Triangle Anomaly

8/13/2019 Triangle Anomaly

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1 Introduction

The triangle anomaly is a quantum effect responsible for the violation of chiral symmetryof charged massless fermions in the presence of a P- and CP-odd conguration of thebackground gauge eld [1, 2]. A massless Dirac fermion, such as a (nearly) massless quarkin QCD, possesses a left- or right-handed chirality. These states are described by the left-and right-handed Weyl spinors giving rise to the U (1)L and U (1)R chiral symmetries. Thefermions with different chiralities contribute to the triangle anomaly with opposite signs as a result, the anomaly is absent for the vector current J V J L + J R , and the electriccharge is conserved. On the other hand, for the axial current J A J L + J R it leads to

J A = e2

22 E B , (1.1)

where e is the charge of the fermion; the sum over different fermion species is implicithere. It will be useful for our purposes to view the above equation in terms of two separateanomaly equations for the left- and right-handed chiral currents:

J L,R = e2

42 E B . (1.2)

Triangle anomaly plays an important role in the chiral dynamics of low energy QCD,explaining in particular the 0 decay. Recently, it has become clear that theanomaly also affects the transport and hydrodynamical macroscopic behavior; much of this work was motivated by the applications to quark-gluon plasma in magnetic eld [3, 4]produced in heavy ion collisions. The novel transport phenomena induced by the anomalyinclude the chiral magnetic effect [ 3, 4, 5, 6], the chiral separation effect [7, 8], and thechiral vortical effect [6, 9, 10, 11, 12]. Closely related phenomena have been discussed inthe physics of neutrinos [ 9], conductors with mirror isomer symmetry [ 13, 14], primordialelectroweak plasma [15] and quantum wires [ 16]. Note that the roles of axial anomalyand the topology of background gauge eld are crucial for the existence of the chiralmagnetic current; without the anomaly, this current has to vanish in thermal equilibrium,

in contrast to some of the early arguments.For systems that contain charged chiral fermions, the chiral magnetic (separation)

effects dictate the existence of vector (axial) charge currents along the direction of anexternal magnetic eld in the presence of the axial (vector) chemical potential. In the

This is the covariant form of anomaly.

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case of the chiral vortical effect, the role of magnetic eld is played by the vorticity of theuid. Due to the topological nature of triangle anomaly, these new transport phenomenahave been shown to be robust and not modied by interactions even in the strong couplinglimit [17, 18, 19], with a possible exception [20, 21] of the temperature-dependent

T 2

term [22, 23] in the chiral vortical conductivity.

The persistence of the anomalous charge transport at strong coupling suggests thepossibility of hydrodynamical formulation, and such formulation was given in Ref [12],see also [24, 25, 26, 27]. The absence of contributions to the local entropy productionrate from the (T-even) anomalous terms has been used to constrain the hydrodynamicalformulation [28]. In heavy ion collisions, the chiral magnetic and chiral vortical effects canpotentially be separated by measuring the electric charge and baryon number asymmetries[29, 30]. The experimental evidence for the chiral magnetic effect in heavy ion collisions

has been presented by RHIC [31, 32, 33] and LHC [34, 35, 36] experiments.Recently, it has been realized that the triangle anomalies and the chiral magnetic effect

can be realized also in a condensed matter system a (3+1)-dimensional Weyl semi-metal[37, 38, 39]. The existence of substances intermediate between metals and dielectricswith the point touchings of the valence and conduction bands in the Brillouin zone wasanticipated long time ago [40]. In the vicinity of the point touching, the dispersionrelation of the quasiparticles is approximately linear, as described by the HamiltonianH = vF k, where vF is the Fermi velocity of the quasi-particle, k is the momentum inthe rst Brillouin zone, and are the Pauli matrices. This Hamiltonian describes masslessparticles with positive or negative (depending on the sign) chiralities, e.g. neutrinos,and the corresponding wave equation is known as the Weyl equation hence the nameWeyl semimetal [37]. Weyl semimetals are closely related to 2D graphene [41], and to thetopological insulators [ 42, 43] 3D materials with a gapped bulk and a surface supportingchiral excitations. Specic realizations of Weyl semimetals have been proposed, includingdoped silver chalcogenides Ag2+ Se and Ag2+ T e [44], pyrochlore irridates A2Ir 2O7 [37],and a multilayer heterostructure composed of identical thin lms of a magnetically doped3D topological insulator, separated by ordinary-insulator spacer layers [38].

