Anomalous transport model with axial magnetic fields

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Physics Letters B

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Anomalous transport model with axial magnetic fields

Karl Landsteiner

^a

, Yan Liu

^b,∗

aInstitutodeFísicaTeóricaUAM/CSIC,C/NicolásCabrera13-15,UniversidadAutónomadeMadrid,Cantoblanco,28049Madrid,Spain bDepartmentofSpaceScience,andInternationalResearchInstituteofMultidisciplinaryScience,BeihangUniversity,Beijing100191,China

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received12November2017

Receivedinrevisedform16March2018 Accepted30April2018

Availableonline4May2018 Editor:N.Lambert

Thetransportpropertiesofmasslessfermionsin3+1 spacetimedimensionhavebeeninthefocusof recenttheoreticalandexperimentalresearch.Newtransportpropertiesappearasconsequencesofchiral anomalies. Themostprominentisthegenerationofacurrentinamagneticfield,the so-calledchiral magnetic effect leading to an enhancement ofthe electric conductivity (negativemagnetoresistivity).

We studytheanalogouseffectforaxialmagneticfieldsthatcouplewithoppositesigns tofermionsof differentchirality.Weemphasizelocalchargeconservationandstudytheinducedmagneto-conductivities proportionaltoanelectricfieldandagradientintemperature.Wefindthatthemagnetoconductivityis enhancedwhereasthemagneto-thermoelectricconductivityisdiminished.Asasideresultweinterpret ananomalouscontributiontotheentropycurrentasageneralizedthermalHalleffect.

©^{2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introductionandmotivation

Chiralanomalies[1,2] andthespecifictransportphenomenain- ducedbythemsuchasthechiralmagneticandthechiralvortical effectshavebeenextensivelydiscussedintherecentyears(see[3, 4] forreviews).

InatheoryofmasslessDiracfermionsthevectorcurrent J μ=

¯

γ

^μandaxialcurrent Jμ

5 = ¯

γ

5

γ

^μcanbedefined.Insucha theorythechiralmagneticeffect(CME)describesthegenerationof anelectriccurrentinamagneticfieldinthepresence ofanaxial chemicalpotential



J

= μ

₅

2

π

^2B

^{ ,}

⁽¹⁾

where

μ

5 is the axial chemical potential conjugate to the axial chargeoperator Q₅=

d³x¯

γ

5

γ

⁰.

This formulahas to be interpreted withcare. At first sight it predicts the generation of a current in equilibrium. It has been pointedouthoweverthatsuchanequilibriumcurrentisforbidden bythe so-calledBloch theorem.InrelationtotheCME thistheo- remhasfirst beeninvokedina condensedmatter context in[5].

A recent discussion ofthe Bloch theorem has beengiven in [6].

Thetheoremcanbeformulatedas

*

Correspondingauthor.

E-mailaddresses:karl.landsteiner@csic.es(K. Landsteiner),yanliu@buaa.edu.cn (Y. Liu).



d³x



J

(

x

) =

⁰

,

(2)

in thermal equilibrium. Seemingly thisis violated by eq. (1) for a homogeneous magnetic field. The important point emphasized in[6] isthattheBlochtheoremisvalidonlyforexactlyconserved currents.Thisallowstoresolvethetensionbetweeneq. (1) andthe Bloch theorem.Morepreciselyeq. (1) holdsonlyfortheso-called covariant version of the current. This covariant current is not a trulyconservedcurrentbutratherhastheanomaly

∂

_μJ^μ

=

¹

8

π

²



^μνρλF_μνF_ρ⁵_λ

,

(3)

whereonealsointroducesaaxialfield A⁵μ assourceforinsertions oftheaxialcurrent Jμ

5.Similarlythecovariantversionoftheaxial anomalyis

∂

_μJ^μ₅

=

¹ 16

π

²



^μνρλ



F_μνF_ρλ

+

^F_μν^{5 F5}_ρλ



.

