Character of matter in holography: Spin–orbit interaction

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Physics Letters B

www.elsevier.com/locate/physletb

Character of matter in holography: Spin–orbit interaction

Yunseok Seo

^a

, Keun-Young Kim

^b

, Kyung Kiu Kim

^b,c

, Sang-Jin Sin

^a,∗

aDepartmentofPhysics,HanyangUniversity,Seoul133-791,RepublicofKorea

bSchoolofPhysicsandChemistry,GwangjuInstituteofScienceandTechnology,Gwangju500-712,RepublicofKorea cDepartmentofPhysics,CollegeofScience,YonseiUniversity,Seoul120-749,RepublicofKorea

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

Articlehistory:

Received21February2016

Receivedinrevisedform13May2016 Accepted20May2016

Availableonline25May2016 Editor:M.Cvetiˇc

Keywords:

Holography Diamagnetism DCconductivity AnomalousHalleffect

Gauge/Gravity duality as a theory of matter needs a systematic way to characterise a system. We suggesta‘dimensionallifting’ oftheleastirrelevantinteractiontothebulktheory.Asanexample,we consider the spin–orbitinteraction,whichcauses magneto-electric interactionterm. We show thatits liftingisanaxioniccoupling.Wepresentanexactandanalyticsolutiondescribingdiamagneticresponse.

Experimentaldataonannealedgraphiteshowsaremarkablesimilaritytoourtheoreticalresult.Wealso findananalyticformulasofDCtransportcoefficients,accordingtowhich,theanomalousHallcoefficient interpolates between thecoherent metallicregimewith ρ_xx² and incoherentmetallicregimewith ρxx as we increasethe disorder parameter β. Thestrength ofthe spin–orbitinteraction also interpolates betweenthetwoscalingregimes.

©^{2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

Overview andsummary: Recently, the gauge/gravity duality [1–3]attractedmuchinterestsasapossiblecandidateforareliable methodtocalculatestronglycorrelated systems.Itisa localfield theoryin onehigherdimensionalspacecalled“bulk”, withafew classicalfieldscoupledwithanti-deSitter (AdS)gravity.Sincethe strongcouplingintheboundary isdualtoaweakcouplinginthe bulk, thebulk fieldscan beconsidered aslocalorder parameters ofameanfield theoryinthe bulk.Italsoprovided anewmech- anismforinstabilitiesingravitylanguage [4]whichisrelevantto thesuperconductivity [5,6] andthe metal insulatortransition[7].

However,asatheoryformaterials,itisstillinlackofoneessen- tialingredient, a way to distinguishone matter fromthe others.

Althoughelectron–electroninteractionistradedforthegravity in the bulk, we still need to specify lattice–electron interactions to characterise thesystem.Withoutit,wewouldnotknowwhatsys- temweareworkingfor.

Naivelyonemaytrytointroducerealisticlatticeatthebound- ary to mimic the reality. However, its effects are mostly irrele- vant inthe infrared(IR) limit. Instrong couplinglimit whereno quasiparticle exists, no Fermi surface (FS) exists either. Actually in the absence of the FS, it is almost impossibleto write down anyrelevantinteractionterminalocalfieldtheoryinhigherthan

*

Correspondingauthor.

E-mailaddresses:yseo@hanyang.ac.kr(Y. Seo),fortoe@gist.ac.kr(K.-Y. Kim), kimkyungkiu@gmail.com(K.K. Kim),sangjin.sin@gmail.com(S.-J. Sin).

1+1 dimension.¹ Therefore non-local effect maybe essential for any interesting physics instrongly interacting system. One inter- esting aspectofa holographictheory isthat anylocalinteraction in thebulk hasnon-local effectinthe boundary[9].Usuallyone characterises amanybodysystemincontinuumlimitbyafewin- teraction terms rather than the detail of structure. Therefore, to characteriseasysteminholographictheory,whatwewanttosug- gest isthe dimensional-lifting,bywhich wemeanpromoting the

“system characterising interaction” of the boundary theory to a term inthe bulk theory using thecovariant form ofthe interac- tion.

