Quantum kinetic equation in phase-space textured multiband systems

Clement H. Wong and Yaroslav TserkovnyakDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA

(Dated: January 11, 2013)

Starting from the density-matrix equation of motion, we derive a semiclassical kinetic equationfor a general two-band electronic Hamiltonian, systematically including quantum-mechanical correc-tions up to second order in space-time gradients. We find, in addition to band-projected correctionsto the single-particle equation of motion due to phase-space Berry curvature, interband terms thatwe attribute to the nonorthorgonality of the projected Hilbert spaces. As examples, we apply ourkinetic equation to electronic systems in the presence of spatially inhomogeneous and dynamical spintextures stemming from electromagnetic gauge potentials, and specifically to the electromagneticresponse of massive 2D Dirac fermions and 3D Weyl fermions.

PACS numbers: 72.10.Bg,72.25.-b,72.20.My,73.43.-f

I. INTRODUCTION

The semiclassical theory of electronic dynamics hasbeen successful in explaining a wide range of transportphenomena in solid-state physics. It is well establishedthat the semiclassical single-particle equations of motion(sEOM) acquire anomalous corrections that stem fromBerry phases1 accumulated in the adiabatic motion ofa wave packet.2–14 Such Berry-phase effects have beensuccessful in explaining the anomalous Hall effect,2,8 cor-rections to semiclassical quantization,6,7 intrinsic mag-netic moments of electronic wave packets,7,9,12,14,15 andanomalous thermoelectric transport16,17 in various elec-tronic systems. The (noncanonical) Hamiltonian per-spective and issues related to the Liouville’s theo-rem and modified electron density of states have alsobeen elucidated recently,11–13 as well as Fermi-liquidgeneralizations.8,18

Collective dynamics in the semiclassical approach isusually described with the Boltzmann equation, whereelectrons drift in phase space according to their single-particle equations of motion projected on respectivebands. While such an approach is intuitive and physicallyappealing, its validity is open to question. The problemis that spatiotemporal inhomogeneities in the Hamilto-nian such as electromagnetic potentials induce interbandcoherences that may not be completely captured by thesEOM. Furthermore, it was noted in Ref. [11] that thesEOM are noncanonical, appearing to violate Liouville’stheorem. As a remedy, the authors introduced a rescal-ing of the phase-space density of states. This modifica-tion was shown to be consistent with a formal requan-tization procedure for band-projected Bloch electrons,which promotes the Poisson brackets of the sEOM tocommutators, resulting in noncanonical commutator re-lations of position and momentum and the correspondingmodified minimum quantum uncertainty in phase-spacevariables.11 However, we find no logical necessity to en-force Liouville’s theorem for the band-projected distri-bution function, and the formal requantization argumentdoes not explain why the density of states is modified bythe band projection.

These fundamental issues motivate us to consider thegeneralized Boltzmann equation for a multiband systemfrom a systematic, ground-up approach. In this paper,for a clean system with nondegenerate bands, we derivea band-projected kinetic equation by performing a semi-classical expansion of the density-matrix equation of mo-tion. A similar Green’s function approach was developedin Ref. 18. However, there it was assumed from the out-set that the distribution function was nonvanishing onlyin a single band. In this paper, we make no a prioriassumptions of decoupled bands and capture systemat-ically all corrections to the Boltzmann equation up tosecond order in space-time gradients. Despite our at-tempt at decoupling by a systematic gradient expansion,we find remaining interband terms that warrant furtherinvestigation.

This paper is organized as follows. In Sec. II A, fora two-band Hamiltonian, we derive a covariant trans-port equation for the Wigner distribution function. InSec. II B, we decouple the transport equation to deriveeffective band-diagonal semiclassical kinetic equations(sKE). In our sKE, in addition to the known Berry curva-ture corrections to the sEOM, we find terms correspond-ing to the aforementioned modified density of states,as well as interband terms not seen before. We inter-pret these terms as representing quasiparticles motionin curved, nonorthogonal subspaces of the total Hilbertspace. In Sec. II C, we extract the hydrodynamic cur-rent from the continuity equation, discuss issues associ-ated with momentum-space ultraviolet cutoffs and band-crossing points. In Sec. II D, we consider minimal cou-pling to electromagnetic gauge fields and express ourtransport equations in terms of the gauge-invariant ki-netic momentum. We then apply these equations tothe 2D Dirac and 3D Weyl Hamiltonians in Sec. III Aand Sec. III C. In the conclusion, Sec. IV, we qualita-tively compare our approach with other methods for de-riving band-projected effective Hamiltonians by canoni-cal transformations on the Hilbert space. Computationaldetails are relegated to the appendices.

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II. SEMICLASSICAL KINETIC EQUATIONS

A. Covariant Transport Equation

Consider the many-particle density matrix,

ραβ(r1, r2, t) ≡ 〈ψ†β(r2, t)ψα(r1, t)〉 , (1)

where ψα(r1, t) are second-quantized field operators, andα is a spin or band index. We will consider only fermionsin this paper, although our formalism is equally appli-cable to bosons. For a quadratic, mean-field (or Fermi-liquid) Hamiltonian,

H(t) =

∫dr1dr2 ψ

†α(r1, t)Hαβ(r1, r2, t)ψβ(r2, t) , (2)

where H is the first-quantized, quasiparticle Hamiltonianexpressed as a kernel (with the implied summation overrepeated indices α, β), the equation of motion for thedensity matrix closes:

∂tραβ +i

~(Hαγ ⊗ ργβ − ραγ ⊗Hγβ) = 0 . (3)

Here, ⊗ denotes a convolution integral in real space:[A ⊗ B](r1, r2) ≡

∫dr′A(r1, r

′)B(r′, r2). We generallyconsider a smooth low-energy effective Hamiltonian

Hαβ(r1, r2, t) = εαβ(−i∂r1 , r1, t)δ(r1 − r2) . (4)

If necessary, the function εαβ(−i∂r1 , r1, t) is properlysymmetrized in its noncommuting arguments ∂r1 and r1.Eq. (4) may represent the continuum Hamiltonian for theslowly-varying envelope fields in the k · p expansion, orthe continuum limit of a tight-binding Hamiltonian.

For slow and long-wavelength spatiotemporal inhomo-geneities, it is useful to define the distribution functionnp(r, t) by the Wigner transform (WT) of the densitymatrix:

np(r, t) ≡∫

dr′e−i~p·r′ ρ

(r +

r′

2, r− r′

2, t

). (5)

In the Wigner representation, the expectation value ofan observable described by the kernel A(r1, r2) is givenby

〈A〉(t) =

∫dr∑p

Tr[np(r, t)Ap(r, t)] , (6)

where∑

p ≡∫dp/(2π~)d, d being the number of spatial

dimensions. Here and henceforth, we make a conventionto denote the spin/band matrix structure by hats. Whennp(r, t) is a smooth function of r and t, one may constructa semiclassical kinetic equation for np(r, t) by taking theWT of Eq. (3) and performing a gradient expansion.

The kinetic equation for np(r, t) is governed by thequasiparticle energy matrix εp(r, t), which is the WT ofthe quasiparticle Hamiltonian kernel (4). Considering,

for simplicity, a two-band system, the energy may beexpanded as

εp(r, t) =1

2εp(r, t) + ∆p(r, t) · τ . (7)

Here, τa = σa/2 are the spin-1/2 matrices satisfying[τa, τb] = iεabcτc and τa, τb = δab/2, σa being the Paulimatrices. εabc is the antisymmetric Levi-Civita tensor,and we will use letters in the beginning of the alphabet forindices representing spin degrees of freedom. Below, wewill be similarly decomposing the components of any 2×2matrix into scalar and vector pieces as M = M/2+M · τ ,

where M ≡ Tr[M ] and M ≡ Tr[M σ]. We parametrizethe gap vector as ∆p(r, t) = ∆p(r, t)mp(r, t), where∆p = |∆p| and mp = (sin θp cosϕp, sin θp sinϕp, cos θp)is a unit-vector field represented by the spherical anglesθp(r, t) and ϕp(r, t).

The gap vector field ∆p(r, t) appears formally as amagnetic field coupled to spin in phase space, whose di-rectional field mp(r, t) we will call the spin texture. Ex-amples of Hamiltonian (7) occur in ferromagnetic semi-conductors, where the gap vector represents the exchangeand spin-orbit fields, and in nonmagnetic semiconductorswith spatially varying spin-orbit coupling. The internaldegrees of freedom need not be the actual electron spin.For example, our kinetic equation can be applied to thepseudospin dynamics near the K(K ′) points of graphene,which is described by the Dirac Hamiltonian.

We will be interested in quasiparticle transport on thesemiclassical spin-orbit bands defined by the eigenvaluesof Eq. (7),

εps(r, t) =1

2[εp(r, t) + s∆p(r, t)] . (8)

where s = ±1. The local spin frame that diagonalizesEq. (7) is defined by an SU(2) rotation Up(r, t) such that

U†p(∆p · τ )Up = ∆pτz . (9)

The basis defined by this rotation is the local energyeigenstates in the sense that the average total energy 〈H〉may be expressed as a spatial integral

〈H〉 =

∫dr∑ps

εpsnps . (10)

nps here are the diagonal elements of the distributionfunction in the local spin frame.