(4)

Inquantumfieldtheorythecurrentsarecompositeoperatorsand havetoberegularized.Thisregularizationintroducescertainambi- guitiesthat havetobe fixedbydemanding certainclassicalprop- erties ofthe currents to holdon thequantum level. Oneway to fix theseambiguities isto define J μ and Jμ

5 tobe invariant ob- jects underboth vector- andaxial-typegaugetransformations[7].

The disadvantage ofthisdefinition isthat it doesnot resultin a conserved vector like currentbut ratherleads to the anomaly in https://doi.org/10.1016/j.physletb.2018.04.068

0370-2693/©^{2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

eq. (3).Ontheotherhandonecaninsistonthevectorlikecurrent tobe exactly conserved ∂_μJ^μ=^{0. The relationbetweenthe two} definitionsofcurrentsis

J

^μ

=

^Jμ

−

¹

4

π

²



^μνρλA⁵_νF_ρλ

.

(5)

Dueto axialanomalythe axialvector Jμ

5 isneverconserved and thereforeitssource A⁵μ cannotbeinterpretedasatruegaugefield.

ThereforetheChern–Simonscurrentin(5) isaphysicalcurrentin acompletelyanalogouswayasthe Chern–Simonscurrentappear- inginthequantumHalleffect.Thisresolvesthetensionbetween the chiral magnetic effect and the Bloch theorem in the follow- ingmanner.Thermalequilibriumisdefinedbythegrandcanonical ensemblewithdensitymatrixexp(−(^H−

μ

5Q5)/T).Thisisequiv- alentto considering the theory in the backgroundof a temporal componentoftheaxialfield A⁵₀=

μ

5.Nowthechiralmagneticef- fectintheexactlyconservedcurrentJ^{μ takestheform[8]}

J = μ

5

2

π

^2B

^{ −}

A⁵₀

2

π

^2B

^{ ,}

⁽⁶⁾

wherethe second termstems fromtheChern–Simons currentin eq. (5). Since in strict equilibrium A⁵₀=

μ

5 this shows that the chiral magnetic effect for the conserved current (5) vanishes as demandedby the Blochtheorem. Theimportance ofdefining the coservedcurrenthas alsobeen discussed inchiral kinetic theory in[9].

Ontheotherhandthecloselyrelatedchiralseparationeffect



_J5

= μ

2

π

²

^

^B

^,

⁽⁷⁾

doesnot sufferanysuch correction.Since theaxial currentisal- waysaffectedbyananomalythereisnocontradictiontotheBloch theoremaspointedoutin[6].

There ishowevera third related effectifone allows foraxial magnetic fields, B₅= ∇ × ^A5. This is a magnetic field that cou- pleswithoppositesignstofermionsofdifferentchirality.Theaxial magneticeffecttakestheform



J

= J = μ

2

π

²

^

^B5

^.

⁽⁸⁾

Formallyitdescribesthegenerationofavector-likecurrentinthe backgroundofanaxialmagneticfieldatfinite(vector-like)chem- icalpotential. Notethat the formulaholds forboththe covariant and the conserved form of the currents. Therefore this formula seemstobeinmuchgreatertensionwiththeBlochtheoremthan thechiralmagneticeffect.One mightdismiss thistensiononthe groundsthat sofaratafundamentallevelnoaxialfieldsseemto existinnature. However, ithas beenargued that suchfields can appearinthe effectivedescription oftheelectronics ofadvanced materials, the so-called Weyl semimetals [10–13]. A low energy field theoretical description of the electronics of these materials givenbytheDiracequation

γ

^μ

(

i D_μ

+

^bμ

γ

5

) =

⁰

.

(9) Here Dμ isthe usual covariant derivative andthe parameter bμ enters justlikethe field A⁵μ couplingto the axialcurrent. It has beenarguedthatstrainingsuchmaterialscanleadtospatialvari- ationoftheparameter bμ andinconsequenceto theappearance ofeffectiveaxial magneticfieldsineq. (9).The reasonwhythere isnocontradictiontotheBlochtheoreminthiscaseisasfollows.

