E + r B 5.32

or

r =E B

B2 . 5.33

The Hall current is simply the sum over r of occupiedstates,

jH = − eE B

B2 dkfk1 + B ·

= − eE B

B2 dkfkDk . 5.34

Therefore in the high-field limit we reach the followingconclusion: the total current in crossed electric and mag-netic fields is the Hall current as if calculated from freeelectron model

j = − eE B

B2 n , 5.35

and it has no dependence on the relaxation time . Thisresult ensures that even in the presence of anomalous

Hall effect, the high-field Hall current gives the “real”electron density.

We now consider the holelike band. The Hall currentis obtained by subtracting the contribution of holes fromthat of the filled band, which is given by −eEdk.Therefore

jhole = eE B

B2 dkDk1 − fk

− eE dk . 5.36

As a result for the holelike band there is an additionalterm in the current expression proportional to the Chernnumber the second integral of the band.

3. The low-field limit

Next we consider the magnetotransport at the low-field limit, i.e., c"1. In particular, we show that theBerry phase induces a linear magnetoresistance. Bysolving the Boltzmann equation, one finds that the diag-onal element of the conductivity is given by

xx = − e2 dkf0

vx2

Dk. 5.37

This is just the zeroth-order expansion based on c.There are four places in this expression depending on B.

1 There is an explicit B dependence in Dk. 2 Theelectron velocity vx is modified by the orbital magneticmoment:

vx =1

0 − mzBkx

= vx0 −

1

mz

kxB . 5.38

3 There is also a modification to the Fermi energy,given by Eq. 5.17. 4 The relaxation time can alsodepend on B. In the presence of the Berry curvature, thecollision term in the Boltzmann equation is given by

f

t

coll= − dkDkWkkfk − fk , 5.39

where Wkk is the transition probability from k to kstate. In the relaxation-time approximation we make theassumption that a characteristic relaxation time exists sothat

f − f0

= Dk dk

Dk

DkWkkfk − fk . 5.40

If we assume k is smooth and Wkk is localized, therelaxation time is given by

=0

Dk 01 −

e

B · . 5.41

More generally, we can always expand the relaxationtime to first order of e /B ·,

12For any periodic function Fk with the periodicity of a re-ciprocal Bravais lattice, the following identity holds for inte-grals taken over a Brillouin zone, BZdkkFk=0. To see this,consider Ik=dkFk+k. Because Fk is periodic in k,Ik should not depend on k. Therefore, kIk=dk ,kFk+k=dkkFk+k=0. Setting k=0 gives thedesired expression. This is also true if Fk is a vector function.



= 0 + 1e

B · , 5.42

where 1 should be regarded as a fitting parameterwithin this theory.

Expanding Eq. 5.37 to first order of B and taking thespherical band approximation, we obtain

xx1 = e20B dk

f0

2ez

vx

2 +2

mz

kxvx

−e

Mxx

−1kF dkfz , 5.43

where M is the effect mass tensor. The zero-field con-ductivity takes the usual form

xx0 = − e20 dk

f0

vx

2. 5.44

The ratio −xx1 /xx

0 will then give us the linear magne-toresistance.

VI. ELECTRON DYNAMICS UNDER GENERALPERTURBATIONS

In this section we present the general theory of elec-tron dynamics in slowly perturbed crystals Sundaramand Niu, 1999; Panati et al., 2003; Shindou and Imura,2005. As expected, the Berry curvature enters into theequations of motion and modifies the density of states.The difference is that one needs to introduce the Berrycurvature defined in the extended parameter spacer ,q , t. Two physical applications are considered: elec-tron dynamics in deformed crystals and adiabatic cur-rent induced by inhomogeneity.

A. Equations of motion

We consider a slowly perturbed crystal whose Hamil-tonian can be expressed in the following form:

Hr,p ;1r,t, . . . ,gr,t , 6.1

where ir , t are the modulation functions characteriz-ing the perturbations. They may represent either gaugepotentials of electromagnetic fields, atomic displace-ments, charge or spin density waves, helical magneticstructures, or compositional gradients. Following thesame procedure as used in the previous section, we ex-pand the Hamiltonian around the wave-packet centerand obtain

H = Hc + H , 6.2

Hc = Hr,p ;irc,t , 6.3

H = i

rcirc,t · r − rc,

H

i . 6.4

Since the local Hamiltonian Hc maintains periodicity ofthe unperturbed crystal, its eigenstates take the Blochform

Hcrc,tqrc,t = crc,q,tqrc,t , 6.5

where q is the Bloch wave vector and crc ,q , t is theband energy. Here we have dropped the band index nfor simplicity.

Following the discussion in Sec. I.D, we switch to theBloch Hamiltonian Hcq ,rc , t=e−iq·rHcrc , teiq·r, whoseeigenstate is the periodic part of the Bloch function,uq ,rc , t=e−iq·rq ,rc , t. The Berry vector potentialscan be defined for each of the coordinates of the param-eter space q ,rc , t; for example,

At = uitu . 6.6

After constructing the wave packet using the localBloch functions qrc , t, one can apply the time-dependent variational principle to find the Lagrangiangoverning the dynamics of the wave packet:

L = − + qc · rc + qc · Aq + rc · Ar + At. 6.7

Note that the wave-packet energy =c+ has a cor-rection from H,

= WHW = − I u

rc · c − Hc u

q . 6.8

From the Lagrangian 6.7 we obtain the following equa-tions:

rc =

qc− J qr · rc +J qq · qc − qt, 6.9a

qc = −

rc+ J rr · rc +J rq · qc + rt, 6.9b

where ’s are the Berry curvatures. For example,

J qr = qAr

− rAk

. 6.10

In the following we also drop the subscript c on rc andqc.

The form of the equations of motion is quite sym-metrical with respect to r and q, and there are Berrycurvatures between every pair of phase-space variablesplus time. The term qt was identified as the adiabaticvelocity vector in Sec. II. In fact, if the perturbation isuniform in space has the same period as the unper-turbed crystal and only varies in time, all the spatialderivatives vanish; we obtain

r =

q− qt, q = 0. 6.11

The first equation is the velocity formula 2.5 obtained

in Sec. II. The term J qq was identified as the Hall con-ductivity tensor. In the presence of electromagnetic per-turbations, we have

H = H0q + eAr − er,t . 6.12

Hence the local basis can be written as urc ,q= uk,where k=q+eAr. One can verify that by using thechain rule q

=kand r

= rAk

, given in Eq.



6.8 becomes −mk ·B, and the equations of motion6.9 reduce to Eq. 5.8. The physics of quantum adia-batic transport and the quantum and anomalous Halleffect can be described from a unified point of view. TheBerry curvature rt plays a role like the electric force.

The antisymmetric tensor J rr is realized in terms of themagnetic field in the Lorenz force and is also seen in thesingular form -function-like distribution of disloca-tions in a deformed crystal Bird and Preston, 1988. Fi-nally, the Berry curvature between r and q can be real-ized in deformed crystals as a quantity proportional tothe strain and the electronic mass renormalization in thecrystal Sundaram and Niu, 1999.

B. Modified density of states

The electron density of states is also modified by theBerry curvature. Consider the time-independent case.To better appreciate the origin of this modification, weintroduce the phase-space coordinates = r ,q. Theequations of motion can be written as

= , 6.13

where J=J −JJ is an antisymmetric matrix with

J = J rr J rq

J qr J qq

, JJ= 0 IJ

− IJ 0 . 6.14

According to standard theory of Hamiltonian dynamicsArnold, 1978, the density of states, which is propor-tional to the phase-space measure, is given by

Dr,q =1

2ddetJ − JJ . 6.15

One can show that in the time-dependent case Dr ,qhas the same form.

Consider the following situations. i If the perturba-tion is electromagnetic field, by the variable substitutionk=q+eAr, Eq. 6.15 reduces to Eq. 5.13. ii In manysituations we are aiming at a first-order calculation inthe spatial gradient. In this case, the density of states isgiven by

D =1

2d 1 + TrJ qr . 6.16

Note that if the Berry curvature vanishes, Eq. 6.13becomes the canonical equations of motion for Hamil-tonian dynamics, and r and q are called canonical vari-ables. The density of states is a constant in this case. Thepresence of the Berry curvature renders the variablesnoncanonical and, as a consequence, modifies the den-sity of states. The noncanonical variables are a commonfeature of Berry phase participated dynamics Littlejohnand Flynn, 1991. The modified density of states alsoarises naturally from a nonequilibrium approach Olsonand Ao, 2007.

To demonstrate the modified density of states, weagain consider the Rice-Mele model discussed in Sec.

II.C.1. Now we introduce the spatial dependence by let-ting the dimerization parameter x vary in space. Us-ing Eq. 1.19 we find

qx =t sin2q/2x

42 + t2 cos2 q/2 + 2 sin2 q/23/2 . 6.17

At half filling, the system is an insulator and its electrondensity is given by

ne = −

dq

2qx. 6.18

We let x have a kink in its profile. Such a domain wallis known to carry fractional charge Su et al., 1979; Riceand Mele, 1982. Figure 11 shows the calculated electrondensity using Eq. 6.18 together with numerical resultobtained by direct diagonalization of the tight-bindingHamiltonian. These two results are virtually indistin-guishable in the plot, which confirms the Berry phasemodification to the density of states.

C. Deformed crystal

In this section we present a general theory of electrondynamics in crystals with deformation Sundaram andNiu, 1999, which could be caused by external pressure,defects in the lattice, or interfacial strain.

First we set up the basic notations for this problem.Consider a deformation described by the atomic dis-placement ul. We denote the deformed crystal poten-tial as Vr ; Rl+ul, where Rl is the atomic position withl labeling the atomic site. Introducing a smooth displace-ment field ur such that uRl+ul=ul, the Hamiltoniancan be written as

H =p2

2m+ Vr − ur + srVr − ur , 6.19

where s=u /r is the unsymmetrized strain andVr−ur=lRl+ur−rV /Rl is a gradient ex-pansion of the crystal potential. The last term, being

0 100 200 300 400 500

Lattice

1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

Density

Numerics

Theory

0 100 200 300 400 500

-1

-0.5

0

0.5

1

FIG. 11. Color online Electron density of the Rice-Melemodel with a spatial varying dimerization parameter. The pa-rameters used are =0.5, t=2, and =tanh0.02x. Inset: Theprofile of x. From Xiao, Clougherty, and Niu, 2007.



proportional to the strain, can be treated perturbatively.The local Hamiltonian is given by

Hc =p2

2m+ Vr − urc , 6.20

with its eigenstates q„r−urc….To write down the equations of motion, two pieces of

information are needed. One is the gradient correctionto the electron energy, given by Eq. 6.8. Sundaram andNiu 1999 found that

= sDq , 6.21

where

D = mvv − vv + V , 6.22

with ¯ the expectation value of the enclosed operatorsin the Bloch state and v is the velocity operator. Notethat in the free electron limit V→0 this quantity van-ishes. This is expected since a wave packet should notfeel the effect of a deformation of the lattice in the ab-sence of electron-phonon coupling. The other piece isthe Berry curvature, which is derived from the Berryvector potentials. For deformed crystals, in addition toAq, there are two other vector potentials

Ar = fur

, At = fut

, 6.23

with

fq =m

q− q . 6.24

This then leads to the following Berry curvatures:

qr= −

ur

fq

, qt =ut

fq

,

6.25rr

=xt = 0.

With the above information we plug in the electron en-ergy as well as the Berry curvatures into Eq. 6.9 toobtain the equations of motion.

We first consider a one-dimensional insulator with lat-tice constant a. Suppose the system is under a uniformstrain with a new lattice constant a+a, i.e., xu=a /a.Assuming one electron per unit cell, the electron densitygoes from 1/a to

1

a + a=

1

a1 −

a

a . 6.26

On the other hand, we can also directly calculate thechange in the electron density using the modified densityof states 6.16, which gives

0

2/a dq

2qx = −

a

a2 . 6.27

From a physical point of view, this change says an insu-lator under a uniform strain remains an insulator.

The above formalism is also applicable to dislocationstrain fields, which are well defined except in a region ofa few atomic spacings around the line of dislocation.Outside this region, the displacement field ur is asmooth but multiple valued function. On account of thismultiple valuedness, a wave packet of incident wave vec-tor q taken around the line of dislocation acquires aBerry phase

= c

dr · Ar = c

du · fk b · fk , 6.28

where b= dru /r is known as the Burgers vector.What we have here is a situation similar to theAharonov-Bohm effect Aharonov and Bohm, 1959,with the dislocation playing the role of the solenoid, andthe Berry curvature rr the role of the magnetic field.Bird and Preston 1988 showed that this Berry phasecan affect the electron diffraction pattern of a deformedcrystal.

The above discussion only touches a few general ideasof the Berry phase effect in deformed crystals. Withcomplete information on the equations of motion, thesemiclassical theory provides a powerful tool to investi-gate the effects of deformation on electron dynamicsand equilibrium properties.

D. Polarization induced by inhomogeneity

In Sec. II.C we discussed the Berry phase theory ofpolarization in crystalline solids, based on the basic ideathat the polarization is identical to the integration of theadiabatic current flow in the bulk. There the system isassumed to be periodic and the perturbation dependsonly on time or any scalar for that matter. In this case,it is straightforward to obtain the polarization based onthe equations of motion 6.11. However, in many physi-cal situations the system is in an inhomogeneous stateand the electric polarization strongly depends on the in-homogeneity. Examples include flexoelectricity where afinite polarization is produced by a strain gradientTagantsev, 1986, 1991 and multiferroic materials wherethe magnetic ordering varies in space and induces a po-larization Fiebig et al., 2002; Kimura et al., 2003; Hur etal., 2004; Cheong and Mostovoy, 2007.

Consider an insulating crystal with an order param-eter that varies slowly in space. We assume that, at leastat the mean-field level, the system can be described by aperfect crystal under the influence of an external fieldhr. If, for example, the order parameter is the magne-tization, then hr can be chosen as the exchange fieldthat yields the corresponding spin configuration. Ourgoal is to calculate the electric polarization to first orderin the spatial gradient as the field hr is graduallyturned on. The Hamiltonian thus takes the formHhr ;, where is the parameter describing the turn-ing on process. Xiao et al. 2009 showed that the first-order contribution to the polarization can be classifiedinto two categories: the perturbative contribution due to



the correction to the wave function and the topologicalcontribution which is from the dynamics of the elec-trons.