To derive the semiclassical kinetic equation, we takethe WT of Eq. (3) up to second order in the gradientexpansion. (See Appendix A for details.) The resultingkinetic equation in the local frame is

Dtnp +i

~[εp, np]− 1

2Diεp, D

inp

− i~8

[DiDj εp, DiDj np] = 0 , (11)

3

where , denotes anticommutator and [, ] commutator,i, j label coordinates of phase-space vector xi ≡ (r,p),and summation over repeated indices is implied. The in-

dices are raised by ∂i = J ij∂j , where J↔

is the symplecticmatrix acting on phase-space derivatives ∂i ≡ (∂r, ∂p) as

J↔(∂r∂p

)=

(∂p−∂r

), J↔

=

(0 1−1 0

), (12)

the latter being written in terms of the d × d matrixblocks.40 In Eq. (11), we introduced the covariant deriva-tives of the distribution function and energy defined by

DIM ≡ ∂IM − i[AI , M ] =1

2∂IM +DIM · τ , (13)

where DIM ≡ ∂IM + AI ×M. Hereafter, the capi-tal letters I, J , and K will be used to denote combinedphase-space and time coordinates. The matrix-valuedgauge fields entering covariant derivatives (13) are de-

fined by AI ≡ iU†p∂I Up ≡ AI · τ = A+I τ+ +A−I τ−+AzI τz,

where A±I = AxI ∓ iAyI and τ± ≡ (τx ± iτy)/2. Inthe Euler-angle parametrization of our local spin frame,U(ϕ, θ, γ) = e−iϕτze−iθτye−iγτz , the gauge fields are

AzI = cos θ∂Iϕ+ ∂Iγ ,

A±I = −e±iγ(sin θ∂Iϕ± i∂Iθ) , (14)

where γ is an arbitrary rotation angle about mp(r, t) andhence a local gauge parameter. The form of AzI reflectsthe north/south pole singularity in the spherical coor-dinate system, where ϕ is not well defined. Near thepoles, we may choose a gauge in which γ = ∓ϕ, whichrenders the gauge fields well behaved either at the northor south poles (θ = 0 or π), respectively, but not both.It is thus necessary to use different gauges locally in re-gions where the texture passes through both north andsouth poles. We emphasize that such singularities havea purely mathematical origin arising from our choice ofcoordinate system, and may occur where the texture isperfectly smooth.

The product of the transverse components in Eq. (14)is a gauge-invariant second-rank tensor:

A+I A−J = sin2 θ∂Iϕ∂Jϕ+ ∂Iθ∂Jθ

+ i sin θ(∂Iθ∂Jϕ− ∂Iϕ∂Jθ)≡GIJ + iFIJ . (15)

The real part GIJ is a kind of metric in spin space, whichwill not appear in the final results of this paper.41 Bygauge invariance, only the Berry curvature, i.e., the curlof the Berry gauge field AI ≡ −AzI , appears in any phys-ical quantities,

FIJ ≡ ImA+I A−J = AxIA

yJ−AyIAxJ = ∂IAJ−∂JAI . (16)

In the rest of the paper, where necessary, we will de-note the Berry gauge fields by A±I = −(cos θp ∓ 1)∂Iϕp

when well defined on the north/south pole, respectively.

Geometrically, the Berry curvature gives the solid an-gle Ω spanned by the spin texture mp(r, t) per areain the (xI , xJ) plane. Nonvanishing Berry curvaturemeans that particles acquire a phase-space Berry phase(s/2)

∮dxIAI ≡ sΩ/2 over a closed trajectory, which

modifies their transport. All the phenomena we will in-vestigate in this paper may be traced back to this phase.

Equation (11) may be viewed as an expansion of Eq. (3)in powers of ~. However, the separation into its classicalO(~0) and quantum O(~) part is not manifest becauseof its matrix structure, which represents the dynamicsof quantum-mechanical internal degrees of freedom. Inthe next section, we will derive a kinetic equation for thedistribution function projected on each band defined byEq. (8), which systematically captures all O(~) quantumcorrections to the classical Boltzmann equation.

B. Decoupling

The diagonal part of kinetic equation (11) reads

∂tnps + ∂iεps∂inps + Fps(∂i,Ai; εs, ns, n) = 0 , (17)

where Fps are the O(~), “anomalous” terms in the Boltz-mann equation. We denote transverse (i.e., x, y) vectorcomponents of the distribution (in the local frame) byn. In the formulae throughout this paper, the partialderivatives ∂i acts only on the symbol to its immedi-ate right, so that in expressions of the form ∂iA∂jB,∂i acts on A only. At this point, the anomalous terms,which are shown explicitly in Eqs. (D2) and (D3) of ap-pendix D, may not appear manifestly gauge invariant(while they certainly should be).2 Let us now decouplethe longitudinal and transverse components in an adia-batic approximation, which will result in a closed, gauge-invariant equation for the diagonal distribution functions.The decoupling procedure may be organized in the fol-lowing way. Suppose that we can solve the transversecomponents in terms of the longitudinal in a gradientexpansion to the (p − 1)th order in space-time deriva-tives: n = n(0) + n(1) + . . . + n(p−1). By substitutingn in Fps, which is at least first order in gradients, anddroppig any (p + 1)th-order terms in Eq. (17), we arriveat a pth-order equation for nps. We will carry out thisprocedure to second order (p = 2), consistent with ourinitial gradient expansion in Eq. (11).

From the off-diagonal part of Eq. (11), in regions where∆ 6= 0, one may readily find the transverse componentsto first-order space-time gradients [see appendix D fordetails]:

np =~2

(npz∆

∂iεp − ∂inp)

Ai − ~npz∆

At , (18)

where we have defined the phase-space particle-numberand longitudinal spin densities, n = n↑ + n↓ and nz =n↑−n↓, respectively. The expression includes terms con-taining the expansion parameters of the adiabatic ap-proximation: (~vF /∆)∂r and (~/∆)∂t, as well as a term

4

O(1/∆0) that shows that interband coherences may begenerated by gradients of the distribution, irrespectiveof the size of the gap and even in the absence of exter-nal fields. We will see an example of this in the Diracequation in Sec.III A.

Let us further consider why, as shown in Eq. (18), inter-band coherences (encoded by the transverse distributionfunction) are proportional to the transverse gauge fields.Recall that the Berry gauge fields are the matrix elementsAss

′

i = i〈m, s|∂i|m, s′〉, where the local eigenspinors of

our spin bands are given by |mp(r, t), s〉 = Up(r, t)|z, s〉.The transverse gauge connection determines the overlapbetween spin up and down states of nearby eigenspinors,

〈m(x, t), s|m(x+ δx, t+ δt), s′〉 = −iAss′I δxI .

Consider the terms O(nz/∆) in Eq. (18), in the presenceof an electric potential in the energy. The particle spinsare locally misaligned with the texture m by an angle χgiven by42

tanχ =|n|nz→ ~

∆|ElApl+vlArl+At| ,

where the applied electric field is E = −∂rε/2, and wedefined group velocity vp = ∂pε/2. (l runs over d spatialdimensions.) A finite χ stems from the slight misalign-ment of electrons spin with the local quantization axis mdue to (i) drift in momentum space with velocity E, (ii)drift in real space with velocity vp, and (iii) dynamicsof the texture. In particular, a locally spin-up electronwill have amplitude on the nearby down band propor-tional to A+−

i . [See Eq. (I1) for a microscopic expressionfor A+−

i .] For a near-equilibrium distribution functionwhich depends only on energy, nps(εps), the O(1/∆0)

term reads ∂εnps∂iεAi. Evidently, in this case, it also

stems from electron drift, but comes from a differentpart of the phase space. For a Fermi-Dirac distribution,this term originates from electrons on the Fermi surfaces,while the O(nz/∆) terms come from the momentum re-gion between the Fermi surfaces.

Substituting Eq. (18) in the anomalous terms inEq. (17), we find

Fs =

[−1

4∂j(∆F) + qs(Ftj −F ij∂iεs)

]∂jns

− 1

4

[∂i∆∂in−s + nz(∂t + ∂iεs∂i)

]F . (19)

Here, the Berry curvature terms are expressed as

FIJ = z ·AI ×AJ , F ≡ ~Frlpl ,

recalling that index l runs over d spatial dimensions. Wehave defined the fictitious charge qs = −s~/2, s = ±1 forthe ↑, ↓ bands, respectively. Equation (19) constitutesa central result of this paper. It is clear that these areO(~) corrections to the Boltzmann equation (17). In thefollowing sections, we will omit ~ for convenience (setting

~ = 1). In the rest of this paper, we will explicate Eq. (19)and apply it to specific examples.

The first line in Eq. (19) represents corrections to thesingle-particle equation of motion. Including only theseterms, the longitudinal transport equation (17) wouldread

∂tns −[(∂iεs + qsFti)J ij + qs∂iεsF ij

]∂jns + . . . = 0 .

(20)We first note that there is a correction to the single-particle energy, εps = εps + δεps, where

δεp = −∆pF4

, (21)

which, in particular, is related to the magnetic momentof a semiclassical wave packet [see Eq. (33)].7,14 The otherBerry-curvature corrections introduce Hall-like terms tothe single-particle equation of motion, which we define sothat the terms shown Eq. (20) constitute the phase-spaceadvective derivative of the distribution function,

Dstns ≡ (∂t + xjs∂j)ns.