Theparameterbμ existsonlywithin thematerialandnecessarily vanishesoutside.Iffordefinitenessweassumetheaxialmagnetic

fieldtobedirectedalongthez directionandwecomputethetotal axialfluxatthroughasurfaceatsomefixedz=^z0



5

=



dxdy B⁵_z

(

x

,

y

,

z₀

) =



∂

d



S

· 

^b

=

⁰

,

(10)

since one canalways take theboundary ofthe surfaceto lie en- tirely outsidethe material where _b=^{0. Thereforethe axial ana-} logueofthechiralmagneticeffect(8) cannotinduceanetcurrent andthisresolvesthetensionwiththeBlochtheoremsincenonet currentcanbegenerated[14,15].

We will take these considerations as motivation to study electro- and thermo-magnetotransport in the background of ax- ial magneticfields underthe assumptionthat the Bloch theorem is implementedby a vanishing netaxial magnetic flux (10). This implies that the net equilibrium electriccurrent vanishes but as wewillseeupon applyingan electricfield (orequivalentlyagra- dient inchemical potential)and a temperaturegradient leads to anomalyinducednetcontributionstothecurrents.

2. Anomaloustransport

Westudyasimplemodelofanomaloustransportwithcoupled energyandcharge transport.Thismeansthatincontrasttoafull hydrodynamicmodelweassumethatnosignificantcollectiveflow parametrizedbyaflowvelocitydevelops.¹ Notonlyisthisasim- plermodelallowingtostudytheeffectsofanomaliesontransport it might also be more directly relevant to systems where elastic scatteringonimpuritiesimpedesthebuildupofcollectiveflow.

Wedevelopnowaformaltransportmodelbasedontheanoma- louscontinuityequations

˙

 + ∇ · 

J_

= 

E

· 

J

,

(11)

˙

ρ + ∇ · 

J

=

c



E

· 

B

,

(12)

where



istheenergydensityand_{J is}theenergycurrent.Charge conservationisaffectedbyananomalywithanomalycoefficientc.

Therighthandsideofequation(11) quantifiestheenergyinjected into the systemby an electricfield (Joule heating) whereas (12) describesthe(covariant)anomaly.Sofarthisisnotspecifictoax- ial magnetic fields but rather relies only on the presence of an anomalyinthecurrent J μ= (

ρ

,J).

To discuss transport we write down constitutive relations for

_{J ,}J andtakeasthermodynamicforcesthegradientsinthether- modynamicpotentialsandexternalelectricandmagneticfields,

 

J_



J



=

^L

·

 ∇

₁

T

E T

− ∇

_μ

T



+

 σ ˆ

B

σ

B



B

 .

(13)

The matrix L encodesresponse due to gradients in chemical potential andtemperature. { ˆ

σ

B,

σ

B} ^{describeresponse dueto the} magnetic field. In principle we could also allow an independent responseduetotheelectricfield.Inouransatzwehavethusan- ticipated that positivity of entropy production is not compatible withsuchadditionaltermsintheconstitutiverelations.

1 Thisdoesnotmeanthatthevelocityorthevariationofthevelocityiszero, justthatitcannotbedeterminedbytheconservedequations.Ourtransportmodel cannotbeobtainedfromhydrodynamicsbysettingtheflowvelocitiestozero.Hy- drodynamicflow(i.e.nonvanishing velocity)appearsalready atzerothorderin derivativesandthisimposesconstraintsonthefirstordertransportcoefficientsthat canappearintheconstitutiverelations[16].Sinceforstrongmomentumrelaxation flowisabsentsuchrelationsarenotpresent.Thismodelhassimilaritytothetreat- mentinthetheoryforincoherentmetalin2+1D[17].

Thetransportcoefficientsareconstrainedbythesecondlawof thermodynamics. Using the thermodynamic relation T ds=^d



₊

μ

d

ρ

asguidelinewedefinetheentropycurrent[18] as



_Js

=

¹ T



_J

− μ

T



_J

+ η

B_B

 .