One may wonder what the gravity dual of the Maxwell the- ory is. In condensed matter, there are two components of elec- tromagnetic interaction. One is electron–electron interaction and the otheris lattice–electroninteraction.While themaindifficulty is coming from the former, system is characterised by the lat- ter. Working hypothesis is that the electron–electron interaction is taken care of by workingin asymptotic AdS gravity. Ourpur- pose is to include the electron–lattice interaction in this holo- graphic scheme, which is possible for two reasons. First, in any boundarysystemwithaconservedglobalU(1)charge,wehavea bulk Maxwell theory, whichcan accommodate usual electromag- netic field asaprobe oran external source.Itwas used tobuild

1 Seehoweverref.[8]forsemi-holographicapproachbasedonIRAdS2andits virtualC F T2,whichisdifferentfromours.

http://dx.doi.org/10.1016/j.physletb.2016.05.059

0370-2693/©^{2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

theholographicversionofsuperconductivitymentionedaboveand also to calculate electric/thermal transport coefficients [10–13].

Second,we canusearelativistictheory foranon-relativisticsys- tem.Therelativisticinvariancehighlyconstrainsthepossibleform ofextension ofinteraction. Apractical way toproceed is toturn on the interaction one by one for technically simplicity.The co- variant form of the interaction is either scalar or top form. The formeristriviallyliftedtohigherdimension,e.g., FμνMμν canbe usedinanydimension.Nowsupposethetopformoftheboundary theoryis Fd andthebulk theoryalreadycontains scalaroperator

ϕ

andone form

ω

1.Then we haveessentially twochoices:

ϕ

dF_d and

ω

1∧^Fd toavoidthetotalderivativeterm.

To discuss the idea in more specific context, we consider thespin–orbit interactionin 2+1 dimensionalsystems.It creates lotsofinterestingphenomenaincludingtopologicalinsulatorsand Weyl semi-metal[14–18] by changing band structures, which in turn causes magneto-electric phenomena [19,20] like anomalous Halleffect.Naively, introducing thespin–orbit interactioninvolve fermions.² However, we can integrate out the massive fermions, thereby avoid dealing with fermions in our theory. Notice that in the absence of Fermi sea as in our strong coupling problem, fermionscan be considered to be massive.It is well known that the fermions integratedout leave the Chern–Simons term A∧^F [21,22], which can be lifted to 4 dimension as F∧^F ∼^E·^B.3 Since it is a total derivative by itself, we have to couple it with anappropriate scalaroperatortohaveanon-trivialdynamicalef- fect. In this paper, we choose it to be the kinetic energy term of the axion scalar fields

χ

I. That is our interaction term is qχ

I=¹,2(∂

χ

I)²F∧^F,where

χ

I wasintroducedtoprovidesome disordergivingmomentum dissipation[32].Noticethat thisterm isodd in time reversal, which is appropriate for the casewhere magnetisation isnon-trivial.⁴

Sincewe wantto have finitetemperature, chemical potential, magneticfields,andfiniteDCconductivity,thesystemshouldcon- tain metric, gauge fields andaxion scalar fields (gμν,Aμ,

χ

I) as theminimal ingredientsinthebulk.Sowehavetostartwiththe Einstein–Maxwell-axionsystem. We have found an exact analytic solutionofsucha non-triviallycoupled systemwitha newinter- actionterm, consequently yielding an explicitandanalytic result fortheDCconductivityusingrecenttechnology[10–12].Whilethe Halleffectisobviouslyconnectedtooursystemfromtheconstruc- tion, the fully back reacted system shows diamagnetic response.