First consider the perturbation contribution, which isbasically a correction to the polarization formula ob-tained by King-Smith and Vanderbilt 1993 for a uni-form system. The perturbative contribution is obtainedby evaluating the Berry curvature qt in Eq. 2.27 tofirst order of the gradient. Remember that we alwaysexpand the Hamiltonian into the form H=Hc+H andchoose the eigenfunctions of Hc as our expansion basis.Hence to calculate the Berry curvature to first order ofthe gradient, one needs to know the form of the wavefunction perturbed by H. This calculation has been dis-cussed in the case of an electric field Nunes and Gonze,2001; Souza et al., 2002.

The topological contribution is of different nature.Starting from Eq. 6.9 and making use of the modifieddensity of states 6.16, one finds the adiabatic currentinduced by inhomogeneity is given by

j2 = e

BZ

dq2d

qqr +

qrq −

qrq .

6.29

We can see that this current is explicitly proportional tothe spatial gradient. Comparing this equation with Eq.2.6 reveals an elegant structure: the zeroth-order con-tribution Eq. 2.6 is given as an integral of the firstChern form, while the first-order contribution Eq.6.29 is given as an integral of the second Chern form.A similar result has been obtained by Qi et al. 2008.

The polarization is obtained by integrating the cur-rent. As usually in the case of multiferroics, we can as-sume the order parameter is periodic in space but ingeneral incommensurate with the crystal lattice. A two-point formula can be written down13

P2 =

e

V drBZ

dq2dA

qr A

q + Aq

qAr

+ Ar

qAq0

1, 6.30

where V is the volume of the periodic structure of theorder parameter. Again, due to the loss of tracking of ,there is an uncertain quantum which is the second Chernnumber. If we assume the order parameter has period lyin the y direction, the polarization quantum in the xdirection is given by

e

lyaz, 6.31

where a is the lattice constant.Kunz 1986 discussed the charge pumping in incom-

mensurate potentials and showed that in general the

charge transport is quantized and given in the form ofChern numbers, which is consistent with what we havederived.

The second Chern form demands that the systemmust be two dimensions or higher, otherwise the secondChern form vanishes. It allows us to determine the gen-eral form of the induced polarization. Consider a two-dimensional minimal model with hr having two com-ponents. If we write Hhr ; as Hhr, i.e., actslike a switch, the polarization can be written as

P2 =e

V dr# · hh − h · h . 6.32

The coefficient # is given by

# =e

8

BZ

dq22

0

1 d

abcdabcd, 6.33

where the Berry curvature is defined on the parameterspace q ,h and abcd is the Levi-Cività antisymmetrictensor.

Xiao et al. 2009 showed how the two-point formulacan be implemented in numerical calculations using adiscretized version Kotiuga, 1989.

1. Magnetic-field-induced polarization

An important application of the above result is themagnetic-field-induced polarization. Essin et al. 2009considered an insulator in the presence of a vector po-tential A=Byz with its associated magnetic field B=h /eazlyx. The inhomogeneity is introduced through thevector potential in the z direction. Note that magneticflux over the supercell az ly in the x direction is exactlyh /e, therefore the system is periodic in the y directionwith period ly. According to our discussion in Sec. V, theeffect of a magnetic field can be counted by the Peierlssubstitution kz→kz+eBy /, hence y= eB /kz. Ap-plying Eq. 6.30, one obtains the induced polarization

Px =e2

2hB , 6.34

with

=1

2

BZdkTrAA − i

23AAA , 6.35

where Amn= iumkun is the non-Abelian Berry

connection, discussed in Sec. IX. Recall that the polar-ization is defined as the response of the total energy toan electric field P=E /E, such a magnetic-field-inducedpolarization implies that there is an electromagneticcoupling of the form

LEM =e2

2hE · B . 6.36

This coupling, labeled “axion electrodynamics,” was dis-cussed by Wilczek 1987. When =, the correspondinginsulator is known as a 3D Z2 topological insulator Qiet al., 2008.

13So far we only considered the Abelian Berry case. The non-Abelian result is obtained by replacing the Chern-Simons formwith its non-Abelian form.



E. Spin texture

So far our discussion has focused on the physical ef-fects of the Berry curvature in the momentum spacekk or in the mixed space of the momentum coordi-nates and some other parameters kr and kt. In thissection we discuss the Berry curvatures which originateonly from the nontrivial real-space configuration of thesystem.

One of such systems is magnetic materials with do-main walls or spin textures. Consider a ferromagneticthin films described by the following Hamiltonian:

H =p2

2m− Jnr,t · , 6.37

where the first term is the bare Hamiltonian for a con-duction electron and the second term is the s-d couplingbetween the conduction electron and the locald-electron spin along the direction nr , t with J the cou-pling strength. Note that we have allowed the spin tex-ture to vary in both space and time. The simple momen-tum dependence of the Hamiltonian dictates that allk-dependent Berry curvatures vanish.

Because of the strong s-d coupling, we adopt the adia-batic approximation which states that the electron spinwill follow the local spin direction during its motion.Then the spatial variation in local spin textures gives riseto the Berry curvature field

rr = 12 sin , 6.38

where and are the spherical angles specifying thedirection of n. According to Eqs. 6.9, this field acts onthe electrons as an effective magnetic field. In addition,the time-dependence of the spin texture also gives risesto

rt = 12 sin t − t . 6.39

Similarly, rt acts on the electrons as an effective elec-tric field. This is in analogy with a moving magnetic fieldrr generating an electric field rt.

The physical consequences of these two fields are ob-vious by analogy with the electromagnetic fields. TheBerry curvature rr will drive a Hall current, just likethe ordinary Hall effect Ye et al., 1999; Bruno et al.,2004. Unlike the anomalous Hall effect discussed inSec. III.D, this mechanism for a nonvanishing Hall effectdoes not require the spin-orbit coupling but does need atopologically nontrivial spin texture, for example, a skyr-mion. On the other hand, for a moving domain wall in athin magnetic wire the Berry curvature rt will inducean electromotive force, which results in a voltage differ-ence between the two ends. This Berry curvature in-duced emf has recently been experimentally measuredYang et al., 2009.

VII. QUANTIZATION OF ELECTRON DYNAMICS

In previous sections we have reviewed several Berryphase effects in solid-state systems. Berry curvature of-

ten appears as a result of restricting or projecting theextent of a theory to its subspace. In particular, theBerry curvature plays a crucial role in the semiclassicaldynamics of electrons, which is valid under the one-bandapproximation. In the following, we explain how thesemiclassical formulation could be requantized. This isnecessary, for example, in studying the quantizedWannier-Stark ladders from the Bloch oscillation, or thequantized Landau levels from the cyclotron orbit Ash-croft and Mermin, 1976. The requantized theory is validin the same subspace of the semiclassical theory. It willbecome clear that, the knowledge of the Bloch energy,the Berry curvature, and the magnetic moment in thesemiclassical theory constitute sufficient information forbuilding the requantized theory. In this section we focuson the following methods of quantization: the Bohr-Sommerfeld quantization and the canonical quantiza-tion.

A. Bohr-Sommerfeld quantization

A method of quantization is a way to select quantummechanically allowed states out of a continuum of clas-sical states. This is often formulated using the general-ized coordinates qi and their conjugate momenta pi. TheBohr-Sommerfeld quantization requires the action inte-gral for each set of the conjugate variables to satisfy

Ci

pidqi = mi +i

4h, i = 1, . . . ,d , 7.1

where Ci are closed trajectories in the phase space withdimension 2d, mi are integers, and i are the so-calledMaslov indices, which are usually integers. Notice thatsince the choice of conjugate variables may not beunique, the Bohr-Sommerfeld quantization methodcould give inequivalent quantization rules. This problemcan be fixed by the following Einstein-Brillouin-KellerEBK quantization rule.

For a completely integrable system, there are d con-stants of motion. As a result, the trajectories in thephase space are confined to a d-dimensional torus. Onsuch a torus, one can have d closed loops that are topo-logically independent of each other Tabor, 1989. Thatis, one cannot be deformed continuously to the other.Since =1

d pdq is invariant under coordinate transfor-mation, instead of Eq. 7.1, one can use the followingEBK quantization condition:

Ck

=1

d

pdq = mk +k

4h, k = 1, . . . ,d , 7.2

where Ck are periodic orbits on invariant tori. Such aformula is geometric in nature i.e., it is coordinate in-dependent. Furthermore, it applies to the invariant toriof systems that may not be completely integrable.Therefore, the EBK formula plays an important role inquantizing chaotic systems Stone, 2005. In the follow-ing, we refer to both types of quantization simply as theBohr-Sommerfeld quantization.



In the wave-packet formulation of Bloch electrons,both rc and qc are treated as generalized coordinates.With the Lagrangian in Eq. 5.7, one can find their con-jugate momenta L /rc and L /qc, which are equal toqc and u iu /qc=A, respectively Sundaram andNiu, 1999. The quantization condition for an orbit withconstant energy thus becomes

C

qc · drc = 2m +

4−C

2 , 7.3

where CCA ·dqc is the Berry phase of an energycontour C see also Wilkinson 1984b and Kuratsuji andIida 1985. Since the Berry phase is path dependent,one may need to solve the equation self-consistently toobtain the quantized orbits.

Before applying the Bohr-Sommerfeld quantization inthe following sections, we point out two disadvantagesof this method. First, the value of the Maslov index isnot always apparent. For example, for a one-dimensional particle bounded by two walls, its valuewould depend on the slopes of the walls van Houten etal., 1989. In fact, a noninteger value may give a moreaccurate prediction of the energy levels Friedrich andTrost, 1996. Second, this method fails if the trajectory inphase space is not closed, or if the dynamic system ischaotic and invariant tori fail to exist. On the otherhand, the method of canonical quantization in Sec.VII.D does not have these problems.

B. Wannier-Stark ladder

Consider an electron moving in a one-dimensional pe-riodic lattice with lattice constant a. Under a weak uni-form electric field E, according to the semiclassical equa-tions of motion, the quasimomentum of an electronwave packet is simply see Eq. 5.8

qct = − eEt . 7.4

It takes the time TB=h /eEa for the electron to traversethe first Brillouin zone. Therefore, the angular fre-quency of the periodic motion is B=eEa /. This is theso-called Bloch oscillation Ashcroft and Mermin, 1976.

Similar to a simple harmonic oscillator, the energy ofthe oscillatory electron is quantized in multiples of B.However, unlike the former, the Bloch oscillator has nozero-point energy that is, the Maslov index is zero.These equally spaced energy levels are called theWannier-Stark ladders. Since the Brillouin zone is peri-odic, the electron orbit is closed. According to the Bohr-Sommerfeld quantization, one has

Cm

rc · dqc = − 2m −Cm

2 . 7.5

For a simple one-dimensional lattice with inversion sym-metry, if the origin is located at a symmetric point, thenthe Berry phase Cm

can only have two values, 0 or Zak, 1989, as discussed in Sec. II.C.

Starting from Eq. 7.5, it is not difficult to find theaverage position of the electron,

rcm = am −C

2 , 7.6

where we have neglected the subscript m in Cmsince all

of the paths in the same energy band have the sameBerry phase here. Such average positions rcm are theaverage positions of the Wannier function Vanderbiltand King-Smith, 1993. Due to the Berry phase, they aredisplaced from the positive ions located at am.

In Sec. II.C the electric polarization is derived usingthe theory of adiabatic transport. It can also be obtainedfrom the expectation value of the position operator di-rectly. Because of the charge separation mentionedabove, the one-dimensional crystal has a polarizationP=ec /2 compared to the state without charge sepa-ration, which is the electric dipole per unit cell. This isconsistent with the result in Eq. 2.28.

After time average, the quantized energies of the elec-tron wave packet are

Em = qc − eErcm = 0 − eEam −C

2 , 7.7

which are the energy levels of the Wannier-Stark lad-ders.

Two short comments are in order: First, beyond theone-band approximation, there exist Zener tunnelingsbetween Bloch bands. Therefore, the quantized levelsare not stationary states of the system. They should beunderstood as resonances with finite lifetimes Avron,1982; Glück et al., 1999. Second, the fascinating phe-nomenon of Bloch oscillation is not commonly observedin laboratory for the following reason: In an usual solid,the electron scattering time is shorter than the period TBby several orders of magnitude. Therefore, the phasecoherence of the electron is destroyed within a smallfraction of a period. Nonetheless, with the help of a su-perlattice that has a much larger lattice constant the pe-riod TB can be reduced by two orders of magnitude,which could make the Bloch oscillation and the accom-panying Wannier-Stark ladders detectable Mendez andBastard, 1993. Alternatively, the Bloch oscillation andWannier-Stark ladders can also be realized in an opticallattice Ben Dahan et al., 1996; Wilkinson and Kay,1996, in which the atom can be coherent over a longperiod of time.

C. de Haas–van Alphen oscillation

When a uniform B field is applied to a solid, the elec-tron would execute a cyclotron motion in both r and thek space. From Eq. 5.25b, it is not difficult to see that anorbit C in k space lies on the intersection of a planeperpendicular to the magnetic field and the constant-energy surface Ashcroft and Mermin, 1976. Withoutquantization, the size of an orbit is determined by theinitial energy of the electron and can be varied continu-ously. One then applies the Bohr-Sommerfeld quantiza-



tion rule, as Onsager did, to quantize the size of theorbit Onsager, 1952. That is, only certain orbits satisfy-ing the quantization rule are allowed. Each orbit corre-sponds to an energy level of the electron i.e., the Lan-dau level. Such a method remains valid in the presenceof the Berry phase.

With the help of the semiclassical equation see Eq.5.8,

kc = − erc B , 7.8

the Bohr-Sommerfeld condition in Eq. 7.3 can be writ-ten as note that qc=kc−eA, and =2

B2

· Cm

rc drc = m +12

−Cm

20, 7.9

where 0h /e is the flux quantum. The integral on theleft-hand side is simply the magnetic flux enclosed by thereal-space orbit allowing a drift along the B direction.Therefore, the enclosed flux has to jump in steps of theflux quantum plus a Berry phase correction.

Similar to the Bohr atom model, in which the electronhas to form a standing wave, here the total phase ac-quired by the electron after one circular motion also hasto be integer multiples of 2. Three types of phases con-tribute to the total phase: a the Aharonov-Bohmphase—an electron circulating a flux quantum picks up aphase of 2; b the phase lag of at each turning pointthere are two of them—this explains why the Maslovindex is two; and c the Berry phase intrinsic to thesolid. Therefore, Eq. 7.9 simply says that the summa-tion of these three phases should be equal to 2m.