(14)

Uptothetermsdependingonthemagneticfieldthisisthestan- dardansatzforcoupledenergyandchargetransport[19].

Following[18,20] weimposethelocalformofthesecond law thermodynamics

˙

s

+ ∇ · 

^Js

≥

⁰

.

(15)

UsingT˙s= ˙



₊

μ ρ

˙ thisleadsto 1

T

 ∂ 

∂

t

+ ∇ · 

^J



− μ

T

 ∂ ρ

∂

t

+ ∇ · 

^J

 + ∇



1 T



· 

^J

− ∇  μ

T

 · 

^J

+ ∇ η

B

· 

^B

+ η

B

∇ · 

^B

≥

⁰

.

(16)

Weassumeabsenceofmagneticmonopolesandthusthelastterm vanishes. Using the constitutive relations we find theconstraints det(L)≥^{0 and L}11≥^{0 and L}22≥^{0.Positivityoftheentropypro-} ductionalsoassuresthattheelectricfielddoesnotgiverisetoad- ditionalresponse notalreadycontainedinL. Entropyisproduced onlybythesymmetricpartofthematrixL.Forthemagneticcon- ductivities one finds a set of one algebraic and two differential equations

σ

_B

₋

c

μ ₌

0

,

(17)

σ

B

+ ∂ η

B

∂ γ

ρ

=

⁰

,

(18)

ˆ σ

B

+ ∂ η

_B

∂ γ

_

⁼

⁰

^,

⁽¹⁹⁾

with

γ

=¹/T and

γ

ρ= −

μ

/T . These equations are the coeffi- cientsofthe terms(E· ^B), ( ∇

γ

_{· }B) and( ∇

γ

_{ρ· }B).These terms canbe either positiveor negative andthereforetheir coefficients mustvanishtoguaranteethelocalformofthesecondlawofther- modynamics.Sincethereisnofurtherdimensionfulparameter

η

B must also fulfill

γ

∂

η

B/∂

γ

= −

η

B as it has to have dimension one, where in our conventions (

μ

,T) have dimension one. The magneticconductivitiesarealmostcompletelydetermined

σ

B

=

c

μ , σ ˆ

B

=

c

μ

²

2

+

cgT²

, η

B

=

c

μ

²

2T

+

cgT

.

(20) Up toambiguitiesarising dueto framechoice theseare basically thesameresultsasinhydrodynamics[18,20,21].

The priori undetermined integration constant cg is relatedto (mixed) gravitational anomalies [22–30]. In holography it was alsoshownrecentlythat therelationto the(mixed) gravitational anomaly is not modified by momentum relaxation in [31]. The intuition that dissipationless transport should not be affected by momentum relaxationtogether withthe results of[31] and [21]

(the case of weak momentum relaxation) we take as evidence that cg=^{0 also inthe caseofstrong momentumrelaxation and} thatit isrelatedtothepresence of(possiblyglobal) gravitational anomalies. For theories containing only spin 1/2 particles and holographictheoriesthisrelationiscg=³²

π

²λwhereλistheco- efficientofthe gravitationalcontributionto theanomaly ∂_{μ J}^μ= λ



^μνρλRα_βμν R^βαμν . Asingle Weylfermion has λ= ±₇₆₈¹_π2 and cg= ±¹/24 with thesign depending on thechirality. Inthe fol- lowingweassumecg=^{0 toberelatedtothemixedaxialgravita-} tionalanomaly asinthecasewithoutmomentum relaxationand

study its implicationsfor thermo-electric transport inaxial mag- neticfields.

Using the results forthe anomalous transport coefficients

σ

B, ˆ

σ

B and

η

B theentropycurrentcanbewrittenas



J_s

= (

¹

/

T

,− μ /

T

) ·

^L

·

 ∇(

¹

/

T

)

E T

− ∇

_μ

T



+

^2cgT



B

.

(21)

Naively one might have expected that the anomalous transport does not contribute to the entropy current. It turns out how- ever that thetemperaturedependence encoding thegravitational anomaliesdoescontributetoentropycurrent.Thishasbeenprevi- ouslyobservedin[21,32].