We thus identify phase-space velocites xjs =(rs, ps),

7,10,18

rs = ∂pεs + qs(Ftp + Fppl∂rlεs −Fprl∂plεs) ,

ps = −∂rεs − qs(Ftr + Frpl∂rlεs −Frrl∂plεs) . (22)

As noted in Refs. 11–13, these equations of motion arenoncanonical and thus appear to violate Louville’s theo-rem. Indeed, the phase-space velocity has a finite phase-space divergence,

∂ixis = qs(∂iFti + ∂iF ik∂kεs) . (23)

From classical considerations of a two-component fluid,we would expect the projected kinetic equation to read:∂tnps + ∇ · (npsxs) = . . ., where we denote the phase-space gradient by ∇ = (∂r, ∂p), and . . . represents pos-sible interband terms. It may be seen, using the identity~∂iFiJ = ∂JF (see appendix E), that terms proportionalto to Eq. (23) indeed appear in the second part of the sec-ond line in Eq. (19), however, multiplied by the longitudi-nal spin density nz which couples the bands.43 To expressour sKE as a kind of phase-space “continuity equation,”we write −nz/4 = (qsns + q−sn−s)/2 = qsns − qsn/2 inthe expression for Fs, then our sKE reads

∂tns + ∂i(nsxis) =

[1

4viz∂in−s + qs

n

2(∂t + ∂iεs∂i)

]F ,(24)

where we have introduced notation for the relative bandvelocity: viz ≡ ∂i∆.

Both sides of the equation contain terms that violateLouville’s theorem. Consider first the LHS. Since thephase-space divergence is in fact the advective derivativeof qsF (up to the order of our gradient expansion)

∂ixis = qs(∂t + ∂iεs∂i)F

= qs(∂t + xis∂i)F +O(∂3µ) ≡ qsDs

tF , (25)

5

we are lead to the following explanation for physical ori-gin of the term ns∂ix

is on the LHS. In a semiclassical

description, spinful particles occupy wave-packet stateslabeled by momentum and position, as well as a spin-coherent state labeled by spherical angles Ω on the unitsphere. A particle is thus specified by a set of continuouscoordinates, (r,p,Ω), which we will call the semiclassicalstate space. However, in our adiabatic approximation, weretained only part of the (matrix) distribution function,nps(r, t), which denotes particle occupation per solid an-gle δΩ subtended by the texture mp(r, t) in a volume∏l δrlδpl. Since this solid angle is determined by the

phase-space Berry curvature, being for example in onespatial dimension δΩ(p, r, t) = Frpδrδp, the changes inthe distribution function due to the modulation of thisspin space volume along the phase-space particle trajec-tories appear in the transport equation (24) as an advec-tive derivative of F ≡ ~Frlpl .

In Ref. 11, it was assumed that ns satisfies Eq. (24)with RHS equal to zero, in which case one could re-cover Louville’s theorem for a rescaled distribution func-tion fs defined by ns ≡ Dsfs,11 where Ds satisfies∂ix

is = −(1/Ds)DtDs, so that the LHS of Eq. (24) reads

DsDst fs. Since Ds satisfies ∂ix

is = −Ds

t lnDs, from Eq.(25) we find

Ds = e−qsF ≈ 1− qsF , (26)

Since we are neglecting terms that are cubic order in spa-tiotemporal derivatives at the level of the sKE, and, tothe lowest order, Dt ∼ ∂µ, we should keep Ds only tolinear order in gradients, as in the approximation above.This expression agrees with the one quoted in Ref. 19.In Ref. 11, Ds was included as part of the phase-spacemeasure representing a modified phase-space density ofstates for band-projected electrons. Here, for our twoband, continuum model, we have given an explanationfor its origin. It accounts for the (pseudo)spin degreesof freedom in an approximation in which the quasipar-ticle spin dynamics is effectively constrained on a sub-manifold of the total state space. This submanifold,which we will call the projected state space, is a hyper-surface r,p,Ω(r,p, t) determined by the spin texturem[Ω(r,p, t)]. The projected state space has a local, dy-namical volume proportional to qsF(r,p, t) and is thuscurved. We note that this curvature vanishes, F = 0, un-less mp(r, t) has both momentum and real-space deriva-tives.

Now consider the terms in the RHS of Eq. (24), whichcontain interband terms. These terms are to be expectedbecause the wave packets in the projected state spaceare not pointlike, but occupy a finite minimum spin solidangle ∆Ω corresponding to a finite minimum volume∏l ∆rl∆pl ∼ ~d in phase space required by the uncer-

tainty principle. However, due to the nonorthorgonalityof the spin-coherent states, the spin up/down (along thetexture) wave-packet states have nonzero overlap, result-ing in interband couplings in the transport equation. Theoverlap amplitudes are proportional to the area ∆Ω ∼ F ,

consistent with the fact that the interband terms are pro-portional to F .

It is clear that these effects are generic to projectedkinetic equation in multiband systems. Since the decou-pling procedure is technically much more difficult in thiscase, let us sketch out qualitatively how our two-bandresult might be generalized. In a low energy, k · p ex-pansion about a point (in the Brillouin zone) of highsymmetry, electronic quasiparticle transforms under anirreducible spinor representation of the crystallographicpoint group, which in some cases may be taken to be ahigher-spin group, as in the case of the Luttinger Hamil-tonian. The multicomponent electronic wave functionmay be viewed as an amplitude on a group manifold (inthe representation space of the symmetry group). There-fore, we may specify quasiparticles as in the two bandcase, except now Ω denotes coordinates on a higher di-mensional group manifold, and Eq. (11), which governscoupled orbital and internal dynamics, may be viewed asquasiparticle motion in (r,p,Ω).

For homogeneous spin-orbit couplings, the eigenfunc-

tions ψs = us(p)eip·r of the k · p Hamiltonian H(p)defines a set of momentum-space spin textures. Whenthe spin-orbit field is inhomogeneous, we can still de-

fine a local, plane-wave-like (overcomplete) basis ψs =us(r,p)eip·r, where us(r,p) is an eigenvector of theWigner-transformed Hamiltonian, and build wave pack-ets out of these states. Thus, the situation is quite similarto the two-band case, and we expect similar argumentspertaining to a curved state space and interband cou-plings due to the nonorthogonality of semiclassical co-herent states to hold.

C. Hydrodynamics

It is natural to expect a continuity equation for the par-ticle density, ρ =

∑p Tr[np] =

∑ps nps, which follows by

taking∑

ps of our sKE’s, resulting in an equation,

∂tρ+ ∂r · j + . . . = 0,

from which we may extract the curl-free part of the par-ticle current. The anomalous, O(~) contribution to ∂tρ isgiven by the terms in Eq. (19), which, upon summation,may be written as [cf. Eq. (D13)]∑

ps

Fps =∑ps

qs∂k

[ns(FitJ ik − ∂iεsF ik)

]− 1

4∂k(∆ns∂

kF)

=∑s

∫ddp

(2π)d

∂r ·

[nps

(δrs −

∆

4∂pF

)]+ ∂p ·

[nps

(δps +

∆

4∂rF

)], (27)

where we have denoted the Berry-curvature correctionsproportional to qs in Eq. (22) by (δrs, δps) (these are

6

the corrections not including the shifts in band energies∂iδεs). The momentum-space integration extends to acutoff pΛ above which our low-energy effective theory isno longer valid.

From the first line in the second equality of Eq. (27)and including the “normal” part of the Boltzmann equa-tion Eq. (17), we identify the particle current

j =∑s

∫ddp

(2π)dnps[∂pεs −

∆

4∂pF

+ qs(Ftp + Fppl∂rlεs −Fprl∂plεs)] . (28)

Note that only part of the energy correction in Eq. (21)and the corresponding contribution to group velocity en-ters into this current. This resulted from cancelationswhich occurred in tracing the RHS of Eq. (24).

The second line in Eq. (27) appears as an anomalousparticle source coming from the boundary in momentumspace (invoking the divergence theorem). Thus the . . . inour continuity equation reads,∑

s

∮dd−1Sp

(2π)dl ·nps

[− ∂rεps +

∆

4∂rF

− qs(Ftr + Frpl∂rlεs −Frrl∂plεs)]

, (29)

where l is the normal to the bounding surface at pΛ.This term represents particles flux coming from outsidethe momentum-space region where our semiclassical the-ory is valid. In principle, if we know the microscopictheory beyond the cutoff and could solve for the distri-bution function there, we may use it as an input in ourlow-energy theory, providing a momentum-space bound-ary condition for our distribution function that is foundbelow the cutoff. In the following, for simplicity, we willtake the cutoff to infinity. It is well known that iso-lated degenerate points (band crossings) where ∆p = 0are monopole sources of Berry curvature1,8,20 (see ap-pendix E). These singularities occur because it is notpossible to choose a spin frame [defined by the SU(2) ro-tation in Eq. (9)] that is continuous, near topological de-fects in the spin textures.44 This occurs, for example, atthe origin of the hedgehog texture in the Hamiltonian ofEq. (49). Strictly speaking, our expression for the trans-verse component in Eq. (18) is not valid at these points(since ∆ = 0).45Generally, we will have to exclude thesepoints by imposing a lower bound in our momentum inte-gration defined by an infinitesimal surface bounding thesingularity. The boundary conditions for nps at these“inner” boundaries may be found by solving the exactquantum-mechanical problem near the crossing point.

We note that since all terms ∝ ∆−1 canceled out inthe final transport equation, we may take the dc limit:ω → 0, ∆ → 0 (in the prescribed order). Thus in someinstances, one may regulate monopole singularities by in-troducing a gap in the Hamiltonian, compute the current,and then take the gap to zero. However, this procedurewill in general introduce some ambiguity in the final an-swer. For example, it may depend on the direction of

the texture at the point where the gap is taken to zero.A well-known example of this problem is the case of themassless 2D Dirac fermion,21 where the vacuum currentdepends on the sign of the mass used to regulate the di-vergences for the massless fermion. Below, we will workout the massive case in our semiclassical approach.