The orbit in k space can be obtained by rescaling ther-space orbit in Eq. 7.9 with a linear factor of B

2 , fol-lowed by a rotation of 90°, where B /eB is the mag-netic length Ashcroft and Mermin, 1976. Therefore,one has

B2

· Cm

kc dkc = 2m +12

−Cm

2 eB

. 7.10

The size of the orbit combined with the knowledge ofthe electron energy Ekc=kc−M ·B help determinethe quantized energy levels. For an electron with a qua-dratic energy dispersion before applying the magneticfield, these levels are equally spaced. However, with theBerry phase correction, which is usually different for dif-ferent orbits, the energy levels are no longer uniformlydistributed. This is related to the discussion in Sec. V onthe relation between the density of states and the Berrycurvature Xiao et al., 2005.

As a demonstration, we apply the quantization rule tographene and calculate the energies of Landau levelsnear the Dirac point. Before applying a magnetic field,the energy dispersion near the Dirac point is linear,k=vFk. It is known that if the energy dispersion

near a degenerate point is linear, then the cyclotron or-bit will acquire a Berry phase C=, independent of theshape of the orbit Blount, 1962b. As a result, the 1/2on the right-hand side of Eq. 7.10 is canceled by theBerry phase term. According to Eq. 7.10, the area of acyclotron orbit is thus k2=2meB /, where m is a non-negative integer, from which one can easily obtain theLandau-level energy m=vF2eBm. The experimentalobservation of a quantum Hall plateau at zero energy isthus a direct consequence of the Berry phase No-voselov et al., 2005, 2006; Zhang et al., 2005.

In addition to point degeneracy, other types of degen-eracy in momentum space can also be a source of theBerry phase. For example, the effect of the Berry phasegenerated by a line of band contact on magneto-oscillations is studied by Mikitik and Sharlai 1999,2004.

The discussion so far is based on the one-band ap-proximation. In reality, the orbit in one band wouldcouple with the orbits in other bands. As a result, theLandau levels are broadened into minibands Wilkinson,1984a. A similar situation occurs in a magnetic Blochband, which is the subject of Sec. VIII.

D. Canonical quantization (Abelian case)

In addition to the Bohr-Sommerfeld quantization, analternative way to quantize a classical theory is by find-ing out position and momentum variables that satisfythe following Poisson brackets:

xi,pj = ij. 7.11

Afterwards, these classical canonical variables are pro-moted to operators that satisfy the commutation relation

xi,pj = iij, 7.12

that is, all we need to do is to substitute the Poissonbracket xi ,pj by the commutator xi ,pj / i. Based onthe commutation relation, these variables can be writtenexplicitly using either the differential-operator represen-tation or the matrix representation. Once this is done,one can proceed to obtain the eigenvalues and eigen-states of the Hamiltonian Hx ,p.

Even though one can always have canonical pairs in aHamiltonian system, as guaranteed by the Darbouxtheorem Arnold, 1989, in practice, however, findingthem may not be a trivial task. For example, the center-of-mass variables rc and kc in the semiclassical dynamicsin Eq. 5.8 are not canonical variables since their Pois-son brackets below are not of the canonical form Xiaoet al., 2005; Duval et al., 2006a,

ri,rj = ijkk/ , 7.13

ki,kj = − ijkeBk/ , 7.14



ri,kj = ij + eBij/ , 7.15

where 1+eBr ·k. In order to carry out the ca-nonical quantization, canonical variables of position andmomentum must be found.

The derivation of Eqs. 7.13–7.15 is outlined as fol-lows: One first writes the equations of motion in Eq.5.7 in the form of Eq. 6.13 = r ,k. In this case, the

only nonzero Berry curvatures are J rrij=−ijkeBk and

J kkij=ijkk, whereas J rk and J kr are zero. The Pois-

son bracket of two functions in the phase space f and g isthen defined as

f,g = f

T

J−1 g

, 7.16

where J=J −JJ see Eq. 6.14. It is defined in such away that the equations of motion can be written in the

standard form: = ,. To linear order of magneticfield or Berry curvature, one can show that

J−1 =1

1 + eB · J kk IJ−J kk

J

rr + eB ·

− IJ+J rrJ

kk − eB · J rr

. 7.17

Equations 7.13–7.15 thus follow when f and g areidentified with the components of .

We start with two special cases. The first is a solid withzero Berry curvature that is under the influence of amagnetic field =0, B0. In this case, the factor inEq. 7.14 reduces to 1 and the position variables com-mute with each other. Obviously, if one assumes kc=p+eAx and requires x and p to be canonical conju-gate variables, then the quantized version of Eq. 7.14with i inserted can easily be satisfied. This is the fa-miliar Peierls substitution Peierls, 1933.

In the second case, consider a system with Berry cur-vature but not in a magnetic field 0, B=0. In thiscase, again we have =1. Now the roles of rc and kc inthe commutators are reversed. The momentum variablescommute with each other but not the coordinates. Onecan apply a Peierls-like substitution to the coordinatevariables and write rc=x+Aq. It is not difficult to seethat the commutation relations arising from Eq. 7.13can indeed be satisfied. After the canonical quantization,x becomes i /q in the quasimomentum representation.Blount 1962b showed that the position operator r inthe one-band approximation acquires a correction,which is our Berry connection A. Therefore, rc can beidentified with the projected position operator PrP,where P projects to the energy band of interest.

When both B and are nonzero, applying both of thePeierls substitutions simultaneously is not enough toproduce the correct commutation relations, mainly be-cause of the nontrivial factor there. In general, exactcanonical variables cannot be found easily. However,since the semiclassical theory itself is valid to linear or-der of field, we only need to find the canonical variablescorrect to the same order in practice. The result isChang and Niu, 2008

rc = x + A + G ,7.18

kc = p + eAx + eBA ,

where =p+eAx and Gkce /AB ·A /kc.This is the generalized Peierls substitution for systemswith Berry connection A and vector potential A. Withthese equations, one can verify Eqs. 7.13 and 7.15 tolinear orders of B and .

A few comments are in order: First, if a physical ob-servable is a product of several canonical variables, theorder of the product may become a problem after thequantization since the variables may not commute witheach other. To preserve the Hermitian property of thephysical observable, the operator product needs to besymmetrized. Second, the Bloch energy, Berry curva-ture, and orbital moment of the semiclassical theorycontains sufficient information for building a quantumtheory that accounts for all physical effects to first orderin external fields. We return to this in Sec. IX, where thenon-Abelian generalization of the canonical quantiza-tion method is addressed.

VIII. MAGNETIC BLOCH BANDS

The semiclassical dynamics in previous sections isvalid when the external field is weak, so that the lattercan be treated as a perturbation to the Bloch states.Such a premise is no longer valid if the external field isso strong that the structure of the Bloch bands is signifi-cantly altered. This happens, for example, in quantumHall systems where the magnetic field is of the order ofTesla and a Bloch band would break into many sub-bands. The translational symmetry and the topologicalproperty of the subband are very different from those ofthe usual Bloch band. To distinguish between the two,the former is called the magnetic Bloch band MBB.

The MBB usually carries nonzero quantum Hall con-ductance and has a nontrivial topology. Compared to theusual Bloch band, the MBB is a more interesting play-ground for many physics phenomena. In fact, nontrivial



topology of magnetic Bloch state was first revealed inthe MBB Thouless et al., 1982. In this section, we re-view some basic facts of the MBB, as well as the semi-classical dynamics of the magnetic Bloch electron whenit is subject to further electromagnetic perturbationChang and Niu, 1995, 1996. Such a formulation pro-vides a clear picture of the hierarchical subbands split bythe strong magnetic field called the Hofstadter spec-trum Hofstadter, 1976, which could also be realized ina Bose-Einstein condensate, e.g., see Umucalılar et al.2008.

A. Magnetic translational symmetry

In the presence of a strong magnetic field, one needsto treat the magnetic field and the lattice potential onequal footing and solve the following Schrodinger equa-tion:

1

2mp + eAr2 + VLrr = Er , 8.1

where VL is the periodic lattice potential. For conve-nience of discussion, we assume the magnetic field isuniform along the z axis and the electron is confined tothe x-y plane. Because of the vector potential, theHamiltonian H above no longer has the lattice transla-tion symmetry.

Since the lattice symmetry of the charge density is notbroken by an uniform magnetic field, one should be ableto define translation operators that differ from the usualones only by phase factors Lifshitz and Landau, 1980.First, consider a system translated by a lattice vector a,

1

2mp + eAr + a2 + VLrr + a = Er + a ,

8.2

where VLr+a=VLr has been used. One can write

Ar + a = Ar + fr , 8.3

where fr=Ar+a−ArAa. In a uniform mag-netic field, one can choose a gauge such that A is per-pendicular to the magnetic field Bz and its componentslinear in x and y. As a result, A is independent of r andf=A ·r. The extra vector potential f can be removedby a gauge transformation,

1

2mp + eAr2 + VLreie/fr + a

= Eeie/fr + a . 8.4

We now identify the state above as the magnetic trans-lated state Tar,

Tar = eie/A·rr + a . 8.5

The operator Ta being defined this way has the desiredproperty that H ,Ta=0.

Unlike usual translation operators, magnetic transla-tions along different directions usually do not commute.For example, let a1 and a2 be lattice vectors, then

Ta2Ta1

= Ta1Ta2

expie

A · dr , 8.6

where A ·dr is the magnetic flux going through the unitcell defined by a1 and a2. That is, the noncommutativityis a result of the Aharonov-Bohm phase. Ta1

commuteswith Ta2

only if the flux is an integer multiple of theflux quantum 0=e /h.

If the magnetic flux enclosed by a plaquette isp /q0, where p and q are coprime integers, then Tqa1would commute with Ta2

see Fig. 12. The simultaneouseigenstate of H, Tqa1

, and Ta2is called a magnetic Bloch

state and its energy the magnetic Bloch energy,

Hnk = Enknk, 8.7

Tqa1nk = eik·qa1nk, 8.8

Ta2nk = eik·a2nk. 8.9

Since the magnetic unit cell is q times larger than theusual unit cell, the magnetic Brillouin zone MBZ hasto be q times smaller. If b1 and b2 are defined as thelattice vectors reciprocal to a1 and a2, then, in this ex-ample, the MBZ is folded back q times along the b1direction.

Magnetic unit cell

plaquette

Mag

netic

Bril

loui

nzo

ne

energycontour

a

2 π /a

FIG. 12. Color online When the magnetic flux per plaquetteis 0 /3, the magnetic unit cell is composed of three plaquettes.The magnetic Brillouin zone is three times smaller than theusual Brillouin zone. Furthermore, the magnetic Bloch statesare threefold degenerate.



In addition, with the help of Eqs. 8.6 and 8.8, onecan show that the eigenvalues of the Ta2

operator for thefollowing translated states,

Ta1nk,T2a1

nk, . . . ,Tq−1a1nk, 8.10

are

eik+b2p/q·a2,eik+2b2p/q·a2, . . . ,eik+q−1b2p/q·a2, 8.11

respectively. These states are not equivalent, but havethe same energy as nk since H ,Ta1

=0. Therefore, theMBZ has a q-fold degeneracy along the b2 direction.Each repetition unit in the MBZ is sometimes called areduced magnetic Brillouin zone. More discussions onthe magnetic translation group can be found in Zak1964a, 1964b, 1964c.

B. Basics of magnetic Bloch band

In this section we review some basic properties of themagnetic Bloch band. These include the pattern of bandsplitting due to a quantizing magnetic field, the phase ofthe magnetic Bloch state, and its connection with theHall conductance.

The rules of band splitting are simple in two oppositelimits, which are characterized by the relative strengthbetween the lattice potential and the magnetic field.When the lattice potential is much stronger than themagnetic field, it is more appropriate to start with thezero-field Bloch band as a reference. It was found that ifeach plaquette encloses a magnetic flux p /q0, theneach Bloch band would split to q subbands Obermairand Wannier, 1976; Schellnhuber and Obermair, 1980;Wannier, 1980; Kohmoto, 1989; Hatsugai and Kohmoto,1990. We know that if N is the total number of latticesites on the two-dimensional plane, then the number ofallowed states in the Brillouin zone and in one Blochband is N. Since the area of the MBZ and the numberof states within is smaller by a factor of q, each MBBhas N /q states, sharing the number of states of the origi-nal Bloch band equally.

On the other hand, if the magnetic field is much stron-ger than the lattice potential, then one should start fromthe Landau level as a reference. In this case, if eachplaquette has a magnetic flux = p /q0, then afterturning on the lattice potential each Landau level LLwill split to p subbands. The state counting is quite dif-ferent from the previous case: The degeneracy of theoriginal LL is /0=Np /q, where =N is the totalmagnetic flux through the two-dimensional sample.Therefore, after splitting, each MBB again has only N /qstates, the number of states in a MBZ.

Between the two limits, when the magnetic field isneither very strong nor very weak, the band splittingdoes not follow a simple pattern. When the field is tunedfrom weak to strong, the subbands will split, merge, andinteract with each other in a complicated manner, suchthat in the end there are only p subbands in the strong-field limit.

According to Laughlin’s gauge-invariance argumentLaughlin, 1981, each of the isolated magnetic Blochbands carries a quantized Hall conductivity see Secs.II.B and III.C. This is closely related to the nontrivialtopological property of the magnetic Bloch stateKohmoto, 1985; Morandi, 1988. Furthermore, the dis-tribution of Hall conductivities among the split subbandsfollows a simple rule first discovered by Thouless et al.1982. This rule can be derived with the help of themagnetic translation symmetry Dana et al., 1985. Weshow the derivation below following the analysis ofDana et al. since it reveals the important role played bythe Berry phase in the magnetic Bloch state.

In general, the phases of Bloch states at different k’sare unrelated and can be defined independently.14 How-ever, the same does not apply to a MBZ. For one thing,the phase has to be nonintegrable in order to accountfor the Hall conductivity. One way to assign the phase ofthe MBS ukr is by imposing the parallel-transport con-dition see Thouless’s article in Prange and Girvin1987,

uk10

k1uk10 = 0, 8.12

uk1k2

k2uk1k2

= 0. 8.13

The first equation defines the phase of the states on thek1 axis; the second equation defines the phase along aline with fixed k1 see Fig. 13a. As a result, the phasesof any two states in the MBZ have a definite relation.