The previous considerations are general andassume only the presenceofananomalyinthechargecurrent.Wecannowspecial- izetothecaseoftheaxialmagneticfield.Inthiscasethecharge conservationtakestheform

˙

ρ + ∇ · 

^J

=

^Nf

2

π

²

^(

^E

^{· }

^B5

^{+ }

^E5

^{· }

^B

^{) ,}

⁽²²⁾

where Nf is the number of Dirac fermions. Rather than an anomaly in this case the right hand side should be interpreted asthedivergenceoftheChern–Simonscurrentineq. (5).Sincewe are mostlyconcerned withtheeffectsofaxialmagneticfieldswe willset E₅= ^B=^{0 inthefollowing.Inthiscasethe}conservation equations areprecisely asinthe generalcasebefore andwe can take overthepreviousresultsby simplyreplacing B withB5 and settingc=₂^N_π^f2 andc_g=^Nf/12.

Nowwewanttogiveaninterpretationfortheanomalouscon- tribution totheentropycurrentineq. (21).We consideran axial magneticfieldconfigurationoftheform

B



5

(

x

) = ˆ

^ez

¯

5

 δ(

x

) − δ(

^x

−

^L

)

.

(23)

AccordingtoourassumptionofcompatibilitywithBloch’stheorem thetotalaxialmagneticfluxalongthez directionvanishesbutthe regions of positive andnegative fluxesare well separated which for simplicity we model by delta-functions distribution localized in x=^{0 and x}=^{L but spread out in the} y direction. The first thing to notice is that according to (21) there is an anomalous entropycurrentlocalizedatthelocationsofaxialmagneticflux.If there isa temperaturegradient suchthat T(x=⁰)=^T+ δ^{T and} T(x=^L)=^{T anetentropycurrentflowsalongthe}z-direction

δ

I_s

=



dx



J_s

=

^2cg

(δ

T

)

5^e

ˆ

z

.

(24) This current flows in a direction orthogonal to the temperature gradient. Heat is not a thermodynamic state variable still it can bedefinedasδQ =^TδS andinananalogouswaywecandefinea heatcurrentasδIQ =^TδIs.Thisleadstothenetheatcurrent

δ

I_Q

=

^2cgT

δ

T

¯

₅e

ˆ

_z

.

(25)

WeinterpretthisasanomalousthermalHalleffect.Inthiswaythe anomalous contributiontotheentropycurrentin(21) canbe un- derstood asageneralizationoftheanomalous thermalHalleffect.

Previous discussions of the relation between thermal Hall effect andgravitationalanomalies are [33,34].Let usalsonote that the very concept of heat current can be questioned on the grounds that heat is not a state variable [35]. In the context of anoma- loustransporttherecertainlyarisesthequestionifinthecommon definitionofheat current JQ = J−

μ

J the current J shouldbe taken to be the covariant orthe conserved current. Defining the heatcurrentasδ J_Q =^Tδ J_s resolvesthisissue.

2.1.Inducedconductivities

Letus nowcome to the main subject: the linearresponse of thissystemtoatemperaturegradientandanexternalelectricfield both alignedwith the axial magnetic field. The continuity equa- tions(11),(12) together withtheconstitutive relations(13) form a dynamical system that allows to compute current and charge distributions givensome initial andboundary conditions.The ef- fective response to an applied electric field and a temperature gradientcanbecomputedbysolvingtheseequations.

Before studying the axial magnetic field caseof interest it is worth to briefly recall how the chiral magnetic effect leads to negativemagneto-resistivity [36–38].Oneassumesahomogeneous magneticfieldandaparallelelectricfield.Axialchargeisnotsub- jectto an exact conservationlaw andthus it isnaturalto intro- duceanaxialchargerelaxationtime

τ

5.Non-conservation ofaxial chargeisprovidede.g.byamasstermintheDiracequationorby inter-valleyscatteringthecontext ofWeyl semimetals.The effec- tiveaxialcharge(non-)conservationisthen

˙

ρ

5

=

^cE

 · 

^B

−

¹

τ

5

ρ

5

.