The triangle anomaly affects Weyl semi-metals [ 45] because the fermionic quasi-particlesaround the Weyl point in momentum space behave like relativistic chiral fermions witha velocity that plays the role of an effective speed of light [ 37, 38, 39, 46]. However, asobserved in Ref.[45], the crux of triangle anomaly in Weyl semimetals is the presence of ahedgehog, or a magnetic monopole, in momentum space [ 43] that leads to the emergence

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of Berrys phase [47] (see also Ref.[48]). The Berrys phase, and the anomaly, thus canaffect the systems that are not truly relativistic.

If the total ux of Berrys phase is an integer k, it induces a non-conservation of thecharged fermion current through the triangle anomaly:

J = ke2

42 E B , (1.3)

where e is the charge of the quasi-particles. The analogy to ( 1.2) is clear; each Weyl pointwith a monopole charge k is similar to a relativistic chiral species with chirality dependenton the sign of k. The total number of electrons in the system should be conserved, andthe sum of monopole charges k over all Weyl points must thus be zero.

As the newly discovered transport phenomena mentioned above rely only on the tri-angle anomaly relation ( 1.3), they should be present also in Weyl semi-metals. This isimportant as it would allow to test the transport phenomena originating from the trian-gle anomaly experimentally in a controlled environment. Our purpose in this paper is toprovide a few examples that may have potential experimental or even practical impor-tance; see Refs.[49, 50] for previous suggestions and Ref.[51] for a discussion of chiralelectronics enabled by Weyl semi-metals.

The new transport phenomena on which we base the subsequent discussion can besummarized as follows. Each Weyl point with k total ux of Berrys phase contributes tothe electromagnetic current through the relation [4, 9, 12, 22, 23]

J = ke2

42 B +

ke42

2 + 2

3 T 2 , (1.4)

where e is the electric charge of the fermionic quasi-particles of the Weyl point, isthe chemical potential of the Fermi surface measured from the Weyl point, and T isthe temperature. The magnetic eld B and the vorticity = 12 v of the velocityeld v should be computed in the local rest frame. In addition, it has been arguedthat the transport phenomena originating from triangle anomaly do not lead to entropyproduction; they are non-dissipative [28]. In Refs.[28, 52, 53] the non-dissipative nature of

anomalous currents and the resulting time-reversal invariance of anomalous conductivitieshave been exploited to extend the rst order anomalous hydrodynamics [ 12] to secondorder in derivatives and to higher dimensional cases. The entropy current originatingfrom the anomaly is given by [12]

S = ke 182

2

T +

T 24

B + k 1122

3

T +

T 12

, (1.5)

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where the B and should be computed in the local rest frame.

2 Chiral magnetic and chiral vortical effects in Weyl

semi-metalsPrior to discussing possible experimental consequences of the anomaly-induced transport,let us make a few cautionary remarks on the application of ( 1.4) to real Weyl semi-metalswhere the electron energy spectrum differs from that of free relativistic chiral fermions.Our discussion is motivated by a recent claim in Ref.[ 54] that one can have a net chiralmagnetic effect even in global equilibrium if the energies of Weyl points are shifted in anasymmetric way by introducing an inversion-symmetry breaking term.