The states on opposite sides of the MBZ boundariesrepresent the same physical state. Therefore, they canonly differ by a phase factor. Following Eqs. 8.12 and8.13, we have

uk1+b1/q,k2= uk1,k2

, 8.14

uk1,k2+b2= eik1uk1,k2

, 8.15

where b1 and b2 are the lengths of the primitive vectorsreciprocal to a1 and a2. That is, the states on the oppo-site sides of the k1 boundaries have the same phase. Thesame cannot also be true for the k2 boundaries, other-wise the topology will be too trivial to accommodate thequantum Hall conductivity.

Periodicity of the MBZ requires that

k1 + b1/q = k1 + 2 integer. 8.16

In order for the integral 1/2MBZdk ·Ak which isnonzero only along the upper horizontal boundary tobe the Hall conductivity H in units of h /e2, the integerin Eq. 8.16 obviously has to be equal to H.

Following the periodicity condition in Eq. 8.16, it ispossible to assign the phase in the form

14In practice, the phases are usually required to be continuousand differentiable so that the Wannier function can behavewell.



k1 = k1 + Hk1qa1, 8.17

where k1+b1 /q= k1. On the other hand, from thediscussion at the end of Sec. VIII.A, we know that

Ta1uk1k2

= eik1uk1k2+2p/qa2. 8.18

Again from the periodicity of the MBZ, one has

k1 + b1/q = k1 + 2m, m Z , 8.19

which gives

k1 = k1 + mk1qa1. 8.20

Choosing k1 and k1 to be 0, one finally gets

Tqa1uk1k2

= eiqmk1qa1uk1k2+2p/a2

= eiqmk1qa1eipHk1qa1uk1k2. 8.21

But this state should also be equal to eiqk1a1uk1k2. There-

fore, the Hall conductivity should satisfy

pH + qm = 1. 8.22

This equation determines the Hall conductivity mod qof a MBB Dana et al., 1985. In Sec. VIII.D, we will seethat the semiclassical analysis can also help us findingout the Hall conductivity of a MBB.

C. Semiclassical picture: Hyperorbits

When a weak magnetic field is applied to a Blochband, the electron experiences a Lorentz force and ex-ecutes a cyclotron motion on the surface of the Fermisea. In the case of the MBB, the magnetic field B0changes the band structure itself. On the other hand, themagnetic quasimomentum k is a good quantum num-

ber with k=0. Therefore, there is no cyclotron motionof k even though there is a magnetic field B0. Similarto the case of the Bloch band, one can construct a wavepacket out of the magnetic Bloch states, and study itsmotion in both r and k spaces when it is subject to anadditional weak electromagnetic field E and B. Thesemiclassical equations of motion that are valid underthe one-band approximation have exactly the same formas Eq. 5.8. One simply needs to reinterpret k, E0k,and B in Eq. 5.8 as the magnetic momentum, the mag-netic band energy, and the extra magnetic field B, re-spectively Chang and Niu, 1995, 1996. As a result,when B is not zero, there exists similar circulating mo-tion in the MBB. This type of orbit will be called “hy-perorbit.”

We first consider the case without the electric fieldthe case with both E and B will be considered in Sec.VIII.D. By combining the following two equations ofmotion cf. Eq. 5.8,

k = − er B , 8.23

r =E

k− k , 8.24

one has

k = −1

E

k B

e

, 8.25

where k=1+kBe /. This determines the k orbitmoving along a path with constant Ek=E0k−Mk ·B, which is the magnetic Bloch band energyshifted by the magnetization energy. Similar to the Blochband case, it is not difficult to see from Eq. 8.23 thatthe r orbit is simply the k orbit rotated by /2 and lin-early scaled by the factor /eB. These orbits in theMBB and their real-space counterparts are the hyperor-bits mentioned above Chambers, 1965.

The size of a real-space hyperorbit may be very largeif phase coherence can be maintained during the circu-lation since it is proportional to the inverse of the re-sidual magnetic field B. Furthermore, since the splitmagnetic subband is narrower and flatter than the origi-nal Bloch band, the electron group velocity is small. Asa result, the frequency of the hyperorbit motion can bevery low. Nevertheless, it is possible to detect the hyper-orbit using, for example, resonant absorption of ultra-sonic wave or conductance oscillation in an electron fo-cusing device.

Similar to the cyclotron orbit, the hyperorbit motioncan also be quantized using the Bohr-Sommerfeld quan-

k2

(a)

(b)

k1

k2

k1

FIG. 13. Color online Reduced magnetic Brillouin zone. aThe phases of the MBS in the reduced MBZ can be assignedusing the parallel transport conditions, first along the k1 axis,then along the paths parallel to the k2 axis. b Hyperorbits ina reduced MBZ. Their sizes are quantized following the Bohr-Sommerfeld quantization condition. The orbit enclosing thelargest area is indicated by solid lines.



tization rule see Eq. 7.3. One only needs to bear inmind that k is confined to the smaller MBZ and themagnetic field in Eq. 7.3 should be B. After the quan-tization, there can only be a finite number of hyperorbitsin the MBZ. The area of the largest hyperorbit shouldbe equal to or slightly smaller assuming B"B0 so thatthe number of hyperorbits is large than the area of theMBZ 2 /a2 /q see Fig. 13b. For such an orbit, theBerry phase correction /2 in Eq. 7.3 is very close tothe integer Hall conductivity H of the MBB. Therefore,it is not difficult to see that the number of hyperorbitsshould be 1/ q+H, where Ba2 /0 is the re-sidual flux per plaquette.

Because the MBZ is q-fold degenerate see Sec.VIII.A, the number of energy levels produced by thesehyperorbits are Chang and Niu, 1995

D =1/q + H

q. 8.26

If one further takes the tunneling between degeneratehyperorbits into account Wilkinson, 1984a, then eachenergy level will be broadened into an energy band.These are the magnetic energy subbands at a finer en-ergy scale compared to the original MBB.

D. Hall conductivity of hyperorbit

According to Laughlin’s argument, each of the iso-lated subband should have its own integer Hall conduc-tivity. That is, as a result of band splitting, the integerHall conductivity H of the parent band is split to adistribution of integers r there are q of them. The sumof these integers should be equal to the original Hallconductivity: H=rr. There is a surprisingly simpleway to determine this distribution using the semiclassicalformulation: one only needs to study the response of thehyperorbit to an electric field.

After adding a term −eE to Eq. 8.23, one obtains

r =

eBk z +

E z

B. 8.27

For a closed orbit, this is just a cyclotron motion super-imposed with a drift along the EB direction. Aftertime average, the former does not contribute to a nettransport. Therefore the Hall current density for a filledmagnetic band in a clean sample is

JH = − eMBZ

d2k22 r = − e

E z

B, 8.28

where is the number of states in the MBZ divided bythe sample area. Therefore, the Hall conductivity isr

close=e /B. If the areal electron density of a sample is0, then after applying a flux =p /q per plaquette theMBZ shrinks by q times and =0 /q.

How can one be sure that both the degeneracy in Eq.8.26 and the Hall conductivity r

close are integers? Thisis closely related to the following question: How doesone divide an uniform magnetic field B into the quantiz-

ing part B0 and the perturbation B? The proper way toseparate them was first proposed by Azbel 1964. Sincethen, such a recipe has been used widely in the analysisof the Hofstadter spectrum Hofstadter, 1976.

One first expands the flux =p /q1 as a continuedfraction,

p

q=

1

f1 +1

f2 +1

f3 +1¯

f1,f2,f3, . . . , 8.29

then the continued fraction is truncated to obtain vari-ous orders of approximate magnetic flux. For example,1= f1p1 /q1, 2= f1 , f2p2 /q2, 3= f1 , f2 , f3p3 /q3 , . . ., etc. What is special about these truncationsis that pr /qr is the closest approximation to p /q amongall fractions with q$qr Khinchin, 1964.

As a reference, we show two identities that will beused below:

qr+1 = fr+1qr + qr−1, 8.30

pr+1qr − prqr+1 = − 1r. 8.31

According to desired accuracy, one chooses a particu-lar r to be the quantizing flux, and takes rr+1−r as a perturbation see Fig. 14. With the help of Eq.8.31 one has

r =− 1r

qrqr+1. 8.32

As a result, the Hall conductivity for a closed hyperorbitproduced by Br−1r−1 /a2 is recall that r=0 /qr

rclose =

er

Br−1= − 1r−1qr−1. 8.33

Substituting this value back into Eq. 8.26 for Drclose the

number of subbands split by r, and using Eq. 8.30,one has

B

Br+1

Br

E

δBr

0r

rqρρ = close

1

rr

r

eBρσ

δ −

=

Dr subbands

close closeopen

1 1,r rρ σ+ +

closeclose

FIG. 14. A parent magnetic Bloch band at magnetic field Brsplits to Dr subbands Dr=5 here due to a perturbation Br+1.The subbands near the band edges of the parent band areusually originated from closed hyperorbits. The subband in themiddle is from an open hyperorbit.



Drclose =

1/qr%r + rclose

qr= fr+1. 8.34

This is the number of subbands split from a parent bandthat is originated from a closed hyperorbit. One can seethat the Hall conductivity and the number of splittingsubbands are indeed integers.

For lattices with square or triangular symmetry, thereis one and only one nesting open hyperorbit in theMBZ for example, see the diamond-shaped energy con-tour in Fig. 13b. Because of its open trajectory, theabove analysis fails for the nesting orbit since the firstterm in Eq. 8.27 also contributes to the Hall conduc-tivity. However, since the total number of hyperorbits inthe parent band can be determined by the quantizationrule, we can easily pin down the value of r

open with thehelp of the sum rule H

parent=rr. Furthermore, Dropen

can be calculated from Eq. 8.26 once ropen is known.

Therefore, both the distribution of the r’s and the pat-tern of splitting can be determined entirely within thesemiclassical formulation. The computation in principlecan be carried out to all orders of r. Interested readersmay consult Chang and Niu 1996 and Bohm et al.2003 Chap. 13 for more details.

IX. NON-ABELIAN FORMULATION

In previous sections we have considered the semiclas-sical electron dynamics with an Abelian Berry curva-ture. Such a formalism can be extended to the caseswhere the energy bands are degenerate or nearly degen-erate e.g., due to spin Culcer et al., 2005; Shindou andImura, 2005 also see Strinati 1978, Gosselin et al.2006, Gosselin et al. 2007, and Dayi 2008. Becausethe degenerate Bloch states have multiple components,the Berry curvature becomes a matrix with non-Abeliangauge structure. We report recent progress on requan-tizing the semiclassical theory that helps turning thewave-packet energy into an effective quantum Hamil-tonian Chang and Niu, 2008. After citing the dynamicsof the Dirac electron as an example, this approach isapplied to semiconductor electrons with spin degrees offreedom. Finally, we point out that the effective Hamil-tonian is only part of an effective theory, and that thevariables in the effective Hamiltonian are often gaugedependent and therefore cannot be physical variables.In order to have a complete effective theory, one alsoneeds to identify the correct physical variables relevantto experiments.

A. Non-Abelian electron wave packet

The wave packet in an energy band with D-fold de-generacy is a superposition of the Bloch states nq cf.Sec. IV,

W = n=1

D d3qaq,t&nq,tnq , 9.1

where n&nq , t2=1 at each q and aq , t is a normal-ized distribution that centers at qct. Furthermore, thewave packet is built to be localized at rc in the r space.One can first obtain an effective Lagrangian for thewave-packet variables rc, qc, and &n, then derive theirdynamical equations of motion. Without going into de-tails, we only review primary results of such an investi-gation Culcer et al., 2005.

Similar to the nondegenerate case, there are three es-sential quantities in such a formulation. In addition tothe Bloch energy E0q, there are the Berry curvatureand the magnetic moment of the wave packet see Sec.IV. However, because of the spinor degree of freedom,the latter two become vector-valued matrices instead ofthe usual vectors. The Berry connection becomes

Rmnq = iumq unq

q . 9.2

In the rest of this section, bold-faced calligraphic fontsare reserved for vector-valued matrices. Therefore, theBerry connection in Eq. 9.2 can simply be written asR.

The Berry curvature is defined as

Fq = qR − iRR . 9.3

Recall that the Berry connection and Berry curvature inthe Abelian case have the same mathematical structuresas the vector potential and the magnetic field in electro-magnetism. Here R and F also have the same structureas the gauge potential and gauge field in the non-Abelian SU2 gauge theory Wilczek and Zee, 1984.Redefining the spinor basis nq amounts to a gaugetransformation. Assuming that the new basis is obtainedfrom the old basis by a gauge transformation U, then Rand F would change in the following way:

R = URU† + iU

U†,

9.4F = UFU†,

where is the parameter of adiabatic change.The magnetic moment of the wave packet can be

found in Eq. 4.6. If the wave packet is narrowly distrib-uted around qc, then it is possible to write it as thespinor average of the following quantity Culcer et al.,2005:

Mnlqc = − ie

2 un

qc H0 − E0qc ul

qc , 9.5

where H0e−iq·rH0eiq·r. That is, M= M=†M=nl&n

Mnl&l. Except for the extension to multiple com-ponents, the form of the magnetic moment remains thesame as its Abelian counterpart see Eq. 4.6.



As a reference, we write down the equations of mo-tion for the non-Abelian wave packet Culcer et al.,2005,

kc = − eE − erc B , 9.6

rc = DDkc

,H − kc F , 9.7

i = − M · B − kc · R , 9.8

where kc=qc+ e /Arc, F= F, and the covariant de-rivative D /Dkc /kc− iR. Again the calligraphic fontsrepresent matrices. A spinor average represented bythe angular bracket is imposed on the commutator ofD /Dkc and H. The semiclassical Hamiltonian matrix in-side the commutator in Eq. 9.7 is

Hrc,kc = E0kc − erc − Mkc · B . 9.9

The spinor-averaged Hamiltonian matrix is nothing butthe wave-packet energy E= H. Like the Abelian case,it has three terms: the Bloch energy, the electrostaticenergy, and the magnetization energy.

Compared to the Abelian case in Eq. 5.8, the kcequation also has the electric force and the Lorentzforce. The rc equation is slightly more complicated: Thederivative in the group velocity E /kc is replaced bythe commutator of the covariant derivative and H. Theanomalous velocity in Eq. 9.7 remains essentially thesame. One only needs to replace the Abelian Berry cur-vature with the spinor average of the non-Abelian one.