(26)

We note that if an external electricfield isabsent butinstead a gradient of the chemical potential is induced the current has a non-vanishinggradient ∇ · ^J5=^c∇

μ

· ^{B which leads toeneffec-} tivelyequivalentequationforthetimedevelopmentofaxialcharge byreplacingE→ − ∇

μ

.Axialchargeisbuiltupuntilasteadystate isreachedwithδ

ρ

5=

τ

5cE· ^{B.The axialchargecanberelatedto} theaxialchemicalpotentialvia

χ

5δ

μ

5= δ

ρ

5where

χ

5istheaxial susceptibility.CombiningOhmicandchiralmagneticcurrentsleads totheenhancedcurrent



J

= σ ^

E

+ τ

5

c²

(

E

· 

^B

) χ

5



B

.

(27)

Forinfiniteaxialchargerelaxationtimetheanomalyinducedmag- netoconductivityisformallyinfiniteandthismightbereferred to aschiralmagnetic superconductivity [39]. Innature fermions are howevermassiveandeffectivechiralfermionsinmaterialssuchas Weylsemimetalsdonotpreservetherechiralityatallenergiesdue tothecompactnatureof theBrillouinzone.

In the case of the axial magnetic field the role of the axial chemicalpotentialisplayedbythe(electric)chemicalpotential

μ

. Electric charge is an exactly conserved quantity due to electro- magnetic gauge invariance. Therefore it is not possible to intro- ducearelaxationtime forelectricchargewithout violatinggauge invariance.Ifitwere possible thento engineer homogeneousax- ialmagneticfieldsan analogousargumentwouldleadnecessarily to infinite axial magneto-conductivity. As we have argued how- everin the introduction the assumption ofsuch a homogeneous axialmagnetic field isby itselfinconsistent withthe Blochtheo- rem,whichbyitselfisaconsequenceofgaugeinvariance[6].Thus weare naturally lead tostudyinduced electro- andthermo-axial magnetoconductivity underthe constraintofvanishing net axial magneticflux. Thismakes theproblemmore complicatedasdif- fusionfromregionswhere chargeisaccumulatedto regionswith charge outflow has to be taken into account. It is this diffusion processthatcanleadtoastationarystateandfiniteinducedaxial magneto-conductivities.

As external drivingforces we assume a homogeneous electric field anda temperature gradient pointingin the z direction. We also assume an axial magnetic field directed along the z direc- tionbutinhomogeneousinthe(x,y)planeandwithzeronetflux 5=

dxdy B₅(x,y)=^{0.Thedynamicalvariablesarethechemical}

potential

μ

andthetemperatureT .Weallowthesystemtoadjust totheexternal forcesbydevelopingnon-trivialprofilesofchemi- calpotentialandtemperatureinthe(x,y)planearoundaconstant backgroundvalue.Thusouransatzis



B₅

=

^B5

(

x

,

y

)ˆ

^ez

, 

E

=

^Ee

ˆ

z

,

(28)

μ = μ

0

+ δ μ (

x

,

y

) ,

T

=

^T0

+ δ

^T

(

x

,

y

) +

^z

∇

zT

.

(29) Theresponseinδ

μ

andδT toE and∇^{T isnowcalculatedinlinear} approximation.

Sincetheaxialmagneticfieldisnotuniforminthe(x,y)plane the system will react to the local charge inflow induced by the anomalous Hall and axial magnetic effects by building up diffu- sioncurrents.Eventuallyastationarystateisreached.Thisstation- arystate can be obtainedfromtheconstitutive relationsandthe conservation equations by dropping the time derivative. We fur- thermoreassumethematrixL tobespatiallyisotropic.Using(11), (12) the constitutive relations(13) withtheanomalous transport coefficients(20) we findthatthefluctuationsδT and δ

μ

haveto fulfillasystemofPoissonequations

L

·

Y



_⊥

 δ

T

(

x_⊥

) δ μ (

x_⊥

)



=

 −

^2cgT₀ c

μ

0

0 c



·

 ∇

zT E



B5

(

x_⊥

) .