Since the energy of each Weyl point, say E i , is now shifted away from the Fermienergy F , it seems naively that each Weyl point has an effective chiral chemical potentiali = F E i measured from the origin (at zero temperature) even in global equilibriumwhen the bands are lled up to the Fermi energy F . This case considered in Ref.[54] isillustrated in Figure 1. If one naively applies (1.4) using these chiral chemical potentialsi , a net chiral magnetic effect would result as claimed in Ref. [54].

However, the existence of the chiral magnetic current in global equilibrium (or vac-uum) at zero temperature would raise a conceptual issue. Suppose one applies a parallelelectric eld in addition to the magnetic eld; because of the chiral magnetic current,

there would be net energy input (or output),dP dt

= J E E B , (2.6)where the sign can be made negative by choosing E appropriately that is, one couldextract energy out of the system. However, it should be impossible to extract energy froma state in global equilibrium (vacuum) at zero temperature since the state is already aminimal energy state by denition; there is simply no energy available. This argumentdictates the absence of the chiral magnetic current in the equilibrium with xed chiral

chemical potentials . This applies not only to Weyl semimetals, but also to neutrinos inmagnetic eld [9], and to conductors with mirror-isomeric structure [ 13] in all of thesecases, the chiral magnetic current has been eventually found to vanish in equilibrium[9, 14].

The chiral magnetic effect discussed in [4] is based on the assumption that the chiralchemical potentials are not xed, i.e. the system is not in the minimal energy state. The

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pz

EFE2

E

E1

E0

Figure 1: A schematic illustration of the energy levels of a Weyl semimetal considered inRef.[54]. The shaded region is lled up by electrons at zero temperature.

chiral magnetic current is thus powered by the energy stored in the difference of the Fermienergies of the left- and right-handed chiral fermions [ 55]. If this difference is xed (forexample, by the geometry of the Weyl semimetal heterostructure, as proposed in [ 54]),the chiral magnetic current cannot exist.

In this section, we will establish the conditions necessary to realize the chiral magneticeffect in a Weyl semimetal. Our results indicate that ( 1.4) is almost correct in realisticWeyl semi-metal systems, but with a few subtle modications that we will discuss indetail. With these modications, we indeed nd that the system in Figure 1 has no netchiral magnetic effect (CME) in global equilibrium. However, the CME can neverthelessbe realized if we generate the initial chiral chemical potential dynamically, e.g. by chirallycharging the Weyl semimetal in parallel electric and magnetic elds.

For our purposes it will be sufficient to use a kinetic approach developed in Refs.[ 45,56, 57, 58, 59]; specically, we will rely on the formulation of Ref. [58]. Each ith Weylpoint of Berry monopole charge ki is assumed to be situated at the energy E i and the

momentum ki , so that the dispersion relation around that point is approximately linear:

E ( p) = E i vi | p ki | . (2.7)It is important to recognize that the states with positive energy (upward from E i , thebranch with the positive sign in ( 2.7)) feel the Berry phase of charge ki while the stateswith negative energy (downward from E i ) feel the opposite charge ki . (It is clear from

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the example of k = 1 with the Hamiltonian H = E i 1 + vi ( p ki )).Note that we have not introduced any notion of holes or anti-particles; it is an

important point that deviates from the usual discussion of relativistic chiral fermions.In the context of condensed matter physics, it is clearly natural as the fermions in thelled Dirac sea are physical electrons. In the relativistic chiral fermion description,one could treat the Dirac sea fermions as electrons with negative energy, but the realdifference comes in the fact that one should then subtract the vacuum (all negative statesare occupied) contribution. This is because the only observable physical effect shouldoriginate from a difference from the vacuum contribution. By the identity

f (|E |) = 1 f (|E |) , (2.8)where f (E ) is the distribution of particles at all energy E

(

, +

) and f (E ), E > 0

is the anti-particle distribution, one can show that this gives the equivalent results to theanti-particle framework. In our condensed matter situation, we can no longer a priori know the results for the vacuum (it is precisely this question that we are addressingnow), so treating both positive and negative branches as electrons will allow to avoidconfusion.