Equation 9.8 governs the dynamics of the spinor,from which we can derive the equation for the spin vec-

tor J, where J= J and J is the spin matrix,

iJ = J,H − kc · R . 9.10

The spin dynamics in Eq. 9.10 is influenced by the Zee-man energy in H, as it should be. However, the signifi-cance of the other term that is proportional to the Berryconnection is less obvious here. Later we will see that itis in fact the spin-orbit coupling.

B. Spin Hall effect

The anomalous velocity in Eq. 9.7 that is propor-tional to the Berry curvature F is of physical signifi-cance. We have seen earlier that it is the transverse cur-rent in the quantum Hall effect and the anomalous Halleffect Sec. III. The latter requires electron spin withspin-orbit coupling and therefore the carrier dynamics issuitably described by Eqs. 9.6, 9.7, and 9.10.

For the non-Abelian case, the Berry curvature F isoften proportional to the spin S see Secs. IX.D andIX.E. If this is true, then in the presence of an electricfield the anomalous velocity is proportional to ES.That is, the trajectories of spin-up and spin-down elec-trons are parted toward opposite directions transverse tothe electric field. There can be a net transverse current if

the populations of spin-up and spin-down electrons aredifferent, as in the case of a ferromagnet. This then leadsto the anomalous Hall effect.

If the populations of different spins are equal, thenthe net electric Hall current is zero. However, the spinHall current can still be nonzero. This is the source ofthe intrinsic spin Hall effect SHE in semiconductorspredicted by Murakami et al. 2003. In the original pro-posal, a four-band Luttinger model is used to describethe heavy-hole HH bands and light-hole LH bands.The Berry curvature emerges when one restricts thewhole Hilbert space to a particular HH or LH sub-space. As shown in Sec. IX.E, such a projection of theHilbert space almost always generates a Berry curva-ture. Therefore, the SHE should be common in semi-conductors or other materials. Indeed, intrinsic SHE hasalso been theoretically predicted in metals Guo et al.,2008. The analysis of the SHE from the semiclassicalpoint of view can also be found in Culcer et al. 2005.

In addition to the Berry curvature, impurity scatteringis another source of the extrinsic SHE. This is first pre-dicted by Dyakonov and Perel 1971a, 1971b see alsoChazalviel 1975 and the same idea is later revived byHirsch 1999. Because of the spin-orbit coupling be-tween the electron and the spinless impurity, the scat-tering amplitude is not symmetric with respect to thetransverse direction. This is the same skew scattering orMott scattering in AHE see Sec. III.D.1.

To date most of the experimental evidences for theSHE belongs to the extrinsic case. They are first ob-served in semiconductors Kato et al., 2004; Sih et al.,2005; Wunderlich et al., 2005 and later in metals Valen-zuela and Tinkham, 2006; Kimura et al., 2007; Seki et al.,2008. The spin Hall conductivity in metals can be de-tected at room temperature and can be several orders ofmagnitude larger than that in semiconductors. Such alarge effect could be due to the resonant Kondo scatter-ing from the Fe impurities Guo et al., 2009. This sub-ject is currently in rapid progress. Complete understand-ing of the intrinsic or extrinsic SHE is crucial to futuredevices that could generate a significant amount of spincurrent.

C. Quantization of electron dynamics

In Sec. VII we have introduced the Bohr-Sommerfeldquantization, which helps predicting quantized energylevels. Such a procedure applies to the Abelian case andis limited to closed orbits in phase space. In this sectionwe report on the method of canonical quantization thatapplies to more general situations. With both the semi-classical theory and the method of requantization athand, one can start from a quantum theory of generalvalidity such as the Dirac theory of electrons and de-scend to an effective quantum theory with a smallerrange of validity. Such a procedure can be applied itera-tively to generate a hierarchy of effective quantum theo-ries.

As mentioned in Sec. VII.D, even though a Hamil-tonian system always admits canonical variables, it is not



always easy to find them. In the wave-packet theory, thevariables rc and kc have clear physical meaning, but theyare not canonical variables. The canonical variables rand p accurate to linear order of the fields are related tothe center-of-mass variables as follows Chang and Niu,2008:

rc = r + R + G ,9.11

kc = p + eAr + eBR ,

where =p+eAr and Ge /RB ·R /.This is a generalization of the Peierls substitution to thenon-Abelian case. The last terms in both equations canbe neglected on some occasions. For example, they willnot change the force and the velocity in Eqs. 9.6 and9.7.

When expressed in the new variables, the semiclassi-cal Hamiltonian in Eq. 9.9 can be written as

Hr,p = E0 − er + eE · R

− B · M − eRE0

, 9.12

where we have used the Taylor expansion and neglectedterms nonlinear in fields. Finally, one promotes the ca-nonical variables to quantum conjugate variables andconverts H to an effective quantum Hamiltonian.

Compared to the semiclassical Hamiltonian in Eq.9.9, the quantum Hamiltonian has two additional termsfrom the Taylor expansion. The dipole-energy termeE ·R is originated from the shift between the chargecenter rc and the center of the canonical variable r. Al-though the exact form of the Berry connection R de-pends on the physical model, we show that for both theDirac electron Sec. IX.D and the semiconductor elec-tron Sec. IX.E the dipole term is closely related to thespin-orbit coupling. The correction to the Zeeman en-ergy is sometimes called the Yafet term, which vanishesnear a band edge Yafet, 1963.

Three remarks are in order. First, the form of theHamiltonian, especially the spin-orbit term and Yafetterm, is clearly gauge dependent because of the gauge-dependent Berry connection. Such gauge dependencehas prevented one from assigning a clear physical signifi-cance to the Yafet term. For that matter, it is also doubt-ful that the electric dipole, or the spin-orbit energy, canbe measured independently. Second, in a neighborhoodof a k point, one can always choose to work within aparticular gauge. However, if the first Chern number orits non-Abelian generalization is not zero, one cannotchoose a global gauge in which R is smooth everywherein the Brillouin zone. In such a nontrivial topologicalsituation one has to work with patches of the Brillouinzone for a single canonical quantum theory. Third, thesemiclassical theory based on the variables F and M,on the other hand, is gauge independent. Therefore, theeffective quantum theory can be smooth globally.

D. Dirac electron

To illustrate the application of the non-Abelian wave-packet theory and its requantization, we use the Diracelectron as an example. The starting quantum Hamil-tonian is

H = c · p + eA + mc2 − er

= H0 + ce · A − er , 9.13

where and are the Dirac matrices Strange, 1998and H0 is the free-particle Hamiltonian. The energyspectrum of H0 has positive-energy branch and negative-energy branch, each with twofold degeneracy due to thespin. These two branches are separated by a large en-ergy gap mc2. One can construct a wave packet out ofthe positive-energy eigenstates and study its dynamicsunder the influence of an external field. The size of thewave packet can be as small as the Compton wavelengthc= /mc but not smaller, which is two orders of mag-nitude smaller than the Bohr radius. Therefore, theadiabatic condition on the external electromagnetic fieldcan be easily satisfied: the spatial variation in the poten-tial only needs to be much smoother than c. In thiscase, even the lattice potential in a solid can be consid-ered as a semiclassical perturbation. Furthermore, be-cause of the large gap between branches, interbranchtunneling happens and the semiclassical theory failsonly if the field is so strong that electron-positron pairproduction can no longer be ignored.

Since the wave packet is living on a branch with two-fold degeneracy, the Berry connection and curvature are22 matrices Chang and Niu, 2008,

Rq =c

2

2 + 1q , 9.14

Fq = −c

2

23 + c2 q ·

+ 1q , 9.15

where q1+ q /mc2 is the relativistic dilation fac-tor. To calculate these quantities, we only need the freeparticle eigenstates of H0 see Eqs. 9.2 and 9.3. Thatis, the nontrivial gauge structure exists in the free par-ticle already.

It may come as a surprise that the free wave packetalso possesses an intrinsic magnetic moment. Straight-forward application of Eq. 9.5 gives Chuu et al., 2010

Mq = −e

2m2 + c2 q ·

+ 1q . 9.16

This result agrees with the one calculated from the ab-

stract spin operator S in the Dirac theory Chuu et al.,2010,

Mq = − ge

2mqWSW , 9.17

in which the g factor is 2. The Zeeman coupling in thewave-packet energy is −M ·B. Therefore, this magneticmoment gives the correct magnitude of the Zeeman en-



ergy with the correct g factor. Recall that Eq. 9.5 isoriginated from Eq. 4.6, which is the magnetic momentdue to a circulating charge current. Therefore, the mag-netic moment here indeed is a result of the spinningwave packet.

The present approach is a revival of Uhlenbeck andGoudsmit’s rotating sphere model of the electron spinbut without its problem. The size of the wave packet cconstructed from the positive-energy states is two ordersof magnitude larger than the classical electron radiuse2 /mc2. Therefore, the wave packet does not have torotate faster than the speed of light to have the correctmagnitude of spin. This semiclassical model for spin ispleasing since it gives a clear and heuristic picture of theelectron spin. Also, one does not have to resort to themore complicated Foldy-Wouthuysen approach to ex-tract the spin from the Dirac Hamiltonian Foldy andWouthuysen, 1950.

From the equation of motion in Eq. 9.10, one ob-tains

=e

mB + E

kc

+ 1mc2 . 9.18

This is the Bargmann-Michel-Telegdi equation for arelativistic electron Bargmann et al., 1959. More dis-cussions on the equations of motion for rc and kc can befound in Chang and Niu 2008.

Finally, substituting the Berry connection and themagnetic moment into Eq. 9.12 and using E0=c22+m2c4, one can obtain the effective quantumHamiltonian,

Hr,p = mc2 − er +B

+ 1

mc2 · E

+B

· B , 9.19

in which all ’s are functions of and B=e /2m. Thisis the relativistic Pauli Hamiltonian. At low velocity, 1, and it reduces to the more familiar form. Noticethat the spin-orbit coupling comes from the dipole en-ergy term with the Berry connection, as mentioned ear-lier also see Mathur 1991 and Shankar and Mathur1994.

E. Semiconductor electron

When studying the transport properties of semicon-ductors, one often only focuses on the carriers near thefundamental gap at the point. In this case, the bandstructure far away from this region is not essential. It isthen a good approximation to use the k ·p expansion andobtain the four-band Luttinger model or the eight-bandKane model Luttinger, 1951; Kane, 1957; Winkler, 2003to replace the more detailed band structure see Fig. 15.In this section, we start from the eight-band Kane modeland study the wave-packet dynamics in one of its sub-space: the conduction band. It is also possible to inves-tigate the wave-packet dynamics in other subspaces: the

HH-LH complex or the spin-orbit split-off band. Theresult of the latter is not reported in this review. Inter-ested readers can consult Chang and Niu 2008 formore details, including the explicit form of the KaneHamiltonian that the calculations are based upon.

To calculate the Berry connection in Eq. 9.2, oneneeds to obtain the eigenstates of the Kane model,which have eight components. Similar to the positive-energy branch of the Dirac electron, the conductionband is twofold degenerate. Detailed calculation showsthat, to linear order in k and up to a SU2 gauge rota-tion, the Berry connection is a 22 matrix of the form

R =V2

3 1

Eg2 −

1

Eg + 2 k , 9.20

where Eg is the fundamental gap, is the spin-orbit spit-off gap, and V= /m0SpxX is a matrix element ofthe momentum operator.

As a result, the dipole term eE ·R becomes

HSO = eE · R = E · k , 9.21

where eV2 /31/Eg2−1/ Eg+2. The coefficient

and the form of the spin-orbit coupling are the same asthe Rashba coupling Rashba, 1960; Bychkov andRashba, 1984. However, unlike the usual Rashba cou-pling that requires structural inversion asymmetry togenerate an internal field, this term exists in a bulk semi-conductor with inversion symmetry but requires an ex-ternal field E.

From the Berry connection, we can calculate theBerry curvature in Eq. 9.3 to the leading order of k as

F =2V2

3 1

Eg2 −

1

Eg + 2 . 9.22

In the presence of an electric field, this would generatethe transverse velocity in Eq. 9.7,

vT = 2eE . 9.23

4-bandLuttingermodel

8-bandKanemodel

E(k)

Eg

∆ HH

LH

SO

CB

k

FIG. 15. Color online Schematic of the semiconductor bandstructure near the fundamental gap. The wave packet in theconduction band is formed from a two-component spinor.



As a result, spin-up and spin-down electrons move to-ward opposite directions, which results in a spin-Halleffect see Sec. IX.B for related discussion.

The wave packet in the conduction band also sponta-neously rotates with respect to its own center of mass.To the lowest order of k, it has the magnetic moment,

M =eV2

3 1

Eg−

1

Eg + . 9.24

With these three basic quantities, R, F, and M, therequantized Hamiltonian in Eq. 9.12 can be establishedas

Hr,p = E0 − er + E · + gBB ·

2,

9.25

where E0 includes the Zeeman energy from the barespin and

g = −43

mV2

2 1

Eg−

1

Eg + . 9.26

In most textbooks on solid-state physics, one can findthis correction of the g factor. However, a clear identifi-cation with electron’s angular momentum is often lack-ing. In the wave-packet formulation, we see that g isindeed originated from the electron’s spinning motion.

F. Incompleteness of effective Hamiltonian

Once the effective Hamiltonian Hr ,p is obtained,one can go on to study its spectra and states, withoutreferring back to the original Hamiltonian. Based on thespectra and states, any physics observables of interestcan be calculated. These physics variables may be posi-tion, momentum, or other related quantities. Neverthe-less, we emphasize that the canonical variables in theeffective Hamiltonian may not be physical observables.They may differ, for example, by a Berry connection inthe case of the position variable. The effective Hamil-tonian itself is not enough for correct prediction if thephysical variables have not been identified properly.

This is best illustrated using the Dirac electron as anexample. At low velocity, the effective Pauli Hamil-tonian is see Eq. 9.19

Hr,p =2

2m− er +

B

2

mc2 · E + B · B ,

9.27

which is a starting point of many solid-state calculations.It is considered accurate for most of the low-energy ap-plications in solid state. When one applies an electricfield, then according to the Heisenberg equation of mo-tion, the velocity of the electron is

r =

m+

ec2

4 E , 9.28

where c is the Compton wavelength.