(30)

Here_⊥ isthetwo dimensionalLaplaceoperator(_⊥= ∂x²+ ∂²y) andY=_T¹2

0

−^{1 0}

μ

0 −^T0



isthetransformationmatrixrelatingthe thermodynamicforcesδ(1/T)andδ(−

μ

/T)tothefluctuationsδT , δ

μ

. Once the fluctuations are determined they can be plugged into the anomalouspart of constitutive relations (13) to find the anomalyinducedcontributiontothecurrents



J^z_ J^z



anom

= −

^B5

(

x_⊥

)

u

(

x_⊥

) ·

 ∇

^z

(

_T¹

)

E

T

− ∇

^z

(

^μ_T

)



,

(31)

withtheconductivitymatrix

 =



2cgT0 c

μ

0

0 c



· (

^L

·

^Y

)

⁻¹

·

 −

2cgT0 c

μ

0

0 c



·



T₀² 0

−

^T0

μ −

^T0



(32)

andthesolutiontothePoissonequation_⊥u(x_⊥)=^B5(x_⊥),i.e.

u

(

x_⊥

) =



dx _⊥G

(

x_⊥

−

^x _⊥

)

B₅

(

x _⊥

) .

(33) Wehavewritten theinducedconductivitymatrixasactingon the naturally definedthermodynamic forces. Thishas the advan- tagethattheOnsagerreciprocityrelationsareautomaticallysatis- fied,i.e.issymmetric,



₁₁

=

¹ det

(

L

)

^T

2



(

L₂₂

(

2c_gT²

+

^c

μ

²

)

²

+

^c2

μ

²L₁₁

)

−

^2c

μ (

2c_gT²

+

^c

μ

²

)

L₁₂



,

(34)



22

=

¹ det

(

L

)

^c

2T²



L₁₁

−

²

μ

L₁₂

+ μ

²L₂₂



,

(35)



₁₂

= 

21

=

¹ det

(

L

)

^cT

2



2c_gT²

( μ

L₂₂

−

^L12

) +

^c

μ (

L₁₁

−

²

μ

L₁₂

+ μ

²L₂₂

)



.

(36)

Using a+^b≥²√

ab for a,b≥^{0 andthe fact L}11≥^{0, L}22≥^0, det(L)≥^{0 oneshowsthat} 11and22 arepositive.Furthermore thetotalcurrentisproportionaltotheexpression

−



d²x_⊥d²x _⊥B5

(

x_⊥

)

u

(

x _⊥

) =



d²q

(

2

π )

²

B

˜

5

( −

^q

) ˜

B5

(

q

)

q² (37)

whichforarealfunction B₅(x_⊥)ispositivedefinite. Thusthere- sponsematrix inthenetcurrentdescribed by (32) hasthesame propertiesasL sinceitsdeterminantisalsopositive

det

() =

¹ det

(

L

)

^4c

2c²_gT⁸

.

(38)

Theelectricandthermoelectric conductivityis definedas J=

σ

E−

α

_∇T . The thermoelectric conductivity

α

is non-vanishing only becauseof the contribution ofthe mixed axial-gravitational anomaly,

σ ₌ σ

0

−

^{u B}5



22

T

,

(39)

α ₌ α

₀

^

1

+

^{u B}5

1

det

(

L

)

^2ccgT

4



,

(40)

with

σ

0=^L22/T and

α

0= (^L12−

μ

L₂₂)/T². Measuring therefore thetotal currentinduced by a temperaturegradient inthe back- groundofanaxial magneticfield isan experimental signatureof themixedaxial-gravitationalanomaly.Notehoweverthattheelec- tricconductivityisenhancedwhereas thethermoelectric conduc- tivitygetsdiminished. Thisisincontrasttotheanomaly induced thermoelectricconductivityinausualmagneticfield[40–43].