For the positive branch, the quasi-electrons are described by the action [45, 58],

I + = dt (p x + A (x ) x E i vi |p k i | a (p ) p ) , (2.9)where we have a Berry monopole

b p a = ki (p k i )2|p k i |3

. (2.10)

Following the steps in Ref.[ 58], the equations of motion become

G+ x = vi(p k i )

|p k i | +

ki vi2

B

|p k i |2 ,

G+ p = vi

(p k i ) B

|p

k

i | , (2.11)

where G+ = (1 + b B ) is the phase-space measure, and the anomaly-induced current is j+ = d

3 p(2)3

f (E ) Gx = B ki42 f (E ) vi|p k i |2 |p k i |

2d|p k i |= B

ki42 f (E )vi d|p k i| = B ki42

E if (E )dE , (2.12)

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where in the last equality, we use E = E i + vi |p k i |. The distribution function f (E ) inglobal equilibrium is

f (E ) = 1

1 + e (E F ) =

1

E

log 1 + e (E F ) , (2.13)

but the above formula is applicable to more general out-of-equilibrium situations, too [58].In the E integration, the upper limit in our situation is not innite and has a physicalcutoff. However, this cutoff is not of signicance at low enough temperature as thesehigh energy states are rarely occupied, f (E ) 1. The above result is a reproduction of previous computations in Refs. [45, 58]. We emphasize that although we assume a speciclinear dispersion relation to derive the result, the nal expression in ( 2.12) can be shownto be universal and is not sensitive to the detailed shape of the dispersion relation.

The small region around the origin where G = (1 + bB )

0 is the quantum region

[58] where the kinetic approach breaks down. Its size scales linearly in B so it gives riseto a small correction (if any) to ( 2.12) of higher orders in B in the small B limit that weassume.

The interesting part is the negative energy branch (note again that we do not haveanti-particles or holes). The action describing these quasiparticles is

I = dt (p x + A (x ) x E i + vi |p k i |+ a (p ) p ) , (2.14)where we have changed signs in two places compared to ( 2.9); the momentum dependentpart of the energy according to the negative branch and the Berry phase are reversedas discussed before, with the same denition of b as in (2.10). Similar steps lead to theequations of motion,

G x = vi (p k i )|p k i |

+ kivi

2B

|p k i |2 ,

G p = vi(p k i ) B

|p k i | , (2.15)

where G = (1

b

B ). Note that the term in the rst equation that is linear in B

which leads to the anomaly-induced current has the same sign as in the positive branch.The contribution to the current from the negative branch then reads as

j = d3 p

(2)3f (E ) G x = B

ki42 f (E ) vi|p k i |2 |p k i |

2d|p k i |= B

ki42 f (E )vi d|p k i | = B ki42

E i

E 0f (E )dE , (2.16)

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where in the last equation, we use E = E i vi |p k i | and E 0 is the physical cutoff of the bottom of the lled Dirac sea that one can see in Figure 1. At the energy E 0,it is intuitively clear that the states from k = 1 Weyl point meet the states from theother k =

1 Weyl point, and become non-chiral massive Dirac states for which the

anomaly-induced transport disappears. The cutoff is therefore physical. Since f (E ) isorder 1 around E 0, this cutoff is important. It is natural to assign the common cutoff E 0to the two Weyl points of k = 1. The distribution f (E ) in equilibrium is the same asin (2.13) that applies to all energies, irrespective of branches.

Expressions ( 2.16) with (2.12) have the same form and differ only by the distributionfunction f (E ). To see that this is a correct result, let us check it for the case of relativisticchiral fermions. The negative branch in that case is the Dirac sea of negative energywhich should be lled in the vacuum state. Using the identity

f (|E |) = 1 f (|E |) , (2.17)where f is the anti-particle distribution, and subtracting 1 from the above preciselycorresponds to subtracting the vacuum contribution since f vac (|E |) = 1, the vacuum-subtracted negative branch contribution reads as

B ki42

0

(f (|E |))dE = B

ki42

0f (E )dE , (2.18)

which is the usual negative contribution from anti-particles.