Thisis becausewe examined metallic state at finitetemperature anddidnotincludespindegreesoffreedomexplicitly.Finally,we commentonthe relevance ofour resultto experimental data.In [33], it was reported that graphite, once annealed to wash out theferromagnetic behaviour, showsa non-lineardiamagnetic re- sponsewhichisverysimilartoouranalyticresult.Alsoitturnsout thatouranalytic conductivityformulas reproducetheexperimen- taldataonthescalingrelationbetweenthenon-linearanomalous Hallcoefficientsandthelongitudinalresistivity.I.e. thenon-linear anomalous Hall coefficients interpolate between the linear and

2 TheChern–Simonstermisderivedfromaminimalinteractionψ¯γ^μψAμ.Ifwe takenon-relativisticlimitfirst,theinteractionLagrangianisLint= μ· ^Bintheelec- tronatrestframe,whichbecomesψ¯γ^μνψFμν incovariantformthatisvalidin anyframe.Whenweincludefermionsexplicitly,wehavetotakeintoaccountthis issue.

3 PreviouslytheChern–Simonsterminthebulkanditshigherdimensionalana- loguewereextensively consideredinholographytodiscussthechiraleffectsor instabilitytotheinhomogeneousphases[23–31].

4 In order to handle time reversal invariant case, one can consider qχ

I=1,2dχI∧A∧F∼qχ

I=1,2χIF∧F.Onecanalsoconsiderthepossibility thatqχcontainsanIsingspinvariable±1 whichisoddundertimereversal.Inthis paperwefocusonthetimereversalbreakingcasetoconsidernon-zeromagnetisa- tion.

quadratic dependence on the longitudinal resistivity. Considering that we added just one interaction term, theseare unexpectedly richconsequences.

The modeland background solution: With motivations de- scribed above, we start from the Einstein–Maxwell-axion action withtheChern–Simonsinteraction

2

κ

²S

=

d⁴x

√

−

^g

⎧ ⎨

⎩

^R

+

⁶ L²

−

¹

4F²

−

I=¹,2

1 2

(∂ χ

I

)

²

⎫ ⎬

⎭

−

¹ 16

q_χ

(∂ χ

I

)

²F

∧

^F

+

^Sc

,

(1) where qχ is a coupling, and

κ

²₌8

π

G and L is the AdS radius andweset2

κ

²₌L=^1.Sc isthecountertermwhichisnecessary to make the action finite. Explicit form of Sc is written in (25) atthe endof thispaper.The axion (

χ

I) which islinear in{^x,y} directionbreakstranslationalsymmetryandhencegivesan effect ofmomentumdissipation[32].Instantondensitycoupledwiththe axion can generate magneto-electric property: if we add charge, non-trivialmagnetisation isgenerated.Theequationsofmotionare ratherlongsowewroteitin(26)attheend.

Asansatztosolutions,weusethefollowingform A

=

^a

(

r

)

dt

+

¹

2H

(

xdy

−

^ydx

) , χ

1

= β

x

, χ

2

= β

y

,

(2)

withthemetricansatz ds²

= −

U

(

r

)

dt²

+

^dr2

U

(

r

) +

r²

(

dx²

+

dy²

) .

(3) Fromtheequationsofmotion,wefoundexactsolution

U

(

r

) =

^r2

− β

² 2

−

^m0

r

+

^q2

+

H²

4r²

+ β

⁴H²q²_χ

20r⁶

− β

²Hqq_χ 6r⁴

,

a

(

r

) = μ −

^q

r

+ β

²Hq_χ 3r³

,

(4)

where

μ

isafreeparameterinterpretedasthechemicalpotential andq andm0 aredeterminedbythecondition At(r0)=^U(r0)=⁰ attheblackholehorizon(r0).q istheconserved U(1)charge in- terpreted asa numberdensityattheboundary system.m₀ turns out to be half ofthe energydensity(9) andβ is relatedto mo- mentumrelaxationrate

q

=

^r0

μ +

¹

3

θ

H with

θ = β

²q_χ r²₀

,

m0

=

^r30

+

^r02

μ

²

+

^H2

−

²

β

²r²₀

4r0

+ θ

²H² 45r0

.

(5)

The solution (4) reproduces the dyonic black hole solution with momentumrelaxation[12]whenqχ vanishes.