D. Coupling to electromagnetic fields

Consider a situation where the real-space texture in-homogeneity stems only from minimal coupling to vec-tor and scalar gauge potentials, a(r, t) and φ(r, t), in thefirst-quantized microscopic Hamiltonian,

H (−i∂r, r, t)→ H (−i∂r − a(r, t), r, t) + φ(r, t) . (30)

The electron charge is absorbed here in the definition ofthe gauge potentials.

From the Wigner transformation of this Hamiltonian,the semiclassical energy becomes

εp → εp−a(r,t) + φ(r, t) ≡ εk(r, t),

where k(r, t) ≡ p − a(r, t) is the kinetic momen-tum. The energy and the canonical momentum arenot gauge invariant. To make the transport equationmanifestly gauge invariant, we must express the dis-tribution function in terms of the kinetic momentum,np = nk+a(r,t)(r, t) ≡ nk(r, t). Note that ∂k = ∂p, butthe space-time derivatives ∂µ = (∂r, ∂t) in the transportequation are taken with fixed canonical momentum p.We have to take account of the implicit space-time de-pendence of kinetic momentum by

∂µ|p = ∂µ|k + ∂µk · ∂k = ∂µ − ∂µa · ∂k , (31)

where it is implicit now that ∂µ is taken with fixed k.We will also impose a gauge invariant momentum-spacecutoff kΛ. The transformation to kinetic momentum maybe viewed as a phase-space coordinate transformation,under which the gradients transform as

∂rεps = ∂rεks − vks,l∂ral ,∂rnp = ∂rnk − ∂klnk∂ral . (32)

Here, the group velocity is denoted by vks ≡ ∂kεs.The transformation of the Berry-curvature correctionsare given in appendix F. We find that F = b · Bk, sothat the energy shift δεk can be interpreted in terms ofa magnetic moment Mk,

δεk ≡ −Mk · b (33)

where Mk = ∆kBk/4. The factor of 1/4 is a result ofour conventions, where we have factored out 1/2 fromthe usual definition of the Berry gauge fields in the lit-erature. We show in appendix H that this agrees with

7

the magnetic moment derived more directly from wave-packet analysis.7 The gauge-invariant transport equationnow reads:(

∂t + rs · ∂r + ks · ∂k)nks

− nz4

(∂t + vks · ∂r + fks · ∂k) (b ·Bk)

+b ·Bk

4(vkz · ∂r + vkz × b · ∂k)nk−s = 0 , (34)

where we have defined the relative velocity vkz ≡ vk↑ −vk↓ and Lorentz force fs = e + vks × b. e ≡ −∂rφ− ∂taand b ≡ ∂r × a are the ordinary electromagnetic fields.The phase-space velocities are7,18

rs = vks + ∂kδεk − qsfks ×Bk

= Zksvks + ∂kδεk − qs[e×Bk + (vks ·Bk)b] , (35)

ks = fs + δfs − qs(fks ×Bk)× b

= Zksfs + δfs − qs(b · e)Bk , (36)

where we define Zks = 1 + qsb ·Bk and

δfs = −∂rδεk + ∂kδεk × b .

We note that Eq. (36) is not simply ps in Eq. (22) trans-formed to kinetic momentum (which would not even begauge invariant ), as it was necessary to transform the en-

tire sKE. In the second equalities for rs, ks, we see thatthe group velocity and electromagnetic force are rescaledand shifted. The shift δfks simply comes from the mag-netic energy and the additional group velocity due to themagnetc moment in Eq. (33). The total O(~) correctionto the group velocity, if we write the first two terms inthe second equality for rs in Eq. (35) as vks + δvks, is

δvks = qs(b ·Bk)vks + ∂kδεk

= −qs(b ·Bk)(vk + svkz)−∆k

4∂k(b ·Bk) . (37)

The third term in the first equality in expression for rsis the well-known anomalous Hall velocity.4 Furthermore,the last terms in the second equality of Eqs. (35) and (36)give an additional phase-space velocity in the directionof the phase-space “magnetic fields.”

Up to the order of our gradient expansion, in whichwe need to keep terms in rs(which multiplies ∂rns) to

linear order in EM fields, and in ks to quadratic order,and defining vks ≡ ∂kεks, eks ≡ −∂rεks − ∂ta (recallthat εks = εks + δεks), the equations of motion may beexpressed as

rs = vks − qsks ×Bk

ks = eks + rs × b , (38)

in agreement with the form of wave-packet equations ofmotion.7 We emphasize, however, that we are retainingterms beyond linear electromagnetic response46 whichare not in the usual wave-packet equations of motion,for example in the b · e term for ks in Eq. (36), and in-cludes forces such as −∂rδεk coming from magnetic fieldgradients.

III. EXAMPLES

A. Massive 2D Dirac fermions

As an application of our transport equation and acheck on our formalism, we consider the electromagneticresponse of massive 2D Dirac fermions, which is relevant,for example, to the gapped surface state of 3D topologicalinsulators.22,23 The Hamiltonian is given by

HD = v∑a=x,y

σa[−i∂r − a(r, t)]a +mv2σz , (39)

where v is a constant with the units of velocity (whichwill be set to unity henceforth) and σ is a vector of Paulimatrices. The particle/hole symmetric dispersions areconveniently expressed in terms of the relativistic ener-gies Ek ≡ ∆k/2 =

√k2 +m2, so that εks = sEk with

corresponding group velocities vks = sk/Ek. In k space,the texture is a vortex (meron) centered at the origin,ϕk = arg(k), with an out-of-plane component given bycos θk = m/Ek. The vortex polarity is given by sgn(m).The spin texture has Berry curvature

Bk = −∂k cos θk × ∂kϕk =m

E3k

z .

The magnetic moment is thusMk = m/2E2k. Both Berry

curvature and magnetic moment are localized near theorigin (see Fig. 1) and have a direction (normal to the xyplane) depending on sgn(m).15

m2Bk

0 1 2 3

0.5

1

mMk

k/m

FIG. 1: In dimensionless variables, a plot of the Berry cur-vature, m2Bk (solid curve), and magnetic moment, mMk

(dashed curve), as functions of k/m.

Consider now applying a static, homogeneous magneticfield b = bz. From Eq. (37), the correction to the groupvelocity is δvks = (bm/2E4

k)k. The equations of motionEq. (35) and Eq. (36) then read

rs =

(s+

bm

2E3k

)k

Ek, ks = brs × z . (40)

8

It is evident that the usual cyclotron motion holds, butthe cyclotron orbits are not exactly traversed in the op-posite sense for the two bands because of the O(b) cor-rection. We also note that to reach this result, we hadto keep the O(b2) term in the Lorentz force, which arisesdue to the magnetic field dependence of the group veloc-ity. The second line in (34) vanishes on account of thespatially uniform static magnetic field and the fact thatfks is transverse to k. The third line reads

B2E4

k

[bk · ∂r + b2k× z · ∂k]nk−s . (41)

For a spatially homogenous distribution (which is validfor an infinite plane), the O(b) term vanishes, and for arotationally invariant distribution in k, the O(b2) termvanishes. Thus, in this case, the transport equation issimply solved by a constant distribution function.

To see some dynamics, consider adiabatically turningon the magnetic field, and let b(t) be a slowly increasingfunction. We choose a rotationally-invariant vector po-tential that produces a uniform magnetic field along thez axis, a = bz×r/2 = b(−y, x)/2, with the accompanying

circulating electric field e = −∂ta = −bz × r/2. First,let us check that our equations agree with the Stredaformula,24 which is a general relationship between theHall conductivity in a gapped system and the changein particle density ρ when a magnetic field is turnedon adiabatically. For the rest of this section, let us re-store ~, the speed of light c, and the electron charge bye = eE, b = eB/c. In 2D, the Streda formula readsσxy = ec∂ρ/∂Bz.

For a homogenous distribution function in linear re-sponse, the transport Eq. (34) reduces to

∂tnks =e

2c

∑s

qsn(0)ks ∂t(B ·Bk) , (42)

where n(0)kz is the unperturbed, longitudinal spin distri-

bution that is zeroth order in B. The resultant change inparticle density δρ for an adiabatic change δBz is givenby

ecδρ

δBz= ν

e2

h, (43)

where

ν = z ·∫d2k n

(0)kz Bk .

The extra particle density comes from particle fluxes atinfinity due to the circulating electric field that producesanomalous Hall velocity rHs = −eqsE×Bk. This resultsin particles on the upper (lower) band to enter (leave)the system. See Fig. 2. One finds the corresponding Hallcurrent

jH = e∑ks

rHsn(0)ks = ν

e2

hE× z , (44)

B

E

Bk

jH

FIG. 2: The anomalous current (G1) comes from particles inthe hole band flowing out of the system due to the anomalousHall velocity (44).

Comparing with Eq. (43), the conductivity σxy = νe2/hsatisfies the Streda formula. Note that this is a generaldynamical result, and, in particular, we do not need toassume a Fermi-Dirac function for the unperturbed dis-tribution. If the unperturbed distribution function de-

scribes the ground state given by n(0)kz = Θ(εF − εk↑) −

Θ(εF − εk↓), then

ν =Φ↑ − Φ↓

4π, (45)

where Φs =∫d2kΘ(εF −εks)Bk are the Berry curvature

fluxes over occupied regions on each band.25 As anothercheck, we show in appendix G that ν indeed gives the cor-rect ground-state current of 2D Dirac massive fermionscomputed from field-theoretical methods.