If one calculates the velocity of a Dirac electron ac-cording to Eq. 9.7, then the result is

rc =km

+ec

2

2 E . 9.29

That is, the transverse velocity is larger by a factor of 2.The source of this discrepancy can be traced back to thedifference between the two position variables: rc and rsee Eq. 9.11. One should regard the equation for rc asthe correct one since it is based on the Dirac theory seealso Bliokh 2005.

Such a discrepancy between the same physical vari-able in different theories can also be understood fromthe perspective of the Foldy-Wouthuysen transforma-tion. The Pauli Hamiltonian can also be obtained fromblock diagonalizing the Dirac Hamiltonian using an uni-tary transformation. Since the basis of states has beenrotated, the explicit representations of all observablesshould be changed as well. For example, rc in Eq. 9.11can be obtained by a Foldy-Wouthuysen rotation, fol-lowed by a projection to the positive-energy subspaceFoldy and Wouthuysen, 1950.

G. Hierarchy structure of effective theories

Finally, we report on the hierarchical relations for thebasic quantities, the Berry curvature F and the mag-netic moment M. We consider theories on three differ-ent levels of hierarchy I, II, and III with progressivelysmaller and smaller Hilbert spaces. These spaces will becalled the full space, the parent space, and the wave-packet space, respectively see Fig. 16.

Alternative to Eqs. 9.3 and 9.5, the Berry curvatureand the magnetic moment can be written in the follow-ing forms Chang and Niu, 2008:

Fmn = i lout

RmlRln, 9.30

Mmn =ie

2 lout

E0,m − E0,lRmlRln, 9.31

where Rml is the Berry connection and l sums over thestates outside of the space of interest. From Eqs. 9.30

parentspace (II)

fullspace (I)

wavepacketspace (III)

FIG. 16. The extent of wave-packet space, parent space, andfull space.



and 9.31, one sees that the Berry curvature and themagnetic moment for theory I are 0 since there is nostate outside the full space. With the help of the states inthe full space, one can calculate the Berry curvaturesand the magnetic moment in theory II and theory III.They are designated as Fp ,Mp and F ,M, respec-tively. These two sets of matrices have different rankssince the parent space and the wave-packet space havedifferent dimensions.

If one starts from the parent space, on the other hand,then the Berry curvature and the magnetic moment fortheory II is 0 instead of Fp and Mp. The Berry cur-vature and the magnetic moment for theory III are nowdesignated as F and M. They are different from Fand M since the former are obtained from the summa-tions with more outside states from the full space. It isstraightforward to see from Eqs. 9.30 and 9.31 that

F = F + PFpP, M = M + PMpP , 9.32

where P is a dimension-reduction projection from theparent space to the wave-packet subspace. This meansthat starting from theory II, instead of theory I, as theparent theory one would have the errors PFpP andPMpP. On the other hand, however, whenever thescope of the parent theory needs to be extended, e.g.,from II to I, instead of starting all of the calculationsanew, one only needs additional input from Fp and Mpand the accuracy can be improved easily.

For example, in the original proposal of Murakami etal. 2003, 2004 of the spin Hall effect of holes, the par-ent space is the HH-LH complex. The heavy hole or thelight hole acquires a nonzero Berry curvature as a resultof the projection from this parent space to the HH bandor the LH band. This Berry curvature corresponds tothe F above. It gives rise to a spin-dependent trans-verse velocity eEF that is crucial to the spin Halleffect.

Instead of the HH-LH complex, if one chooses theeight bands in Fig. 15 as the full space, then the Berrycurvatures of the heavy hole and the light hole will getnew contributions from PFpP. The projection from thefull space with eight bands to the HH-LH complexof four bands generates a Berry curvature Fp

=−2V2 /3Eg2J Chang and Niu, 2008, where J is the

spin-3 /2 matrix. Therefore, after further projections, wewould get additional anomalous velocities eV2 /Eg

2E and eV2 /3Eg

2E for HH and LH, respectively.

X. OUTLOOK

In most of the researches mentioned in this review,the Berry phase and semiclassical theory are explored inthe single-particle context. The fact that they are so use-ful and that in some of the materials the many-bodyeffect is crucial naturally motivates one to extend thisapproach to many-body regime. For example, the Berryphase effect has been explored in the density functionaltheory with spin degrees of freedom Niu and Kleinman,1998; Niu et al., 1999. Also, the Berry phase and rel-

evant quantities are investigated in the context of Fermi-liquid theory Haldane, 2004. Furthermore, the Berrycurvature on the Fermi surface, if strong enough, is pre-dicted to modify a repulsive interaction between elec-trons to an attractive interaction and causes pairing in-stability Shi and Niu, 2006. In addition to the artificialmagnetic field generated by the monopole of Berry cur-vature, a slightly different Berry curvature involving thetime component is predicted to generate an artificialelectric field, which would affect the normalization fac-tor and the transverse conductivity Shindou andBalents, 2006. This latter work has henceforth beengeneralized to multiple-band Fermi liquid with non-Abelian Berry phase Shindou and Balents, 2008. Re-cently, the ferrotoroidic moment in multiferroic materi-als is also found to be a quantum geometric phaseBatista et al., 2008. Research along such a path is ex-citing and still at its early stage.

There has been a growing amount of research on theBerry phase effect in light-matter interaction. The Berrycurvature is responsible for a transverse shift side jumpof a light beam reflecting off an interface Onoda et al.,2004a; Sawada and Nagaosa, 2005; Onoda, Murakami,and Nagaosa, 2006. The shift is of the order of thewavelength and is a result of the conservation of angularmomentum. The direction of the shift depends on thecircular polarization of the incident beam. This “opticalHall effect” can be seen as a rediscovery of the Imbert-Federov effect Federov, 1955; Imbert, 1972. More de-tailed study of the optical transport involving spin hasalso been carried out by Bliokh and others Bliokh,2006a; Bliokh and Bliokh, 2006; Duval et al., 2006a. Thesimilarity between the side jump of a light beam andanalogous “jump” of an electron scattering off an impu-rity has been noticed quite early in Berger and Berg-mann’s review on anomalous Hall effect Chien andWestgate, 1980. In fact, the side jump of the electro-magnetic wave and the electron can be unified usingsimilar dynamical equations. This shows that the equa-tion of motion approach in this review has general va-lidity. Indeed, a similar approach has also been extendedto the quasiparticle dynamics in Bose-Einstein conden-sate Zhang et al., 2006.

Even though the Berry curvature plays a crucial rolein the electronic structure and electron dynamics of crys-tals, direct measurement of such a quantity is still lack-ing. There does exist sporadic and indirect evidence ofthe effect of the Berry phase or the Berry curvaturethrough the measurement of, for example, the quantumHall conductance, the anomalous Hall effect, or the Hallplateau in graphene. However, this is just a beginning. Inthis review, one can see that in many circumstances theBerry curvature should be as important as the Blochenergy. Condensed-matter physicists over the years havecompiled a large database on the band structures andFermi surfaces of all kinds of materials. It is about timeto add theoretical and experimental results of the Berrycurvature that will deepen our understanding of materialproperties. There is still plenty of room in the quasimo-mentum space.



ACKNOWLEDGMENTS

D.X. was supported by the Division of Materials Sci-ence and Engineering, Office of Basic Energy Sciences,Department of Energy. M.-C.C. was supported by theNSC of Taiwan. Q.N. acknowledges the support fromNSF, DOE, the Welch Foundation F-1255, and theTexas Advanced Research Program.

APPENDIX: ADIABATIC EVOLUTION

Suppose the Hamiltonian HRt depends on a set ofparameters Rt. Consider an adiabatic process in whichRt changes slowly in time. The wave function tmust satisfy the time-dependent Schrödinger equation

i

tt = Htt . A1

If we expand the wave function using the instantaneouseigenstates nt of Ht as

t = n

exp−i

t0

t

dtEntantnt , A2

then the coefficients satisfy

ant = − l

altnt

tlt

exp−i

t0

t

dtElt − Ent . A3

So far we have not made any choice regarding the phaseof the instantaneous eigenstates. For our purpose here,it is convenient to impose the condition of parallel trans-port,

nt

tnt = Rtnt

Rnt = 0. A4

We denote the wave function chosen this way as n. Wenote that this condition is different from the single-valued phase choice we used in the main text: For aclosed path in the parameter space, i.e., Rtf=Rt0,there is no guarantee that the phase at the final time tf isthe same as the phase at the beginning t0. In otherwords, under the parallel transport condition, eventhough nt is uniquely determined as a function of t, itcan still be a multivalued function of R. The phase dif-ference n in ntf=einnt0 is precisely the Berryphase of the closed path. In fact, this can be consideredas another definition of the Berry phase.

In the limit of R→0, we have, to zeroth order,

ant = 0. A5

Therefore if the system is initially in the nth eigenstate,it will stay in that state afterwards. This is the quantumadiabatic theorem. Now consider the first-order correc-tion. We have an0=1 and an0=0 for nn. For the

nth state, we still have an=0, therefore an=1. However,for nn,

tan = − n

tnexp−

i

t0

t

dtEnt − Ent .

A6

Then, since the exponential factor oscillates while its co-efficient slowly varies in time, we can integrate theabove equation by parts, yielding

an = −n/tnEn − En

i

exp−i

t0

t

dtEnt − Ent . A7

The wave function including the first-order approxima-tion is given by

t = exp−i

t0

t

dtEntn − i

nn

nn/tnEn − En

. A8

When applying the above result, as long as the expres-sions involving n are gauge independent, we can alwaysreplace n with eigenstates under another phase choice.

Finally, we mentioned that if the phase of the eigen-states is required to be single valued as a function of R,Eq. A8 can still be obtained with an extra overall phaseein, where

n = it0

t

dtn„Rt…

tn„Rt… . A9

If the path Rt is closed, n gives the Berry phase.

REFERENCES

Adams, E. N., 1952, Phys. Rev. 85, 41.Adams, E. N., and E. I. Blount, 1959, J. Phys. Chem. Solids 10,

286.Aharonov, Y., and D. Bohm, 1959, Phys. Rev. 115, 485.Akhmerov, A. R., and C. W. J. Beenakker, 2007, Phys. Rev.

Lett. 98, 157003.Arnold, V. I., 1978, Mathematical Methods of Classical Me-

chanics Springer, New York.Arnold, V. I., 1989, Mathematical Methods of Classical Me-

chanics, 2nd ed. Springer-Verlag, New York.Ashcroft, N. W., and N. D. Mermin, 1976, Solid State Physics

Saunders, Philadelphia.Avron, J. E., 1982, Ann. Phys. Paris 143, 33.Avron, J. E., A. Elgart, G. M. Graf, and L. Sadun, 2001, Phys.

Rev. Lett. 87, 236601.Avron, J. E., E. Elgart, G. M. Graf, and L. Sadun, 2004, J. Stat.

Phys. 116, 425.Avron, J. E., and R. Seiler, 1985, Phys. Rev. Lett. 54, 259.Avron, J. E., R. Seiler, and B. Simon, 1983, Phys. Rev. Lett. 51,

51.Azbel, M. Y., 1964, Sov. Phys. JETP 19, 634.



http://dx.doi.org/10.1103/PhysRev.85.41

http://dx.doi.org/10.1016/0022-3697(59)90004-6

http://dx.doi.org/10.1016/0022-3697(59)90004-6


http://dx.doi.org/10.1103/PhysRevLett.98.157003


http://dx.doi.org/10.1016/0003-4916(82)90213-5



http://dx.doi.org/10.1023/B:JOSS.0000037245.45780.e1

http://dx.doi.org/10.1023/B:JOSS.0000037245.45780.e1




Bargmann, V., L. Michel, and V. L. Telegdi, 1959, Phys. Rev.Lett. 2, 435.

Batista, C. D., G. Ortiz, and A. A. Aligia, 2008, Phys. Rev.Lett. 101, 077203.

Ben Dahan, M., E. Peik, J. Reichel, Y. Castin, and C. Salomon,1996, Phys. Rev. Lett. 76, 4508.

Berger, L., 1970, Phys. Rev. B 2, 4559.Berger, L., 1972, Phys. Rev. B 5, 1862.Berry, M. V., 1984, Proc. R. Soc. London, Ser. A 392, 45.Bird, D. M., and A. R. Preston, 1988, Phys. Rev. Lett. 61, 2863.Bliokh, K. Y., 2005, Europhys. Lett. 72, 7.Bliokh, K. Y., 2006a, Phys. Rev. Lett. 97, 043901.Bliokh, K. Y., 2006b, Phys. Lett. A 351, 123.Bliokh, K. Y., and Y. P. Bliokh, 2006, Phys. Rev. Lett. 96,

073903.Bliokh, K. Y., A. Niv, V. Kleiner, and E. Hasman, 2008, Nat.

Photonics 2, 748.Blount, E. I., 1962a, Phys. Rev. 126, 1636.Blount, E. I., 1962b, Solid State Physics Academic, New

York, Vol. 13.Bohm, A., A. Mostafazadeh, H. Koizumi, Q. Niu, and J. Zwan-

ziger, 2003, The Geometric Phase in Quantum Systems: Foun-dations, Mathematical Concepts, and Applications in Molecu-lar and Condensed Matter Physics Springer-Verlag, Berlin.

Brouwer, P. W., 1998, Phys. Rev. B 58, R10135.Brown, E., 1968, Phys. Rev. 166, 626.Bruno, P., V. K. Dugaev, and M. Taillefumier, 2004, Phys. Rev.

Lett. 93, 096806.Bychkov, Y., and E. Rashba, 1984, JETP Lett. 39, 78.Ceresoli, D., T. Thonhauser, D. Vanderbilt, and R. Resta, 2006,

Phys. Rev. B 74, 024408.Chambers, W., 1965, Phys. Rev. 140, A135.Chang, M.-C., and Q. Niu, 1995, Phys. Rev. Lett. 75, 1348.Chang, M.-C., and Q. Niu, 1996, Phys. Rev. B 53, 7010.Chang, M.-C., and Q. Niu, 2008, J. Phys.: Condens. Matter 20,

193202.Chazalviel, J.-N., 1975, Phys. Rev. B 11, 3918.Cheong, S.-W., and M. Mostovoy, 2007, Nature Mater. 6, 13.Chien, C. L., and C. R. Westgate, 1980, Eds., The Hall Effect

and Its Application Plenum, New York.Chun, S. H., Y. S. Kim, H. K. Choi, I. T. Jeong, W. O. Lee, K.