Diamagneticresponse: The thermodynamic potential density W ^{intheboundarytheory iscomputedbytheEuclideanon-shell} action S^E of(25): S^E≡V2W/T,V2=

dxdy usingthesolutions (2)–(3)

W

= −

^r3₀

−

¹ 4r₀

μ

²r²₀

+

²

β

²r²₀

−

^3H2

+

²

3

μ θ

H

+

⁷ 45r₀

θ

²H²

.

(6) ThesystemtemperatureT isidentifiedwiththeHawkingtemper- atureoftheblackhole,

T

=

³ 4

π

^r0

−

1 16

π

r₀³

(

q

− θ

H

)

²

+

^H2

+

^2r20

β

²

,

(7)

Fig. 1.(a)Theexternalmagneticfielddependencesofthemagnetisation atT=1 (purple),T=1.2 (blue),T=1.5 (green)andT=2 (red)withqχ=10,β=2.2,μ=0.

(b) Theorycurvecomparedwithexperimentaldataofannealedhighlyorientedpyrolyticgraphite(HOPG)sampleforT=150 K (rectangle)andT=300 K (circle)in[33].

Forcomparison,theblueandredcurvesobtainedbyourformula(14)areadded.TheredoneisforT=^1,β=^{2 andtheblueoneisfor T}=⁰.8,β=².2.qχ=^{10 and} μ=^{0 forbothcases.(For}interpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

andtheentropydensityisgivenbytheareaofthehorizon

s

=

⁴

π

r²₀

.

(8)

Wehavenumericallycheckedthattheentropyisamonotonically increasing function of temperature for the parameters analysed in thispaper. The energy density

ε

is one point function ofthe boundaryenergymomentumtensor T₀₀,whichisholographically encodedinthemetric(5):

ε =

T00

=

^2m0

.

(9) It is remarkable that the complicated expression of the ther- modynamic potential density (6) gives a simple thermodynamic relation

W

= ε −

^sT

− μ

q

,

(10)

with energy, temperature andentropy given by (7), (8) and (9).

Thevariationofthepotentialdensity(6)boilsdownto

δ

W

= − ˜

M

δ

H

−

s

δ

T

−

q

δ μ ,

(11) where

M

˜ ≡

¹ r0

−

^H

+

¹ 3

θ

q

−

¹

5

θ

²H

.

(12)

Notice that the (10) and(11) impliesthe first law ofthermody- namics;

δ ε = − ˜

^M

δ

H

+

^T

δ

s

+ μ δ

q

.

(13) Twoimportantremarks areinorder:First, H isinterpretedasan externallyappliedfield,althoughitisafullybackreactedobjectin thebulk.Histhemagneticfieldgeneratedbyfreecurrent,notthe magneticinductionwhichisusuallydenotedbyB.Thisisbecause wedidnot encodeanyspindynamicsinthebulk andwe donot have fully dynamical gauge fields at the boundary. The Maxwell fields atthe boundary enter asan external source oras a weak probefield.

Second, M˜ hasdimension 1 anddescribes a genuine 2+^{1 di-} mensionalsystem. Therefore,itcannot beidentified asthe mag- netisation M ofa physicalsystemwhichisa 2dimensionalarray in3spatial dimension.Furthermore,since M˜ andH are different in mass dimension, they cannot be added to form magnetic in- ductionB.ThemagneticfieldHandmagnetisationMarethoseof spatial3dimension,thereforebothHandMshouldhavethesame massdimension2.IfwejustmultiplyM˜ byr0,amassdimension 1parameterwhichisaconstantforadiabaticprocesses,wewould getB=^H+^M=^{0 forq}χ=^{0.Itmeansthatthedyonicblackhole}

exhibitstheMeisnereffect,whichisnotphysical.Theproblemcan betracedtothefactthatafterwescale

ε

byr0tobalancethedi- mensionofM andH,thefreeenergycontains H²/2,whichisthe fieldenergyofmagneticfieldappliedonvacuum.Whenwecalcu- latethemagnetisation bytakingitsderivative,weshouldsubtract itfromthefreeenergyassuggestedbyLandauandLifshitzinsec- tion32ofref.[34].Therefore,wecalculatethemagnetisation from F≡^r0

ε

−¹₂^H2:

M

= − ∂

F

∂

H

fixedr₀,q

= θ

q

/

3

− θ

²H

/

5

.