Let us now consider our results in light of the well-known physics of the Dirac equation. In this case,the adiabatic approximation corresponds to the semi-classical limit, valid in the presence of weak externalfields in which particle/antiparticle pair creation may beneglected.14 Historically, the single-particle interpreta-tion of the Dirac equation showed some puzzling features.The velocity operator given by cσ in 2D has discreteeigenvalues ±c [restoring units here for the elementaryelectron, for which v = c in Eq. (39)], which may seem tocontradict the experimental fact that the electron’s ac-tual velocity is much less than the speed of light. Futher-more, it has noncommuting components and, therefore,does not lend itself to a classical interpretation. Diracfirst resolved this apparent paradox by demonstratingthat the electron trajectory exhibits a trembling mo-tion called the “Zitterbewegung:”26 The electron posi-tion moves with mean (group) velocity pc2/Ep and fastoscillations with frequency of the order of the mass gap∆ = 2mc2/h and amplitude of the order of the Comp-ton wavelength λc = h/mc. However, when the electron’srest-mass frequency and Compton wavelength are beyondthe finite time and spatial resolutions of real experiments,only the mean velocity and position are measured. Our

9

transport equation correctly captures the transport cur-rent due to the mean velocity.

By considering the current of an electron wave packet,one can show that the group velocity pc2/Ep comesfrom the positive-energy band, while the Zitterbewegungcomes from mixing with the negative energy band.27 Fur-thermore, it can be shown that the minimum size of awave packet constructed from only positive-energy statesis (in the nonrelativistic limit) of the order of λc.

28 Inother words, it is necessary to have a mixture of bothpositive- and negative-energy states to localize a wavepacket in a spatial region smaller than λc.

27 Therefore,a semiclassical theory where the electron velocities aregiven by pc2/Ep requires a one-band description, whichis possible only if the size of electronic wave packets ismuch larger than λc, even in the absence of externalfields.

This requirement shows up in our formalism as follows.In the expression for the transverse distribution functionn in Eq. (18), the term ∼ ~∂rlnApl must be small forour approximations to hold. Consider the magnitude oftransverse gauge field in a relativistic expansion in theparticle speed, βp ≡ |vps|/c = pc/Ep = sin θ,

~|Apl | = ~√

sin2 θ(∂pϕ)2 + (∂pθ)2

= ~cos θ√

tan2 θ(∂pϕ)2 + cos2 θ(∂p tan θ)2

=λc2π

cos θ√

1 + cos2 θ =λc2π

√(1− β2

p)(2− β2p)

= λc

[√2

2π+O(β2

p)

]≈ λc , (46)

where we used tan θ = p/mc, (∂pϕ)2 = 1/p2. Thus we

require |np/n| ∼ ~|∂rlnApl/n| ∼ λc/l 1, where l isthe lengthscale for spatial gradients of np. That is, thedistribution function must be smooth on the scale of theCompton wavelength, consistent with the size require-ment on nonrelativistic wave packets. Note that in theultrarelativistic limit, the right-hand side of Eq. (46) ap-proaches the de Broglie wavelength h/p, which then hasto be much smaller than l. This is always required inthe gradient expansion, even in the absence of interbandterms.

The microscopic currents associated with the finite sizeof the electronic wave packet lead to an appearance of themagnetic moment [see Eq. (33)] Mk = EkBk/2. In fact,the anomalous velocity may be understood from the rel-ativistic physics of this magnetic moment, as follows.14

Under an applied E field, in the frame of an electronmoving with velocity vks = skc2/Ek, by the Lorentztransformation, there is a magnetic field given (for smallvelocities) by B(vks) = −vks × E/c. The magnetic en-ergy in the electron rest frame then reads

δεks =~ec2

Mk · vks ×E =~e2sk ·E×Bk , (47)

which in the lab frame may be interpreted as the energyof an effective electric dipole Pks = (~e/c2)vks ×Mk.

The contribution to the group velocity ∂kδεks due to thisextra energy is exactly the anomalous velocity.

Finally, we point out some surprising features in ourband-diagonal transport equation that call for further in-vestigation. Equation (34) (where the second line is pro-portional to nz) shows that the Streda formula does nothold separately on each band because half of the expectedintraband flux goes into interband flux, which seems tosuggest the following phenomenon. Suppose the mag-netic field smoothly falls off to zero in an outside regionfar away from the origin, where there are well-definedenergy eigenstates. Due to the anomalous Hall velocity,particles on the upper (lower) band flow in (out) of thesample. Inside the region with magnetic field, particlesare transferred between bands. If we started with thelower band occupied in the outside region, and slowlyturn on the magnetic field, we expect a pumping effectwhereby some of the particles traveling into the originwould come out on the upper band (as defined by ourWT basis). However, we note that the interband termsin the second and third lines of Eq. (34) that are inducedby spatial inhomegeneities would also be present in thissituation and need to be taken into account. The ques-tions of whether the adiabatic interband fluxes in thepresence of a finite gap can be physically manifested or ifone could contrive a transformation that eliminates suchinterband terms in our transport equation (24) altogetherremain open.

B. Faraday’s law in momentum space

As another instructive example, consider a slowly-varying gap m = m(t) of 2D Dirac fermions, which, forexample, would occur due to a Zeeman splitting in thez direction in topological insulators. For simplicity, weassume here the magnetic field is static. Then the trans-port equation is

(∂t + rs · ∂r + ks · ∂k)nks −nz4

(bB) = 0 , (48)

where B = m/E3k. The time-dependent Berry flux causes

an anomalous velocity [cf. Eq. (22)] δrs = qsFkt ≡ qsEks,where we have defined the momentum-space fictitiouselectric field which in this example is given by

Ek = ∂t cos θ∂kϕ =m

E3k

z× k .

This anomalous velocity is transverse to the cyclotronorbits and gives a radial particle flow in opposite di-rections on the two bands, similar to the previous ex-ample, except that it is peaked at k = m/

√2. This

radial flow causes changes in particle density, deter-mined by the time dependence of the Berry curvature:B = m(k2 − 2m2)/E5

k. See Fig. 3. It may readily beverified the momentum-space, fictitious electromagneticfields satisfies Faraday’s law, ∂tBk + ∂k × Ek = 0 [SeeEq. (E4)].

10

1 2 3

2

1

0

0.5

m2Ek

m

k

m

m3Bk

m

FIG. 3: A plot in dimensionless variables of the fictitious elec-tric field, (m2/m)Ek (solid curve), and Berry flux variation,

(m3/m)Bk (dashed curve), versus k/m.

C. 3D Weyl fermions

As another illumimating example, we consider 3DWeyl fermions, which represent 3D band crossing points.In the presence of electromagnetic fields along the z axis,the Hamiltonian for a right-handed (RH) Weyl fermionis

HW = vσ · [−i∂r − a(r)] . (49)

Here, a = axx + ayy is the vector potential associatedwith a magnetic field b > 0 (Bz < 0 for electrons) alongthe z axis, using the same gauge as in Sec. III A. Theelectric field is given by ez = −∂taz(t).

The Hamiltonian in Eq. (49) has semiclassical disper-sion εk = ±|k| and the Berry curvature of a monopolesource at the origin: Bk = k/k3, which causes amomentum-space particle flux represented by the b · eterm for ks in Eq. (36). As discussed in Sec. II C, to ex-clude the singularity at the origin, we must introducean inner boundary in the momentum-space flux integralin Eq. (29). The particle fluxes through this boundaryproduce an anomalous source of particle density,

∂tρ =

[e2

h2

1

4π

∮S2

dSk ·Bk nkz

]B ·E . (50)

Up to quadratic electromagnetic response, the integralin Eq. (50) depends on the zeroth-order (in E and B)

distribution function n(0)kz on the boundary surface, which

we take to be S2. Assuming a ground-state distributiongiven by the Fermi level at zero energy, so that only the

lower band is filled, n(0)kz = −Θ(εF − εk↓), the integral

gives 4π.47 In units where ~ = 1, this results may beexpressed as ∂tρ = −(e2/4π2) B · E, which is known inparticle physics literature as the (3 + 1)D Adler-Bell-Jackiw anomaly,29 and, strictly speaking, cannot occuralone in nature because it violates charge conservation.

Let us examine the source of this particle flux in thequantum mechanical solution of Eq. (49). In the presenceof a magnetic field along z, the motion in the x−y planeis quantized into Landau levels. By translational invari-ance, the momentum along the magnetic field remainsa good quantum number. The spinor eigenfunctions arethus of the form Ψn(r) = eipzzψpzn(x, y). Since

HWΨn(r) = eipzzHD[m→ pz]ψpzn(x, y) ,

ψpzn(x, y) are the eigenfunctions of the Dirac Hamilto-nian (39) with a mass pz. The square of Hamiltonian(49) determines the eigenvalues (up to a sign), which arethe relativistic Landau levels,30

Enσ = σv√

2|eB|n+ p2z , E0 = vpzsgn(eB) .