In our condensed matter case, there is nothing to subtract since the Dirac sea isphysical; the vacuum contribution is very important to keep. Summing ( 2.16) and(2.12), the total result simplies as

j = j + + j = B ki42

E 0f (E )dE , (2.19)

which is a single integral of f (E ) from the bottom of the lled sea to the high energycutoff. Note that there is no dependence left on the energy E i of the Weyl point sincef (E ) in equilibrium does not depend on it. Using ( 2.13) we can evaluate the expression

above in equilibrium as

j i = B ki42

1

log 1 + e (E 0 F ) , (2.20)

with the zero temperature limit

j i (T = 0) = B ki42

(F E 0) . (2.21)8


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To summarize this discussion, the effective chiral chemical potential that enters theformula (1.4) should be measured from the bottom of the lled sea E 0, not from the originof Weyl points. It is clear from the above result that after summing over all Weyl points,the net chiral magnetic current vanishes in global equilibrium, since

i ki = 0. This is

true at any temperature. To have a net chiral magnetic current, one has to asymmetricallydistort the distribution f (E ) around each Weyl point. The above formula also indicatesthat the chiral magnetic conductivity depends non-trivially on the temperature. This isunderstandable since we have a physical cutoff ( E 0) for the bottom of the lled sea.

We can repeat similar steps for the derivation of the chiral vortical effect, following thesuggestion in Ref.[58] that the rotation can be included as a Coriolis force in the rotatingframe; the equation for p becomes

p = 2vi |p k i| x = 2 (E E i ) x , (2.22)where is for positive and negative branches respectively, and the last equation is trueirrespective of branches. We restrict our attention to the case where and k i are parallelso that Weyl points remain static in the rotating frame. The above implies that themagnetic eld B can be replaced by [58]

B 2 (E E i ) , (2.23)which leads to the nal result

j i = ki22

E 0dE (E E i ) f (E ) , (2.24)

with the zero temperature limit

j i = ki42

(F E 0) (F + E 0 2E i ) . (2.25)After summing over all Weyl points, the net current is

j total =

i

ki E i

22 (F

E 0) , (2.26)

which may not be zero even in equilibrium. In this case, the necessary energy when weapply a parallel electric eld can be provided by the external rotation so the argumentthat we gave at the beginning of the section does not apply.

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Rotation

Figure 2: Anomaly cooling of Weyl semimetal by rotation and electric eld. Thesemimetal could be either cooled or heated depending on the relative orientation of electriceld and the angular momentum.

3 Anomaly cooling of Weyl semi-metal

Quantum anomaly in Weyl semimetals leads to an interesting phenomenon - it appearsthat one may use the combination of rotation and external electric eld to cool thismaterial. To see this, let us use the set-up explained in Figure 2. We rotate the Weylsemi-metal in a two dimensional plane, say ( x1, x2), with angular velocity , so that

= x3. From (1.4), this induces the current along x3 direction as

J 3 = e42 i

ki 2i + 2

3 T 2 , (3.27)

where we sum over all Weyl points of the system labeled by index i. We then apply anexternal electric eld along x3 direction E = E x3, and the total power the system absorbswill be given by

d

E dt = J E = eE 42 i k

i 2i + 2

3 T 2 , (3.28)

which can be either positive or negative depending on the sign of E . If it is negative thesystem will be cooled down.

More intuitively, we can understand the cooling in terms of the entropy current ( 1.5).We will now show that there exists an entropy ow directed radially outwards from the

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system S = S r r , and since no net entropy is generated by the anomaly-induced transport,the entropy of any bounded region around the center should decrease and the systemshould indeed cool down. For a nite-size system with radius R, the entropy extractedfrom the cooled central region will accumulate around the boundary r = R, causing atemperature gradient between the center and the boundary and a compensating heatow will develop. The system will eventually reach an equilibrium with a stationarytemperature gradient along the radial direction.