2.2. Example

Finallywewouldliketodiscussasimpleexampledemonstrat- ing the finiteness of the total induced current. We assume the periodicaxialmagneticfieldconfigurationoftheform



B₅

= ˆ

^ezB

¯

₅sin

(

2

π

x

/

L

) .

(41)

Integratingoveraperiodthenetfluxvanishes.Thesolutiontothe Poissonequationisnow

u

= − ¯

^B5

L²

4

π

^2sin

⁽

²

π

x

/

L

) .

(42)

Asboundaryconditionswehaveimposedthatnochemicalpoten- tialisinducedoverone period.Now thenetcurrentdensityover oneperiodofoscillationisproportionalto

−

¹ L



L 0

dxu

(

x

)

B_5,z

(

x

) = ( ¯

^B5

)

^{2 L}

2

8

π

²

^.

⁽⁴³⁾

The gradient inthis field configurationis proportional to the in- verse of the period L. The current density is therefore inversely proportional to the square of the field gradient as expected and diverges in the limit of homogeneous field L→ ∞^{. In thislimit} diffusion is not effective andsince there is no relaxation of the exactly conserved electric charge we end up again with infinite conductivities.

3. Discussion

We have developed a simple model of coupled energy and charge transport for chiral fermions in the background of axial magneticfields.Ourstudywasmotivatedbyconsiderationsbased on compatibility with the Bloch theorem that forbids net cur- rents in thermalequilibrium. Inorder to circumvent this we as- sumedaxialmagneticfield configurationswithvanishingnetflux suchthatinequilibriumtheintegratedtotalcurrentvanishes.The

anomaloustransportmodelwasconstructeddemandingapositive definite entropyproduction.Evenwithoutassuming fullhydrody- namics,i.e.assumingthatnosignificantcollectiveflowcandevelop wefoundthatanomaliesinducechiralmagneticchargeandenergy currents. The form of the chiral magnetic transport coefficients contain a prioriundeterminedintegration constant depending on thetemperaturewhichcanberelatedtothepresencemixedgauge perturbative andglobalgravitationalanomalies.As previouslyob- served in [21,32] the entropy current contains somewhat unex- pectedly an anomalous term. We gave a physical interpretation relatingittoageneralizedformofthethermalHalleffect,i.e.the generationofaheatcurrentperpendiculartoatemperaturegradi- ent.

Then we studied electro- and thermo-magneto conductivities.

We found that the assumption of vanishing net axial magnetic fluxactivatesthediffusiontermsintheconstitutiverelationslead- ing to finite induced conductivities. Despitethe fact there is no net magnetic flux a net electriccurrent is induced either by an external electric field or by a temperature gradient. The result- ing netaxialmagnetoconductivityisenhancedwhereas theaxial thermo-magnetoconductivityisdiminishedandisproportionalto thecoefficientofthemixedaxial-gravitationalanomaly.

The focus in the previous literature is considers the effects due toanomalies in the presence ofbackground magneticfields.

Anomaly relatedenhancement ofelectricandthermoelectriccon- ductivities in magnetic fields have indeed been observed in [38, 43].Incontrastinthisworkwehaveconcentratedontheobserv- able effects in the presence of background axial magnetic fields.

Ourstudydiffersintwoimportantpointsfromprevious ones[11, 13] inthatwetaketheBlochtheoremintoaccountandalsostudy the thermo-electricconductivity.Wehope that theeffectscanbe measuredinthefutureandwillenrichourcurrentunderstanding oftheroleofchiralanomaly.

Acknowledgements

We thank Y.-W. Sun, S. Golkar and S. Sethi foruseful discus- sions.TheresearchofK.L.hasbeensupportedbyFPA2015-65480-P andby the Centrode ExcelenciaSeveroOchoa Programmeunder grantSEV-2012-0249andSEV-2016-0597.TheresearchofY.L.has beensupported bytheThousandYoungTalents ProgramofChina andgrantsZG216S17A5andKG12003301fromBeihangUniversity.

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