Here n = 1, 2, . . . is a Landau-level index and σ = ±1labels the particle/hole branches. The Landau levelsspread out into bands in the z direction, all of whichare gapped, except for the “zero mode” E0, which has alinear, chiral dispersion along the direction of magneticfield. Starting with an equilibrium distribution with theFermi level at zero energy, if we apply an electric fieldalong the z axis by adiabatically turning on az, particlesare transported above the Fermi level by spectral flow,locally populating states near the momentum-space ori-gin. These particles originate at pz = −∞ (outside themomentum-space cutoff).48

The particle production rate may be computed fromthe rate at which the Fermi surface of the chiral branchchanges, pFz = eEz, taking into account the zerothLandau-level degeneracy. The degeneracy of the Landaulevels are given by Φ/Φ0, where Φ = |BzA| is the mag-netic flux (A is the transverse area) and, in units where~ = 1 and c = 1, the flux quantum is Φ0 = 2π/e. Thusthe density of states per area in the transverse directionis eB/2π, and per length along z is 1/2π. Therefore29

∂tρ =eEz2π

e|Bz|2π

= −e2EzBz4π2

. (51)

We may thus match this quantum-mechanical solutionfor ∂tρ with the semiclassical result given in Eq. (50).We stress that our semiclassical analysis here involvesresponse quadratic in electromagnetic fields and is, there-fore, beyond the scope of the wave-packet analysis foundin the literature.7

IV. CONCLUSION

In summary, we have derived a band-diagonal, semi-classical kinetic equation (sKE) for electrons in a two-band, (pseudo)spin-orbit coupled system, which takes theform of a Boltzmann equation [see Eqs. (17) and (19)]for a collisionless plasma. In addition to the correctionsto quasiparticle equations of motion, we find terms pro-portional to the distribution function that we attribute

11

to single-particle motion constrained on a curved statespace. We also find interband couplings that representcoherences due to the nonorthogonal nature of the pro-jected Hilbert spaces. As a check on our formalism, wefind our kinetic equation reproduces the well-known elec-tromagnetic response of 2D Dirac fermions and 3D Weylfermions. For Hamiltonians with less symmetry, morecomplicated inhomogeneous spin-orbit couplings, exactquantum-mechanical solutions are hard to find, while oursKE remains valid and a useful analytical tool.

At first sight, the interband terms in our sKE seempuzzling, given that there are well-defined proceduresfor decoupling multiband Hamiltonian in the presence ofweak external fields.4,5 However, these methods employcomplicated quantum-mechanical transformations on theHilbert space which generally do not constitute a simplechange of basis. Blount5 used a mixed representationwhere operators are specified by both Bloch (or canoni-cal) momentum and position. By successive transforma-tions, that author derives an effective Hamiltonian forBloch (Dirac) electrons in a magnetic field as a functionof kinetic momentum H(k), which is a power series inB. The transformations are functions of k and becomenonunitary in the presence of a magnetic field becauseof the commutator: [ki, kj ] = iεijkb

k. As pointed out bythe author, qualitatively, these transformations amountto a “local” diagonalization of the Hamiltonian in phasespace, similar in spirit to our approach.

For the (3+1)D Dirac equation, one may apply theFoldy-Wouthuysen (FW) transformation,31 to decouplethe positive/negative energy bands in the presence ofweak external fields in a power series of inverse mass1/m, resulting in the nonrelativistic Pauli Hamiltonian.The FW representation has the advantage that the op-erators retain their classical meaning. In the absence ofexternal fields, one can show that the position operatorr satisfies r = ±p/Ep (for the positive/negative energybands), and the orbital and spin angular-momentum op-erators are separately conserved. However, these opera-tors correspond to the original Dirac operators evaluatedat the mean position of the electron, which, as discussedin Sec. III A, is spread out in a region of (at least) thesize of the Compton wavelength λc. In the presence ofexternal fields, in principle, the meaning of the operatorsalso changes and it may be seen explicitly that part ofthe FW transformation becomes nonunitary.

The key features of these methods are consistent withour formalism. Exact diagonalization of spin-orbit cou-pled Hamiltonian H(p) is possible only in the absenceof inhomogenieties. When the bands are gapped, oneintuitively expects that in the presence of smooth space-time inhomogeneities, quasiparticle motion could be con-fined to separate subspaces, which may be called field-modified energy bands.4 One may indeed find represen-tations in which the Hamiltonian is approximately block-diagonalized in the field-modified band space, but at theexpense of altering the Hilbert space with nonunitarytransformation, which may render the subspaces of the

field-modified bands nonorthogonal and change the phys-ical meaning of operators. This caveat is consistent withour explanation of the interband terms in Sec. II B. Incases where the Hilbert space may be specified by contin-uous coordinates, we find it useful to visualize the field-modified bands as submanifolds with possible nontrivialinterband orthorgonality relations.

The Wigner distribution function gives us an exactphase-space representation of nonequilibrium quantumdynamics, where we can treat r,p as real numbers fromthe get-go, allowing us to diagonalize the semiclassicalHamiltonian εp(r, t) in a straightforward manner. Find-ing the exact correspondance between our matrix trans-formation of the distribution function and the aforemen-tioned transformation on the Hilbert space (where r,pare operators) seems to be a nontrivial task, which wewill relegate to future work. Our approach based on thedensity-matrix formalism is quite general, allowing forthe treatment of multicomponent fermions or bosons, andin principle allows for a straightforward incorporation ofelectron interaction effects in the spirit of Fermi-liquidtheory. Lastly, our projection process described belowEq. (17) is valid up to higher orders, and, in particular,with some additional labor, one may carry out the gra-dient expansion to 3rd order, which may warrant furtherstudy.49

Acknowledgments

We are grateful to A. Kovalev and P. Krauss for stimu-lating discussions. This work was supported by the NSFunder Grant No. DMR-0840965 and DARPA.

Appendix A: Gradient expansion

The gradient expansion expresses the WT of a convo-lution of two kernels in terms of gradients of the WT ofthe individual kernels through the following formula.32

Let a(b) be the WT of A(B), then WT of [A⊗B] is,

WT[A⊗B] = a

[exp

i~2

(∂←

r · ∂→

p − ∂←

p · ∂→

r)

]b

= ab+i~2∂ia∂

ib− ~2

8∂i∂ja∂

i∂jb+ . . . ,

(A1)

where . . . indicate higher order terms. For fermions, thegradient expansion is valid when the length scale ξ ofspatial inhomogeneities is much longer than the Fermiwavelength λF ∼ 1/qF , where qF is the Fermi wave vec-tor. One may consider it as an expansion in q/qF , andq ∼ 1/ξ is the characteristic wave vector of the spatialinhomogeneities.

In fact, in the gradient expansion applied to A⊗ B −B ⊗ A in Eq. (3), all even powers vanish if A and B

12

commute because pairs of contracted indices are sym-metric. Therefore, in the scalar (single band) case, theleading quantum mechanical corrections are third orderand O(~2).33 However, when Eq. (3) has nontrivial ma-trix structure, the second order expansion is required forcapturing all semiclassical O(~) terms.

Appendix B: Covariance of the kinetic equation

The transport equation (11) is covariant in the sensethat it remains the same form in an arbitrary spinframe. Specifically, under a local SU(2) transforma-

tion np → U†pnpUp, Eq. (11) represented symbolically

as dn/dt transforms as dn/dt→ U†dn/dtU . The covari-ant derivatives transform similarly, with the appropriatechange in the gauge potentials given by

Din→ U†DinU ,

Ai → U†AiU + iU†∂iU . (B1)

Expressing the matrix gauge potentials as U =exp(iηaτa), the infinitesimal transformation with ηa 1is given by

n→ n− η × n,

Ai → Ai − η ×Ai + ∂iη . (B2)

Evidently, the spin components of the distribution func-tion transforms in the adjoint representation of SU(2).

Appendix C: Integrability condtion

By their definition, the matrix Berry gauge potentialsare pure gauge and so that their SU(2) field strengthvanishes, which provides an integrability condition:

∂iAj − ∂jAi = −Ai ×Aj . (C1)

This is a partial differential equation which relates thethree component of the gauge field. Mathematically, ina simply connected region where Eq. (C1) holds, onecan always find a frame in which the gauge potentialvanishes.34 Contracting indices in Eq. (C1) gives an iden-tity,

2∂iAi = −Ai ×Ai , (C2)

which was used repeatedly to arrive at the final gauge-invariant expression in Eq. (D13).

Appendix D: Decoupling procedure

In this appendix, we show the computations that leadto the anomalous terms in Eq. (19) in projected kinetic

equation. It will be convenient to first express the lon-gitudinal equation in terms of n and nz. The trace andthe τz component of Eq. (11) reads

∂tn−1

2(∂iε∂

in+ ∂i∆∂inz) + F = 0 ,

∂tnz −1

2[∂iε∂

inz + ∂i∆∂in] + Fz = 0 , (D1)

where

F ≡− 1

2

[∂i∆(Ai × z · n) + ∆(z · Ai × ∂in)

+∆Ai · nAzi], (D2)

Fz ≡z · At × n− 1

2∂iε(z ·Ai × n)

+~8

z ·(D2ijε×D2ijn

)(D3)

The vector part of the second covariant derivative, de-fined by DiDj n ≡ ∂i∂jn/2 +D2

ijn · τ , reads

D2ijn =∂i∂jn + ∂iAj × n + Aj × ∂in

+ Ai × ∂jn + Ai × (Aj × n) . (D4)

Let us first simplify the last term of Eq. (D3). The an-tisymmetric part of of the second covariant derivativeEq. (D4) vanishes, as one would expect since the SU(2)field strength is zero. Indeed, the antisymmetric part ofthe last term in (D4) reads

(A[i ×Aj] × n)b =1

2[(Ai ×Aj)× n]b ,

where we define anti-symmetrization by A[iBj] ≡(AiBj − BiAj)/2. Therefore, the commutator of covari-ant derivatives vanishes

[Di, Dj ]n = [∂iAj − ∂jAi + Ai×Aj ]×n ≡ Fij ×n = 0 .(D5)

where we’ve used the identity Eq. (C1). The remain-ing symmetric part of the covariant derivative Eq. (D4)reads,

D(iDj)n = ∂2ijn + ∂(iAj) × n + 2A(i × ∂j)n

+ A(i · n)Aj) − (Ai ·Aj)n , (D6)

where we defined the symmetrization symbol A(iBj) ≡(AiBj + BiAj)/2. Therefore, we will only need to com-pute that quantity