Having a constant = x3 means that the uid at a position ( x1, x2) = r r has atangential velocity (counter-clockwise) v = rt, where r and t are radial and tangentialunit vectors respectively. In the local rest frame of that uid cell, the uid experiences aradial magnetic eld

B = E v = Er r , (3.29)via Lorentz transformation of eld strengths. By ( 1.5) this gives a radial entropy ow

S = ei

ki 182

2iT

+ T 24

B = eEr i ki 182

2iT

+ T 24

r , (3.30)

with divergence given by

S =

1r

r (rS r ) = 2eE i ki 182

2iT

+ T 24

= eE 42T i

ki 2i + 2

3 T 2 .

(3.31)Since no net entropy should be produced, we have dS dt + S = 0 where S is the entropydensity of the uid; this tells us that the local entropy density changes as

dS dt

= S = eE 42T i

ki 2i + 2

3 T 2 . (3.32)

The relation dE = T dS precisely reproduces the previous power formula ( 3.28) from theabove.

Using the fact that i ki = 0 for Weyl semi-metals, the T 2 term in the cooling rate

drops in the nal result, and one needs an asymmetric distribution of i s to get a non-vanishing effect. This can be achieved by shifting the energy of Weyl points, as discussed

previously. Note also that applying the electric eld would induce the ordinary current E where is the conductivity, which leads to a dissipative heating of the system

dE dt

= E 2 , (3.33)

which is quadratic in E . For the anomaly cooling, which is linear in E , to dominate overthis dissipative heating, one therefore needs a smaller E and a larger .

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Burgers solitary wave of charge

Rotation

Figure 3: Anomaly induced solitary wave of charge governed by the Burgers equation.

4 Charge transport in rotating hot Weyl semi-metaland the Burgers equation

Let us consider the long wavelength collective charge transport in a rotating Weyl semimetalas in Figure 3. Our starting point is again ( 1.4) with a xed angular momentum = x3

J 3 = ke

422 +

2

3 T 2 . (4.34)

Although the total current is the sum over all Weyl points, one can treat contributionsfrom each Weyl point independently, to a good approximation; anomaly-induced collectivecharge transports from each Weyl point behave independently of other Weyl points. Inmore explicit terms, one can introduce U (1) i charge symmetry for each ith Weyl pointseparately, and these U (1) i s are approximately conserved. Each U (1) i has its own triangleanomaly with coefficient ki , and its anomaly-induced current J i is given by (1.4) withk ki and i . Note that this is a non-trivial statement because ordinarily the quasi-particles from different Weyl points interact with each other, and this interaction affectstheir normal (non-anomalous) transport properties. What protects the independencyof anomaly-induced transports between different Weyl points is the triangle anomaly of U (1) iU (1) j U (1)k which is not zero only if i = j = k.

An analogy to a massless quark in QCD may be helpful: there we have two Weylpoints, one left-handed ( k = 1) and the other right-handed ( k = 1). The left-handed

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fermions interact strongly with the right-handed fermions in general, and their normaltransport properties are not independent at all. Yet, their anomaly-induced transportsare independent:

J L,R = e42 L,R B 142 2L,R +

2

3 T 2 . (4.35)

This feature is dictated by the two separate triangle anomalies of U (1)3L and U (1)3R withoutcrossing. This independency of collective charge transports of two chiralities is the essenceof chiral magnetic waves proposed in Ref.[ 60, 61]. Weak residual interactions betweenthese two chiral collective transports (analogs of sphalerons in the case of QCD) lead todiffusive chiral magnetic waves.