S. Suh, Y. S. Oh, K. H. Kim, Z. G. Khim, J. C. Woo, and Y. D.Park, 2007, Phys. Rev. Lett. 98, 026601.

Chuu, C., M.-C. Chang, and Q. Niu, 2010, Solid State Com-mun. 150, 533.

Cooper, N. R., B. I. Halperin, and I. M. Ruzin, 1997, Phys. Rev.B 55, 2344.

Culcer, D., A. MacDonald, and Q. Niu, 2003, Phys. Rev. B 68,045327.

Culcer, D., J. Sinova, N. A. Sinitsyn, T. Jungwirth, A. H. Mac-Donald, and Q. Niu, 2004, Phys. Rev. Lett. 93, 046602.

Culcer, D., Y. Yao, and Q. Niu, 2005, Phys. Rev. B 72, 085110.Dana, I., Y. Avron, and J. Zak, 1985, J. Phys. C 18, L679.Dayi, O. F., 2008, J. Phys. A: Math. Theor. 41, 315204.Dirac, P. A. M., 1931, Proc. R. Soc. London, Ser. A 133, 60.Dugaev, V. K., P. Bruno, M. Taillefumier, B. Canals, and C.

Lacroix, 2005, Phys. Rev. B 71, 224423.Duval, C., Z. Horváth, P. A. Horváthy, L. Martina, and P. C.

Stichel, 2006a, Mod. Phys. Lett. B 20, 373.Duval, C., Z. Horváth, P. A. Horváthy, L. Martina, and P. C.

Stichel, 2006b, Phys. Rev. Lett. 96, 099701.Dyakonov, M. I., and V. I. Perel, 1971a, JETP Lett. 13, 467.Dyakonov, M. I., and V. I. Perel, 1971b, Phys. Lett. 35A, 459.

Essin, A. M., J. E. Moore, and D. Vanderbilt, 2009, Phys. Rev.Lett. 102, 146805.

Fang, Z., N. Nagaosa, K. S. Takahashi, A. Asamitsu, R.Mathieu, T. Ogasawara, H. Yamada, M. Kawasaki, Y. Tokura,and K. Terakura, 2003, Science 302, 92.

Federov, F. I., 1955, Dokl. Akad. Nauk SSSR 105, 465.Fiebig, M., R. Lottermoser, D. Frohlich, A. V. Goltsev, and R.

V. Pisarev, 2002, Nature London 419, 818.Fock, V., 1928, Z. Phys. 49, 323.Foldy, L. L., and S. A. Wouthuysen, 1950, Phys. Rev. 78, 29.Friedrich, H., and J. Trost, 1996, Phys. Rev. Lett. 76, 4869.Gat, O., and J. E. Avron, 2003a, New J. Phys. 5, 44.Gat, O., and J. E. Avron, 2003b, Phys. Rev. Lett. 91, 186801.Glück, M., A. R. Kolovsky, and H. J. Korsch, 1999, Phys. Rev.

Lett. 83, 891.Goodrich, R. G., D. L. Maslov, A. F. Hebard, J. L. Sarrao, D.

Hall, and Z. Fisk, 2002, Phys. Rev. Lett. 89, 026401.Gorbachev, R. V., F. V. Tikhonenko, A. S. Mayorov, D. W.

Horsell, and A. K. Savchenko, 2007, Phys. Rev. Lett. 98,176805.

Gosselin, P., A. Bérard, and H. J. Mohrbach, 2007, Eur. Phys.J. B 58, 137.

Gosselin, P., F. Ménas, A. Bérard, and H. Mohrbach, 2006,Europhys. Lett. 76, 651.

Gunawan, O., Y. P. Shkolnikov, K. Vakili, T. Gokmen, E. P. D.Poortere, and M. Shayegan, 2006, Phys. Rev. Lett. 97, 186404.

Guo, G.-Y., S. Maekawa, and N. Nagaosa, 2009, Phys. Rev.Lett. 102, 036401.

Guo, G. Y., S. Murakami, T.-W. Chen, and N. Nagaosa, 2008,Phys. Rev. Lett. 100, 096401.

Hagedorn, G. A., 1980, Commun. Math. Phys. 71, 77.Haldane, F. D. M., 1988, Phys. Rev. Lett. 61, 2015.Haldane, F. D. M., 2004, Phys. Rev. Lett. 93, 206602.Halperin, B. I., 1982, Phys. Rev. B 25, 2185.Hanasaki, N., K. Sano, Y. Onose, T. Ohtsuka, S. Iguchi, I.

Kézsmárki, S. Miyasaka, S. Onoda, N. Nagaosa, and Y.Tokura, 2008, Phys. Rev. Lett. 100, 106601.

Hatsugai, Y., and M. Kohmoto, 1990, Phys. Rev. B 42, 8282.Hirsch, J. E., 1999, Phys. Rev. Lett. 83, 1834.Hirst, L. L., 1997, Rev. Mod. Phys. 69, 607.Hofstadter, D. R., 1976, Phys. Rev. B 14, 2239.Hur, N., S. Park, P. A. Sharma, J. S. Ahn, S. Guha, and S.-W.

Cheong, 2004, Nature London 429, 392.Imbert, C., 1972, Phys. Rev. D 5, 787.Inoue, J.-I., T. Kato, Y. Ishikawa, H. Itoh, G. E. W. Bauer, and

L. W. Molenkamp, 2006, Phys. Rev. Lett. 97, 046604.Jackson, J. D., 1998, Classical Electrodynamics, 3rd ed. Wiley,

New York.Jungwirth, T., Q. Niu, and A. H. MacDonald, 2002, Phys. Rev.

Lett. 88, 207208.Kane, E. O., 1957, J. Phys. Chem. Solids 1, 249.Karplus, R., and J. M. Luttinger, 1954, Phys. Rev. 95, 1154.Kato, T., 1950, J. Phys. Soc. Jpn. 5, 435.Kato, T., Y. Ishikawa, H. Itoh, and J. Inoue, 2007, New J. Phys.

9, 350.Kato, Y. K., R. C. Myers, A. C. Gossard, and D. D. Awscha-

lom, 2004, Science 306, 1910.Khinchin, A. Y., 1964, Continued Fractions The University of

Chicago Press, Chicago.Kimura, T., T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y.

Tokura, 2003, Nature London 426, 55.Kimura, T., Y. Otani, T. Sato, S. Takahashi, and S. Maekawa,

2007, Phys. Rev. Lett. 98, 156601.








http://dx.doi.org/10.1103/PhysRevB.2.4559


http://dx.doi.org/10.1098/rspa.1984.0023


http://dx.doi.org/10.1209/epl/i2005-10205-1


http://dx.doi.org/10.1016/j.physleta.2005.10.087



http://dx.doi.org/10.1038/nphoton.2008.229

http://dx.doi.org/10.1038/nphoton.2008.229


http://dx.doi.org/10.1103/PhysRevB.58.R10135





http://dx.doi.org/10.1103/PhysRev.140.A135



http://dx.doi.org/10.1088/0953-8984/20/19/193202

http://dx.doi.org/10.1088/0953-8984/20/19/193202


http://dx.doi.org/10.1038/nmat1804


http://dx.doi.org/10.1016/j.ssc.2009.10.039

http://dx.doi.org/10.1016/j.ssc.2009.10.039







http://dx.doi.org/10.1088/0022-3719/18/22/004

http://dx.doi.org/10.1088/1751-8113/41/31/315204



http://dx.doi.org/10.1142/S0217984906010573


http://dx.doi.org/10.1016/0375-9601(71)90196-4



http://dx.doi.org/10.1126/science.1089408

http://dx.doi.org/10.1038/nature01077

http://dx.doi.org/10.1007/BF01337922



http://dx.doi.org/10.1088/1367-2630/5/1/344







http://dx.doi.org/10.1140/epjb/e2007-00212-6

http://dx.doi.org/10.1140/epjb/e2007-00212-6

http://dx.doi.org/10.1209/epl/i2006-10321-4





http://dx.doi.org/10.1007/BF01230088










http://dx.doi.org/10.1103/PhysRevD.5.787




http://dx.doi.org/10.1016/0022-3697(57)90013-6


http://dx.doi.org/10.1143/JPSJ.5.435

http://dx.doi.org/10.1088/1367-2630/9/9/350

http://dx.doi.org/10.1088/1367-2630/9/9/350




King-Smith, R. D., and D. Vanderbilt, 1993, Phys. Rev. B 47,1651.

Klitzing, K. v., G. Dorda, and M. Pepper, 1980, Phys. Rev. Lett.45, 494.

Kohmoto, M., 1985, Ann. Phys. Paris 160, 343.Kohmoto, M., 1989, Phys. Rev. B 39, 11943.Kohn, W., 1959, Phys. Rev. 115, 809.Kohn, W., 1964, Phys. Rev. 133, A171.Kohn, W., and J. M. Luttinger, 1957, Phys. Rev. 108, 590.Kotiuga, P. R., 1989, IEEE Trans. Magn. 25, 2813.Kramer, P., and M. Saraceno, 1981, Geometry of the Time-

Dependent Variational Principle in Quantum MechanicsSpringer-Verlag, Berlin, Vol. 140.

Kunz, H., 1986, Phys. Rev. Lett. 57, 1095.Kuratsuji, H., and S. Iida, 1985, Prog. Theor. Phys. 74, 439.Laughlin, R. B., 1981, Phys. Rev. B 23, 5632.Lee, W.-L., S. Watauchi, V. L. Miller, R. J. Cava, and N. P. Ong,

2004a, Phys. Rev. Lett. 93, 226601.Lee, W.-L., S. Watauchi, V. L. Miller, R. J. Cava, and N. P. Ong,

2004b, Science 303, 1647.Lifshitz, E. M., and L. D. Landau, 1980, Statistical Physics: Part

2 (Course of Theoretical Physics) Butterworth-Heinemann,Washington, D.C., Vol. 9.

Littlejohn, R. G., and W. G. Flynn, 1991, Phys. Rev. A 44,5239.

Liu, C.-X., X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, 2008,Phys. Rev. Lett. 101, 146802.

Luttinger, J. M., 1951, Phys. Rev. 84, 814.Luttinger, J. M., 1958, Phys. Rev. 112, 739.Marder, M. P., 2000, Condensed-Matter Physics Wiley, New

York.Martin, R. M., 1972, Phys. Rev. B 5, 1607.Martin, R. M., 1974, Phys. Rev. B 9, 1998.Marzari, N., and D. Vanderbilt, 1997, Phys. Rev. B 56, 12847.Mathieu, R., A. Asamitsu, H. Yamada, K. S. Takahashi, M.

Kawasaki, Z. Fang, N. Nagaosa, and Y. Tokura, 2004, Phys.Rev. Lett. 93, 016602.

Mathur, H., 1991, Phys. Rev. Lett. 67, 3325.McCann, E., and V. I. Fal’ko, 2006, Phys. Rev. Lett. 96, 086805.Mendez, E., and G. Bastard, 1993, Phys. Today 46 6, 34.Messiah, A., 1962, Quantum Mechanics North Holland, Am-

sterdam, Vol. II.Mikitik, G. P., and Y. V. Sharlai, 1999, Phys. Rev. Lett. 82, 2147.Mikitik, G. P., and Y. V. Sharlai, 2004, Phys. Rev. Lett. 93,

106403.Mikitik, G. P., and Y. V. Sharlai, 2007, Low Temp. Phys. 33,

439.Min, H., B. Sahu, S. K. Banerjee, and A. H. MacDonald, 2007,

Phys. Rev. B 75, 155115.Miyasato, T., N. Abe, T. Fujii, A. Asamitsu, S. Onoda, Y.

Onose, N. Nagaosa, and Y. Tokura, 2007, Phys. Rev. Lett. 99,086602.

Morandi, G., 1988, Quantum Hall Effect: Topological Prob-lems in Condensed-Matter Physics Bibliopolis, Naples.

Morozov, S. V., K. S. Novoselov, M. I. Katsnelson, F. Schedin,L. A. Ponomarenko, D. Jiang, and A. K. Geim, 2006, Phys.Rev. Lett. 97, 016801.

Morpurgo A. F., and F. Guinea, 2006, Phys. Rev. Lett. 97,196804.

Mucciolo E. R., C. Chamon, and C. M. Marcus, 2002, Phys.Rev. Lett. 89, 146802.

Murakami S., N. Nagaosa, and S.-C. Zhang, 2003, Science 301,1348.

Murakami S., N. Nagosa, and S.-C. Zhang, 2004, Phys. Rev. B69, 235206.

Nagaosa, N., J. Sinova, S. Onoda, A. H. MacDonald, and N. P.Ong, 2010, Rev. Mod. Phys. 82, 1539.

Nayak, C., S. H. Simon, A. Stern, M. Freedman, and S. D.Sarma, 2008, Rev. Mod. Phys. 80, 1083.

Nenciu, G., 1991, Rev. Mod. Phys. 63, 91.Niu, Q., 1990, Phys. Rev. Lett. 64, 1812.Niu, Q., and L. Kleinman, 1998, Phys. Rev. Lett. 80, 2205.Niu, Q., and G. Sundaram, 2001, Physica E Amsterdam 9,

327.Niu, Q., and D. J. Thouless, 1984, J. Phys. A 17, 2453.Niu, Q., D. J. Thouless, and Y.-S. Wu, 1985, Phys. Rev. B 31,

3372.Niu, Q., X. Wang, L. Kleinman, W.-M. Liu, D. M. C. Nichol-

son, and G. M. Stocks, 1999, Phys. Rev. Lett. 83, 207.Novoselov, K. S., A. K. Geim, S. V. Morozov, D. Jiang, M. I.

Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov,2005, Nature London 438, 197.

Novoselov, K. S., E. McCann, S. V. Morozov, V. I. Fal’ko, M. I.Katsnelson, U. Zeitler, D. Jiang, F. Schedin, and A. K. Geim,2006, Nat. Phys. 2, 177.

Nozières, P., and C. Lewiner, 1973, J. Phys. France 34, 901.Nunes, R. W., and X. Gonze, 2001, Phys. Rev. B 63, 155107.Nunes, R. W., and D. Vanderbilt, 1994, Phys. Rev. Lett. 73, 712.Obermair, G., and G. Wannier, 1976, Phys. Status Solidi B 76,

217.Ohta, T., A. Bostwick, T. Seyller, K. Horn, and E. Rotenberg,

2006, Science 313, 951.Olson, J. C., and P. Ao, 2007, Phys. Rev. B 75, 035114.Onoda, M., S. Murakami, and N. Nagaosa, 2004a, Phys. Rev.