P (2)z ≡ 1

8z ·(D2

(ij)ε×D2(ij)n), (D7)

of which we will only need the terms containing nz, sincethe terms containing n will be third order in gradients.Taking n → nz z, noting that in the rotated frame wealready have ε = ∆z, and that the vector product (D7)above involves only the transverse components of the sec-ond covariant derivative (D6), we find

13

P (2)z =

1

8z ·[

∆∂(iAj) + 2∂(i∆Aj)

]× z−∆A(iAj)

×[

nz∂(iAj) + 2∂(inzA

j)]× z− nzA(iAj)

=

1

8(∂i∆nz −∆∂inz)z · Aj × (∂iAj + ∂jAi −AjGji)

+1

4(∂i∆∂

inzFjj + ∂j∆∂inzFij)

=1

4(∂i∆nz −∆∂inz)∂

jFji

− 1

2∂inz∂

i∆F − 1

4∂inz∂j∆F ij . (D8)

In the third line, we use identity (C2) and (E2). Thisexpression is gauge invariant, as it should be. Next, wewant to solve for n to first order gradients. Consider thetransverse part of the transport Eq. (11) to the first-ordergradient expansion,50

Dtn−∆

~z× n− 1

2

[∂iεD

in + ∂inDiε]

= 0 , (D9)

where the transverse covariant derivatives are defined by

DIn = ∂I n +AzIz× n + nzAI × z ,

Diε = ∆Ai × z . (D10)

In regions where there is a finite gap, ∆p 6= 0, Eq. (D9)

shows that n(0) = 0, so that the lowest nonvanishingorder is n(1). Therefore, to first order in Eq. (D9) we candrop terms ∼ ∂I n, AzI n, then

∆

~z× n(1) = nzAt × z

− 1

2

[∂kε(nzA

k × z) + ∂kn∆Ak × z]. (D11)

Solving for n results in the solution Eq. (18).In approximating Eq. (D9) with Eq. (D11), we have

neglected terms of O(∂tn), O(n ∂pε ·Azr , n ∂rε ·Azp), andO(∂pε · ∂rn, ∂rε · ∂pn), in comparison with the commu-tator term which is O(∆n). Considering the near equi-librium case when n is localized on the Fermi surface,our approximation implies the limit [see also Sec. III Afor additional constraint related to the second term inEq. (18)]

~ω ∆ , ξ ~vF∆

, λF |∂rε| ∆ , (D12)

where vF is the Fermi velocity, ω (ξ) are the charac-teristic frequency (length scale) of the dynamics (inho-mogeneities) of the system. The last condition on en-ergy gradients is a requirement on the size of drivingelectromagnetic fields. The limits above define the adi-abatic approximation. Also, same conditions as abovewith ∆→ EF (λF ξ etc.), for intraband adiabaticity.

Substituting the expression for n in (D11) into theanomalous terms of Eqs. (D2) and (D3), we find

F =1

4F ij(∂iε∂jnz + ∂i∆∂jn)− 1

2Fti∂inz

+1

4(nz∂jε−∆∂jn)∂iFij −

1

2nz∂

iFit

Fz =1

4F ij(∂iε∂jn+ ∂i∆∂jnz)−

1

2Fti∂in

+1

4(nz∂j∆−∆∂jnz)∂

iFij −1

2F∂inz∂i∆ .

(D13)

The anomalous terms in Eq. (19) are given by Fs = (F +sFz)/2.

Appendix E: Bianchi identity

The Bianchi identity for the Berry curvature reads

∂IFJK + ∂JFKI + ∂KFIJ = 0, I 6= J 6= K, (E1)

and follows directly from the definition of Berry cur-vature in terms of gauge fields if they are nonsingular,since ∂[IFJK] = ∂[I∂JAK] ≡ 0. Let IJ = ij be phase-space indices and contracting identity (E1) by multiply-ing

∑ij J

ij , then

∂iF iK + ∂KF = 0 . (E2)

We have used (E2) repeatedly, for example, we trans-formed Eq. (23) by,

∂iFti + ∂iF ik∂kεs = [∂t + ∂iεs∂i]F , (E3)

We have used this identity in the second line of (19).In the following, it will be necessary to introduce dif-

ferential forms in order to use Stokes theorem in a spacewith more than three dimensions.35 The Berry gauge field(connection) is a 1-form A = AIdxI and the Berry cur-vature is a two-form F = 1

2FJKdxJ ∧ dxK , where sum-mation over repeated indices are implied, and the com-ponents of these forms are given in Eq. (15) and the textfollowing Eq. (16). Denoting the exterior derivative byd, we may write F = dA = d(cos θ) ∧ dϕ, which is thearea 2-form representing surface elements on the sphere.The Bianchi identity (E1) states that dF = 0 and rep-resents a set of homogenous Maxwell equations for thephase-space, fictitious electromagnetic fields defined by

Fqiqj ≡ εijlBql , Fqt ≡ Eq ,

where q ∈ r,p is a 3D vector in phase space which mayhave indices in both r and p. In differential forms no-tation, Eq. (E1) are the components of a 3-form, whichmay be expressed in terms of the fictitious electromag-

14

netic fields,

dF =1

2∂IFJK dxI ∧ dxJ ∧ dxK

=∑

q∈r,p

(∂q · Bq) dq1 ∧ dq2 ∧ dq3

+εlmn

2(∂tBq + ∂q × Eq)l dt ∧ dqm ∧ dqn

= 0. (E4)

Here (lmn) runs through three dimensions. The firstterm represent the absence of monopoles and the secondis the phase-space Faraday’s law for the fictitious electro-magnetic fields.

However, near crossing points, (E1) needs to be recon-sidered, since point degeneracies of the Hamiltonian aremonopole sources of Berry flux1 and Eq. (E1) assumesthe absence of such singularities. Consider first the Berryflux over a sphere containing the crossing point in phasespace. Modulo 4π, it is an integer topological invariantknown as the Chern number: Φ =

∫S2 F = 4πN , N ∈

π2(S2) = Z, and represents the winding number of themapping between spherical surfaces from (r,p, t) spaceto spin space defined by the texture, mp(r, t) : S2 → S2,analogous to the skyrmion number in ferromagnets. Us-ing Stokes theorem in the form

∫MdF =

∫∂M

F, with

∂M = S2 being the surface of the sphere M , we have

Φ =

∫M

dF = 4πN . (E5)

If we take the limit of M being infinitesimally small andcontaining a topological defect of the texture at qd in a3D q space, only the first term of dF in Eq. (E4) con-tributes to Eq. (E5), which implies that8

∂q · Bq = ±4πδ3(q− qd) . (E6)

Here we have used the fact that near a crossing point qd,one may make a linear expansion of the gap vector

∆q = [(q− qd) · ∂q]∆q ≡∑i

(qi − qid)ei,

thus the texture has winding number N = ±1, with thesign determined by det(eai ), where eai ≡ ∂i∆

a(qd) are aset of basis vectors defined by the gap gradients. TheWeyl equation in Sec. III C is an example of a monopoletexture.

If, due to time-dependent parameters in the Hamilto-nian, the monopole position changes in time, qd = qd(t),the current associated with the monopole motion willappear as a source term in Faraday’s law. To see this,consider the plane defined by ql = qld(t0) in the 3D qspace. Near t0, we may approximate

ql − qld(t) = (ql − qld(t0)) + qld(t0)(t− t0),

Substituting this expression in the expansion above forthe gap vector, it is evident that there is a monopoletexture located at (qmd , q

nd , t0) in (qm, qn, t) space, and

that the basis vector in the time direction is given byet = qldel. The Berry flux over an infinitesimal surfacecontaining this monopole in (qm, qn, t) space, computedusing Eq. (E5), will have contributions only from thesecond term of dF in Eq. (E4), thus18

[∂tBq+∂q × Eq]l ± 4πεlmn

δ(t− t0)δ(qm − qmd (t0))δ(qn − qnd (t0)) . (E7)

The sign here is determined by sgn[qld det(eai )], and LHSof Eq.(E7) is evaluated at ql = qld(t0). In terms ofmonopole motion, this nonzero winding number of thetexture comes from the fact that, on the ql = qld(t0)plane, the texture ∆(qm, qn, qld(t0)) is a vortex whosepolarity, given by sgn(qld(t0) − qld(t)), switches sign dur-ing the course of time as the monopole passes the planeat t = t0. To express the RHS of Eq. (E7) in terms ofthe monopole current, we note that the linear expansionqld(t) near t0 may be inverted to yield a mapping t(qld),which we may use to change variables: δ(t(qld) − t0) =|qld|δ(qld(t) − qld(t0)), and that on account of the deltafunction δ(t− t0), we may evaluated the LHS of Eq.(E7)at ql = qld(t). Then,

[∂tBq+∂q × Eq](t0) = sgn[det(eai )]4πqdδ3(q− qmd (t0)) .

(E8)

Similarly, we note that the 2D Dirac equation (SeeSec.III B) with a time-dependent mass m(t) ∝ t wouldgive a source term in Faraday’s law.

Lastly, the magnetic flux through an open 2D hyper-surface S may be converted to a contour integral of thevector potential around the boundary ∂M , provided thatwe use the appropriate gauge,

Φ =

∫S

F =

∮∂S

dlIA±I = −∫ 1

0

[cos θ(l)∓ 1]∂Iϕ(l)dlI .

(E9)

The integral depends only on the path [θ(l), ϕ(l)] tracedout by the texture in spin space space (which canbe many closed loops) as the phase-space contour ∂Mparametrized by l ∈ [0, 1] is traversed. Except when thecone angle θ = π/2, this flux is generally not quantized.