Let us therefore restrict ourselves to a single Weyl point, omitting i, with anomaly-

induced current at xed is

J 3 = ke42

2 + 2

3 T 2 D 3 + O 2 , (4.36)

where we include the usual diffusion term in the derivative expansion up to rst orderwith diffusion coefficient D . We focus on the regime T 1 of a hot Weyl semi-metal.In this case, is approximately proportional to the charge density by

1 + O 3 , (4.37)with susceptibility , so that the ( 4.36) becomes

J 3 = ke42

22 + 2

3 T 2 D 3 + O 2, 3 . (4.38)

Using this in the charge conservation equation t + 3J 3 = 0 leads to

t + C x D 2x = 0 , x x3 , (4.39)with C = ke2 2 2 . This is the Burgers equation, a prototypical integrable partial differential

equation in 1+1 dimensions. It is completely soluble given an initial data, and manyanalytic solutions are known. Shockwave-type solutions are not acceptable in our problembecause of our small amplitude approximation ( 4.37), but solitary traveling waves withnite amplitudes are relevant. One can tune the constant C by varying .

The total electromagnetic current is given by the sum over each Weyl point contribu-tions i J i . In practice one may perturb the system by injecting a net electromagnetic

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charge, and it may be hard to excite charge uctuations of each U (1) i individually; onegenerally excites a superposition of all ith charges. Since each ith uctuations are prop-agating independently, one would observe a splitting of charge transport. This is veryinteresting as it would allow to study the properties of each Weyl point separately; chargetransport is a prism for probing different Weyl points in a Weyl semi-metal.

5 Quantized vortices in a Weyl superuid

Let us now move to a slightly different topic and consider a superuid system which cou-ples to Weyl fermions as in [62, 63]. This type of conguration was studied in [64, 65] asa phenomenological description of quark matter at nite isospin chemical potential. Fur-thermore, it was recently observed that Weyl fermion excitations can be realized in ultra-

cold fermionic gases in the presence of Zeeman eld and Rashba spin-orbit coupling[ 66, 67].We start with the following Lagrangian:

L= i (D + i ) + 12|D |

2 (5.40)

Here = | |ei is the bosonic eld which condenses and forms the superuid and D = ieA . The phase is related with the superuid velocity as follows:

u = 1m

. (5.41)

Here m is the scale that xes the magnitude of u . The superuid has vortex line cong-urations which are of the form

= f (x)x1 + ix2

r (5.42)

where x = ( x1, x2) denotes the transverse plane where the uid rotates and r = x21 + x22.The vortex is centered at the origin x1 = x2 = 0 and is elongated along a line in x3direction. The function f (r ) is constant at large values of r, f () (T )/m andvanishes at the center r = 0. The constant value is associated with the superuid density(T ) which decreases with temperature and vanishes above the critical temperature T c.The distance r0, in which f (r ) changes from 0 to its constant value is the radius of the coreof vortex. We will assume that r 0 is smaller than any macroscopic scale in our problem.

The identication ( 5.41) leads to the Onsager-Feynman quantization

( ).d l = 2n , n = 0, 1, 2,... (5.43)14


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The integer n is the winding number of the vortex and is thus a topological invariant.From now on we will focus on a single vortex with unit vorticity n = 1 since a vortex witha multiple winding number is unstable against decaying into multiple vortices with unitwinding.

Assuming that the distance between the two vortices is much larger than the coreradius ( r 12 >> r 0) and neglecting the contribution from the cores, the interaction energybetween the two vortices with the same vorticity is given by

E K = 2L d2x u1.u2 L 2m2 ln(r 12/r 0) (5.44)where L is the size of the sample along the vortex line. The chemical potential for fermionscan be realized as a shift in the superuid phase

+ t (5.45)In [62] it was shown that the existence of a chemical potential induces a charge currentthrough the triangle anomaly ,

J 3 = 2

(5.46)

This current is localized inside the vortex core which can be treated point-like in x1x2plane. The existence of the anomalous current will modify the interaction between thevortices since the two currents repel each other. The energy of this current-current inter-

action has the same form as the vortex-vortex interactionE cc = L

J 31 J 322

ln(r12r 0

) = 2

83 ln(r 12/r 0) (5.47)