Lett. 93, 083901.Onoda, M., S. Murakami, and N. Nagaosa, 2006, Phys. Rev. E

74, 066610.Onoda, M., and N. Nagaosa, 2002, J. Phys. Soc. Jpn. 71, 19.Onoda, S., S. Murakami, and N. Nagaosa, 2004b, Phys. Rev.

Lett. 93, 167602.Onoda, S., N. Sugimoto, and N. Nagaosa, 2006, Phys. Rev.

Lett. 97, 126602.Onoda, S., N. Sugimoto, and N. Nagaosa, 2008, Phys. Rev. B

77, 165103.Onsager, L., 1952, Philos. Mag. 43, 1006.Ortiz, G., and R. M. Martin, 1994, Phys. Rev. B 49, 14202.Panati, G., H. Spohn, and S. Teufel, 2003, Commun. Math.

Phys. 242, 547.Peierls, R. E., 1933, Z. Phys. 80, 763.Prange, R., and S. Girvin, 1987, The Quantum Hall Effect

Springer, New York.Pu, Y., D. Chiba, F. Matsukura, H. Ohno, and J. Shi, 2008,

Phys. Rev. Lett. 101, 117208.Qi, X.-L., T. Hughes, and S.-C. Zhang, 2008, Phys. Rev. B 78,

195424.Qi, X.-L., Y.-S. Wu, and S.-C. Zhang, 2006, Phys. Rev. B 74,

085308.Rammal, R., and J. Bellissard, 1990, J. Phys. France 51, 1803.Rashba, E. I., 1960, Sov. Phys. Solid State 2, 1224.Resta, R., 1992, Ferroelectrics 136, 51.Resta, R., 1994, Rev. Mod. Phys. 66, 899.Resta, R., 1998, Phys. Rev. Lett. 80, 1800.Resta, R., 2000, J. Phys.: Condens. Matter 12, R107.Resta, R., and S. Sorella, 1999, Phys. Rev. Lett. 82, 370.Resta, R., and D. Vanderbilt, 2007, in Physics of Ferroelectrics:

A Modern Perspective, edited by K. Rabe, C. H. Ahn, and







http://dx.doi.org/10.1016/0003-4916(85)90148-4





http://dx.doi.org/10.1109/20.34293


http://dx.doi.org/10.1143/PTP.74.439




http://dx.doi.org/10.1103/PhysRevA.44.5239

http://dx.doi.org/10.1103/PhysRevA.44.5239











http://dx.doi.org/10.1063/1.881353




http://dx.doi.org/10.1063/1.2737555

http://dx.doi.org/10.1063/1.2737555



















http://dx.doi.org/10.1016/S1386-9477(00)00223-X

http://dx.doi.org/10.1016/S1386-9477(00)00223-X

http://dx.doi.org/10.1088/0305-4470/17/12/016





http://dx.doi.org/10.1038/nphys245

http://dx.doi.org/10.1051/jphys:019730034010090100



http://dx.doi.org/10.1002/pssb.2220760122

http://dx.doi.org/10.1002/pssb.2220760122





http://dx.doi.org/10.1103/PhysRevE.74.066610

http://dx.doi.org/10.1103/PhysRevE.74.066610









http://dx.doi.org/10.1007/s00220-003-0950-1

http://dx.doi.org/10.1007/s00220-003-0950-1

http://dx.doi.org/10.1007/BF01342591






http://dx.doi.org/10.1051/jphys:0199000510190215300

http://dx.doi.org/10.1080/00150199208016065



http://dx.doi.org/10.1088/0953-8984/12/9/201


J.-M. Triscone Springer-Verlag, Berlin, p. 31.Rice, M. J., and E. J. Mele, 1982, Phys. Rev. Lett. 49, 1455.Rycerz, A., J. Tworzydio, and C. W. J. Beenakker, 2007, Nat.

Phys. 3, 172.Sakurai, J. J., 1993, Modern Quantum Mechanics, 2nd ed. Ad-

dison Wesley, Reading, MA.Sales, B. C., R. Jin, D. Mandrus, and P. Khalifah, 2006, Phys.

Rev. B 73, 224435.Sawada, K., and N. Nagaosa, 2005, Phys. Rev. Lett. 95, 237402.Schellnhuber, H. J., and G. M. Obermair, 1980, Phys. Rev.

Lett. 45, 276.Seki, T., Y. Hasegawa, S. Mitani, S. Takahashi, H. Imamura, S.

Maekawa, J. Nitta, and K. Takanashi, 2008, Nature Mater. 7,125.

Semenoff, G. W., 1984, Phys. Rev. Lett. 53, 2449.Shankar, R., and H. Mathur, 1994, Phys. Rev. Lett. 73, 1565.Shapere, A., and F. Wilczek, 1989, Eds., Geometric Phases in

Physics World Scientific, Singapore.Sharma, P., and C. Chamon, 2001, Phys. Rev. Lett. 87, 096401.Shi, J., and Q. Niu, 2006, e-print arXiv:cond-mat/0601531.Shi, J., G. Vignale, D. Xiao, and Q. Niu, 2007, Phys. Rev. Lett.

99, 197202.Shindou, R., 2005, J. Phys. Soc. Jpn. 74, 1214.Shindou, R., and L. Balents, 2006, Phys. Rev. Lett. 97, 216601.Shindou, R., and L. Balents, 2008, Phys. Rev. B 77, 035110.Shindou, R., and K.-I. Imura, 2005, Nucl. Phys. B 720, 399.Sih, V., R. C. Myers, Y. K. Kato, W. H. Lau, A. C. Gossard, and

D. D. Awschalom, 2005, Nat. Phys. 1, 31.Simon, B., 1983, Phys. Rev. Lett. 51, 2167.Sinitsyn, N. A., 2008, J. Phys.: Condens. Matter 20, 023201.Sinitsyn, N. A., A. H. MacDonald, T. Jungwirth, V. K. Dugaev,

and J. Sinova, 2007, Phys. Rev. B 75, 045315.Sinitsyn, N. A., Q. Niu, J. Sinova, and K. Nomura, 2005, Phys.

Rev. B 72, 045346.Sipe, J. E., and J. Zak, 1999, Phys. Lett. A 258, 406.Slater, J. C., 1949, Phys. Rev. 76, 1592.Smit, J., 1958, Physica Amsterdam 24, 39.Souza, I., J. Íñiguez, and D. Vanderbilt, 2002, Phys. Rev. Lett.

89, 117602.Souza, I., and D. Vanderbilt, 2008, Phys. Rev. B 77, 054438.Souza, I., T. Wilkens, and R. M. Martin, 2000, Phys. Rev. B 62,

1666.Stone, A. D., 2005, Phys. Today 58 8, 37.Strange, P., 1998, Relativistic Quantum Mechanics: With Appli-

cations in Condensed Matter and Atomic Physics CambridgeUniversity Press, Cambridge.

Streda, P., 1982, J. Phys. C 15, L717.Strinati, G., 1978, Phys. Rev. B 18, 4104.Su, W. P., J. R. Schrieffer, and A. J. Heeger, 1979, Phys. Rev.

Lett. 42, 1698.Sundaram, G., and Q. Niu, 1999, Phys. Rev. B 59, 14915.Switkes, M., C. M. Marcus, K. Campman, and A. C. Gossard,

1999, Science 283, 1905.Tabor, M., 1989, Chaos and Integrability in Nonlinear Dynam-

ics: An Introduction Wiley-Interscience, New York.Tagantsev, A. K., 1986, Phys. Rev. B 34, 5883.Tagantsev, A. K., 1991, Phase Transitions 35, 119.Talyanskii, V. I., J. M. Shilton, M. Pepper, C. G. Smith, C. J. B.

Ford, E. H. Linfield, D. A. Ritchie, and G. A. C. Jones, 1997,Phys. Rev. B 56, 15180.

Teufel, S., 2003, Adiabatic Perturbation Theory in QuantumDynamics Springer-Verlag, New York.

Thonhauser, T., D. Ceresoli, D. Vanderbilt, and R. Resta, 2005,

Phys. Rev. Lett. 95, 137205.Thouless, D. J., 1981, J. Phys. C 14, 3475.Thouless, D. J., 1983, Phys. Rev. B 27, 6083.Thouless, D. J., 1998, Topological Quantum Numbers in Non-

relativistic Physics World Scientific, Singapore.Thouless, D. J., M. Kohmoto, M. P. Nightingale, and M. den

Nijs, 1982, Phys. Rev. Lett. 49, 405.Tian, Y. L. Ye, and X. Jin, 2009, Phys. Rev. Lett. 103, 087206.Umucalılar, R. O., H. Zhai, and M. O. Oktel, 2008, Phys. Rev.

Lett. 100, 070402.Valenzuela, S. O., and M. Tinkham, 2006, Nature London

442, 176.Vanderbilt, D., and R. D. King-Smith, 1993, Phys. Rev. B 48,

4442.van Houten, H., C. W. J. Beenakker, J. G. Williamson, M. E. I.

Broekaart, P. H. M. van Loosdrecht, B. J. van Wees, J. E.Mooij, C. T. Foxon, and J. J. Harris, 1989, Phys. Rev. B 39,8556.

Vignale, G., and M. Rasolt, 1988, Phys. Rev. B 37, 10685.Wang, X., D. Vanderbilt, J. R. Yates, and I. Souza, 2007, Phys.

Rev. B 76, 195109.Wannier, G. H., 1962, Rev. Mod. Phys. 34, 645.Wannier, G. H., 1980, J. Math. Phys. 21, 2844.Watson, S. K., R. M. Potok, C. M. Marcus, and V. Umansky,

2003, Phys. Rev. Lett. 91, 258301.Wen, X. G., and Q. Niu, 1990, Phys. Rev. B 41, 9377.Wilczek, F., 1987, Phys. Rev. Lett. 58, 1799.Wilczek, F., and A. Zee, 1984, Phys. Rev. Lett. 52, 2111.Wilkinson, M., 1984a, Proc. R. Soc. London, Ser. A 391, 305.Wilkinson, M., 1984b, J. Phys. A 17, 3459.Wilkinson, M., and R. J. Kay, 1996, Phys. Rev. Lett. 76, 1896.Winkler, R., 2003, Spin-Orbit Coupling Effects in Two-

Dimensional Electron and Hole Systems Springer, NewYork.

Wu, T. T., and C. N. Yang, 1975, Phys. Rev. D 12, 3845.Wunderlich, J., B. Kaestner, J. Sinova, and T. Jungwirth, 2005,

Phys. Rev. Lett. 94, 047204.Xiao, D., D. Clougherty, and Q. Niu, 2007, unpublished.Xiao, D., J. Shi, D. P. Clougherty, and Q. Niu, 2009, Phys. Rev.

Lett. 102, 087602.Xiao, D., J. Shi, and Q. Niu, 2005, Phys. Rev. Lett. 95, 137204.Xiao, D., J. Shi, and Q. Niu, 2006, Phys. Rev. Lett. 96, 099702.Xiao, D., W. Yao, and Q. Niu, 2007, Phys. Rev. Lett. 99,

236809.Xiao, D., Y. Yao, Z. Fang, and Q. Niu, 2006, Phys. Rev. Lett.

97, 026603.Yafet, Y., 1963, in Solid State Physics: Advances in Research

and Applications, edited by F. Seitz and D. Turnbull Aca-demic, New York, Vol. 14, p. 1.

Yang, S. A., G. S. D. Beach, C. Knutson, D. Xiao, Q. Niu, M.Tsoi, and J. L. Erskine, 2009, Phys. Rev. Lett. 102, 067201.

Yao, W., D. Xiao, and Q. Niu, 2008, Phys. Rev. B 77, 235406.Yao, Y., L. Kleinman, A. H. MacDonald, J. Sinova, T. Jung-

wirth, D.-S. Wang, E. Wang, and Q. Niu, 2004, Phys. Rev.Lett. 92, 037204.

Yao, Y., Y. Liang, D. Xiao, Q. Niu, S.-Q. Shen, X. Dai, and Z.Fang, 2007, Phys. Rev. B 75, 020401.

Ye, J., Y. B. Kim, A. J. Millis, B. I. Shraiman, P. Majumdar, andZ. Tesanovic, 1999, Phys. Rev. Lett. 83, 3737.

Zak, J., 1964a, Phys. Rev. 136, A776.Zak, J., 1964b, Phys. Rev. 134, A1602.Zak, J., 1964c, Phys. Rev. 134, A1607.Zak, J., 1977, Phys. Rev. B 15, 771.
















http://arXiv.org/abs/arXiv:cond-mat/0601531






http://dx.doi.org/10.1016/j.nuclphysb.2005.05.019



http://dx.doi.org/10.1088/0953-8984/20/02/023201




http://dx.doi.org/10.1016/S0375-9601(99)00308-4


http://dx.doi.org/10.1016/S0031-8914(58)93541-9






http://dx.doi.org/10.1063/1.2062917

http://dx.doi.org/10.1088/0022-3719/15/22/005





http://dx.doi.org/10.1126/science.283.5409.1905


http://dx.doi.org/10.1080/01411599108213201



http://dx.doi.org/10.1088/0022-3719/14/23/022
















http://dx.doi.org/10.1063/1.524384






http://dx.doi.org/10.1088/0305-4470/17/18/016


http://dx.doi.org/10.1103/PhysRevD.12.3845




















Zak, J., 1989, Phys. Rev. Lett. 62, 2747.Zeng, C., Y. Yao, Q. Niu, and H. H. Weitering, 2006, Phys.

Rev. Lett. 96, 037204.Zhang, C., A. M. Dudarev, and Q. Niu, 2006, Phys. Rev. Lett.

97, 040401.Zhang, Y., Y.-W. Tan, H. L. Stormer, and P. Kim, 2005, Nature

London 438, 201.Zheng, W., J. Wu, B. Wang, J. Wang, Q. Sun, and H. Guo,

2003, Phys. Rev. B 68, 113306.Zhou, F., B. Spivak, and B. Altshuler, 1999, Phys. Rev. Lett. 82,

608.Zhou, H.-Q., S. Y. Cho, and R. H. McKenzie, 2003, Phys. Rev.

Lett. 91, 186803.Zhou, S. Y., G.-H. Gweon, A. V. Fedorov, P. N. First, W. A. de

Heer, D.-H. Lee, F. Guinea, A. H. C. Neto, and A. Lanzara,2007, Nature Mater. 6, 770.
















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