Appendix F: Berry curvature in general coordinates

Under a general phase-space coordinate transforma-tion xi → xi(x), the energy and the distribution func-tion transforms as a scalar ε(x) = ε(x(x)) ≡ ε(x), whilederivatives transforms as a covariant phase-space vector

∂

∂xj→ Λij

∂

∂xi, Λij ≡

∂xi

∂xj. (F1)

Since the Berry curvature is a product of derivatives, ittransforms as a second rank covariant tensor

Fij = (ΛT FΛ)ij ,

15

where the bar denotes the Berry curvature with deriva-tives w.r.t. the coordinates x. In particular, when trans-forming to the kinetic momentum for electromagneticperturbations, the spatial derivatives transforms as (31),resulting in the following transformations of the Berrycurvature

Frit = ∂ia · ∂ta×Bk Frirj = ∂ia · ∂ja×Bk

Fpit = −(∂ta×Bk)i Fpirj = −(∂ja×Bk)i

Fpipj = Fkikj ≡ εijl(Bk)l Fripj = (∂ia×Bk)j , (F2)

while F transforms as

F =∑i

(∂a

∂ri×Bk

)i

=∑ijl

εijl∂aj

∂ri(Bk)l = b ·Bk .

(F3)The simplest example of the transformations in Eq. (F2)is in the presence of an electric field in the vector po-tential gauge. Then Ep = Fpt = e × Bk representsthe anomalous Hall velocity. This may be contrastedwith the scalar potential gauge, in which the kinetic andcanonical momentum are the same.

In the presence of spatiotemporal inhomogeneities inthe spin textures other than gauge potentials, the Berrycurvatures will have space-time dependence not relatedto kinetic momentum, and there will be additional termsin the equations of motion Eq. (35) and Eq. (36). The ad-ditional terms come simply from the Berry curvatures inEq. (22) with space-time indices, with p→ k. For exam-ple, if the texture is due to a ferromagnetic exchange field,the Berry curvatures Frr and Frt are fictitious electro-magnetic fields that are known to mediate spin-transfertorques in itinerant ferromagnets.36

Appendix G: Parity anomaly

For the example in Sec. III A, it is instructive to com-pute ν when the Fermi level lies in the gap, which givesthe well known half-integer conductivity of 2D Diracfermions.30 Although the Berry flux in Eq. (45) is con-centrated near the origin, up to a sign they dependonly on the texture far away from the origin. To em-phasize the topological nature of the number ν, we ap-ply Stokes theorem to convert integrals to Fermi sur-face Berry phases which determines the solid angle sub-tended by the texture, taking care to use the appro-priate gauge, Ak = −(cos θk − sgn(m))∂kϕk. Thenν = −(1/4π)

∮Ak · dk = −sgn(m)/2, in agreement

with the formula well known in particle physics litera-ture for anomalous charge current associated with theparity anomaly of the massive 2D Dirac equation,21,37

〈jµ〉 = sgn(m)e2

8πεµαβFαβ . (G1)

In the formula above, Fαβ is the electromagnetic fieldstrength tensor, Fµν = ∂µAν − ∂νAµ, Aµ = (Φ,−A),

F0i = Ei, Fij = −εijkBk , the expectation value is takenin the vacuum state, which in the single particle pictureis the “Dirac sea,” with all negative energy levels occu-pied and thus corresponds to Fermi level in the gap. Oneshould restore units in (G1) by setting 2π = h to com-pare with our result for ν. The dependence on sgn(m)is a signature of the infrared singularity at k = 0 whenm = 0, as discussed in Sec. II C. The 2D Dirac massterm explicitly breaks parity, defined by inversion of onespatial coordinate. In field theory, even when m = 0 inthe Hamiltonian, gauge-invariant regularization of the ul-traviolet divergence may violate parity (for example, inthe Pauli-Villars regularization), leading to the parity-violating anomalous current.38

In field-theoretical methods, one calculates the anoma-lous current (G1) by the coupling of fermionic vacuumenergy to an external vector potential, represented bya fermionic functional determinant. While the resultis precise, its physical origin is somewhat mysterious.The semiclassical approach gives an intuitive but rigor-ous single-particle picture, in which the vacuum currentsimply represents the flow of quasiparticles.

Appendix H: Comparison with wave-packet theory

According to wave-packet analysis,7 the correction tothe single particle energy due to inhomogeneous pertur-bations is given by,

δεs = Im 〈∂rlus|Hc(r, t)− εs(r, t)|∂plus〉 , (H1)

where Hc(r, t) is the local Hamiltonian, with the positionr evaluated at the center of the wave packet, and εs(r, t)is the band s local eigenvalue. We emphasize that inthe wavepacket approach, r is a c-number parameter of aquantum Hamiltonian, whose complete set of eigenstatesis known. The local Hamiltonian is the zeroth order termin an expansion of a Hamiltonian with smooth inhomo-geneous perturbations. The perturbations to first orderin gradients has the form δH ∼ (r − r) · ∂rH and it’sexpectation value in a wave packet state gives the en-ergy correction in first order perturbation theory. Fromthe semiclassical transport equation approach, the localHamiltonian is equivilant to the semiclassical quasiparti-cle energy (7),

Hc = εp(r, t)

To compare with formulae from wave-packet analysis, wefirst define the local eigenkets by a rotation of the labframe spin eigenkets pointing in the ±z direction:

|ups(r, t)〉 ≡ Up(r, t)|z; s〉 .

They satisfy the local eigenvalue equation:

[Hc(r, t)− εs(r, t)]|ups(r, t)〉 = 0 , (H2)

16

where εs is given by (10). For brevity, we will denote“lab” frame spinor basis by |s〉, s = ±1. The matrixelements of the gauge fields are given by

[Ai · τ ]ss′ ≡ Ass′i = i〈s|U†∂iU |s′〉 = i〈us|∂ius′〉 , (H3)

where we’ve omitted phase-space subscripts for brevity.From (H1), we find

δεs =1

2i〈s|∂iU†[∆mp · τ − s∆/2]∂iU |s〉

=1

2i〈s|∂iU†U [∆τz − s∆/2]U†∂iU |s〉

=1

2i

[Ai

∆

2

(1− s 0

0 −1− s

)Ai]ss

=∆

2i

∑s′ 6=s

s′

2Ass

′

i As′si

=∆F

4(H4)

where F ≡ Fii/2. Note that the energy shift is indepen-dent of s.

Appendix I: Relation of gauge fields to velocityoperator

We note that the off-diagonal gauge field is related tothe phase-space velocity operator, vi = ∂iH. Differen-tiating the eigenvalue equation (H2) and projecting on|us′〉 for s 6= s′, we find an equation for the off-diagonalgauge fields [cf. (H3)]:

Ass′

i = −i 〈us|∂iH|us′〉εs − εs′

. (I1)

Consider going into the frame of the electron movingin phase space, then the phase-space dependence of thetexture appears to the electron as dynamics of a localphase-space magnetic field. Therefore, to make heuristiccomparison with the quantum adiabatic theorem,39 wetake ∂i → ∂t in Eq. (I1). Then the theorem shows thatthe amplitude for transitions to other states is given byAss

′

i /∆ in agreement with Eq. (18).

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(Prentice Hall, 2000), 2nd ed.40 Since J ij is antisymmetric, interchanging the positions of

contracted raised and lower indices introduces an extrasign:

AiBi = AiJ

ijBj = −AiJjiBj = −AiBi .

17

In particular, AiAi = 0. Note that since J

↔2 = −1, the

inverse is J↔−1 = −J

↔.

41 GIJ ≡ ReA+I A−J is related to the field strength by

FIJ = εIJ√

(detG)IJ ,

where we consider the metric tensor GIJ as a matrix in the(xI , xJ) plane and defined its determinant by (detG)IJ ≡GIIGJJ −G2IJ . This tensor appears in intermediate, gauge-dependent steps in our computations, but not in the final,gauge-invariant results.

42 The first term proportional to the electric field was firstderived in the original paper of Karplus and Luttinger2 onthe anomalous Hall effect in ferromagnets.

43 The appearance of interband couplings at the second-ordergradient expansion is not inconsistent with the quantumadiabatic theorem.39 The latter allows transitions to otherstates to occur with a probability quadratic in the adi-abatic parameter, which are space-time gradients in ourproblem. See appendix I.

44 One should distinguish this from the case when ∆p = 0but the texture is topologically trivial.

45 Furthemore, the Bianchi identity which was used in thederivation of Eq. (19) will acquire source terms at these

points, cf. appendix E.46 In the wave-packet formalism of Ref. 7, the Hamiltonian

is expanded to linear order in gradients, and therefore tolinear order in electromagnetic fields.

47 Strictly speaking, the ground state is a Fermi sphere inthe p space, and the corresponding distribution is slightlyshifted by a in the k space.

48 Left-handed (LH) Weyl fermions [Eq. (49) with a minussign] has a zero mode with the opposite chirality so thatparticles leave the system under spectral flow. Therefore,Weyl fermions may occur in pairs with opposite chiral-ity, so that particles are transferred between the crossingpoints in momentum space, conserving particle number.This creates a nonequilibrium distribution in momentumspace, which would relax back to equilibrium through scat-tering between crossing points.29

49 At this order, second-order derivatives of the diagonal dis-tribution function will appear, and the transport equationcannot be interpreted in terms of continuous quasiparticlemotion in phase space.

50 The off-diagonal, second order term in Eq. (11) are neces-

sary only if we want to solve for n(2).