As a result, the total interaction energy between the two vortices is modied as

E L

=2m2

+ 2

83ln(

r12r 0

) (5.48)

This anomalous contribution to the vortex dynamics has interesting consequences.For example, consider a very thin sample, such as a thin lm which can be treatedeffectively as two dimensional. In this case, above a certain temperature T BKT creationof vortices becomes energetically favorable. This is the famous Berezinskii-Kosterlitz-Thouless (BKT) transition [ 68]. We now show that the existence of the anomalous current

Note that we are considering Weyl fermions. Therefore there is no distinction between axial currentand charge current since there is only a single anomalous U(1) symmetry in our problem.

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modies the BKT transition. Following the standard argument, let us calculate the freeenergy of a single vortex conguration

F = E TS (5.49)where S is the entropy. If we assume that the vortices are distributed in the two dimen-sional plane of the thin lm of area R2 much larger than the vortex size, we can haveR2/r 20 possible congurations and the entropy is simply S = ln( R2/r 20). The energy E has two terms, the kinetic term due to the velocity eld and the magnetic energy inducedby the anomalous current:

E = 12

L d2x( u2 + B 2) = L R

r 0rdr

r 2m2

+ 2

42r 2 Lm2

+ L2

43ln(R/r 0)

(5.50)

We again neglected the contribution of the core which is valid in the thermodynamic limitR >> r 0. The free energy in this limit is

F =Lm2

+ L2

43 2T ln(R/r 0) (5.51)The phase transition occurs when the free energy changes sign and when it is negative itis preferable to create vortices. The phase transition temperature is

T BKT = L2m2

(T BKT ) + L2

83 (5.52)

The rst term is the well known expression for the temperature of the BKT phase tran-sition, whereas the second term is the modication due to the anomaly. It is possible toconsider the full dynamics by taking into account the screening of the vortices and solvingthe gap equation as in [68]. In the thermodynamic limit the vortex interaction ( 5.48) willbe made stronger by the additional anomalous term with the same spatial dependence therefore the effect of anomaly can be included in a straightforward way.

6 Summary

Let us briey summarize our results:

The Chiral Magnetic Effect (CME) does not appear in Weyl semimetals in equi-librium with a xed chiral chemical potential. However, the CME can be realizedwhen the initial chiral chemical potentials are induced dynamically, e.g. by parallel

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electric and magnetic elds. The chiral magnetic current is given by ( 2.20) and by(2.21) for the case of zero temperature.

Contrary to the chiral magnetic effect, the chiral vortical effect in Weyl semimetalscan exist even in equilibrium, with xed chiral chemical potentials. The chiralvortical current is given by ( 2.24) and by ( 2.26) for the case of zero temperature.

The chiral vortical effect in a rotating Weyl semimetal leads to a very interestingphenomenon the anomaly cooling, when the temperature of the material with anon-zero chiral chemical potential can be reduced as a result of rotation. The localentropy density changes according to ( 3.32).

The anomaly-induced transport of charge in rotating hot (with chemical potentialmuch smaller than temperature) Weyl semimetals is described by the integrableBurgers equation ( 4.39) that admits solitary wave solutions.

The anomaly induces a new term in the interaction energy of quantized vortices ina superuid coupled to Weyl fermions. This shifts the energy of the BKT phasetransition according to ( 5.52).

Quantum anomalies are among the most subtle and beautiful effects in relativistic eldtheory. Weyl semimetals open an intriguing possibility to study the effects of quantumanomalies experimentally, in a controlled setting. Such studies are of fundamental interest,and may lead to practical applications as well.

Acknowledgement

This work was supported by the U.S. Department of Energy under Contracts No.DE-FG-88ER40388 and DE-AC02-98CH10886 (GB and DK). We thank L. Levitov, M.Stephanov, Y. Yin and I. Zahed for useful discussions.

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