(14)

Bothtermsherearetheconsequencesoftheaxioniccoupling.

Thefirsttermisthemagnetisation atH=^{0,whichwillbede-} noted by M₀. It is proportional to the charge of the systemand givesferromagnetism.Moreexplicitly,

M₀

=

¹

3

θ

q

= μ β

²q_χ

3r₀

,

(15)

withr0=(4

π

T+

16

π

²T²+³(

μ

²₊2β²))/6 at H=^0.Forgiven

μ

,β andqχ,M0hasthemaximumvalueatzerotemperatureand decreases as1/T for large temperature. In the coherent metallic regime [35] β/

μ

^{1, M}0∼β²qχ.The second term in(14) rep- resents theback reaction ofthe systemto theexternal magnetic fieldandgivesdiamagnetism.

We want to analyse the magnetisation as a function of the magneticfield withtheother parametersfixed.Notice that,atfix temperature,r₀hastobecomputedfrom(7).InFig. 1(a),wedraw the magnetic field dependence of the magnetisation at different temperatures for

μ

=^{0, q}χ=^{10, and} β=^{2. The} magnetisation seems to be saturatedfor large magneticfield andthe magnetic susceptibilityisdecreasingfunctionoftemperature.Ourresultsare verysimilartothegraphitedatainFig. 1(b)[33].Here,inaddition toexperimentaldata,weaddedtheblueandredcurvesusingour formula(14)forcomparison,wheretheredoneisforT=^1,β=² andthe blueone is forT =⁰.8, β=².2.qχ=^{10 and}

μ

=^{0 for} bothcases.

DCtransportcoefficients:Recently, a systematicway tocom- putetheDCtransportcoefficientshasbeendevelopedin[11,12,36]

bywhichwecancomputethelongitudinalandtransverseelectric andthermoelectricconductivities.Herewewriteonlyresult.

σ

xx

= (

F

−

^H2

)(

F

+

G²

) (

F²

+

^H2G²

) ,

σ

xy

=

^HG

(

2F

+

G²

−

^H2

) + θ (

F²

+

^H2G²

) (

F²

+

^H2G²

) , α

xx

=

^sG

(

F

−

H²

)

F²

+

H²G²

, α

xy

=

^{s H}

(

F

+

G²

)

F²

+

H²G²

−

^H

r₀

,

(16)

where

F

=

^r2₀

β

²

+

^H2

1

+ θ

²

− θ

qH

,

G

=

q

− θ

H

,

s

=

4

π

r₀²

.

(17)

Wegavesomedetailsattheend.Wechecktwolimits:i)forβ=^0, theDCconductivities(16)become

σ

xx=^0,

σ

xy=^q/H,

α

xx=^{0 and}

α

xy=^s/H,whichagreeswith[37]; ii)forqχ=^{0(16)reproduces} theresultobtainedin[12,36].

Atfiniteβ andfiniteqχ butwithH=^{0 theelectricconductiv-} itiesreduceto:

σ

_xx

=

1

+ μ

²

β

²

, σ

_xy

= θ =

^3M0

q

.

(18)

Notice that

σ

xx is a known result [32], but, interestingly,

σ

xy is non-zero evenwhen H=^{0.Thisphenomenaisrelatedtoanoma-} lousHalleffect,whichwillbediscussednext.Itistheresultofthe axioncouplingweintroduced,whichgivesaferromagnetismwith themagnetisationM₀ (15).

AnomalousHall effect: In ferromagnetic conductor, the Hall effect is about 10 times bigger than in non-magnetic material.

Thisstronger Hall effect inferromagnetic conductoris known as the anomalous Hall (AH) effect [38,39]. The precise mechanism for AH effect has a century-long history of debates [38]. Three mechanismshavebeensuggested:i) intrinsiconedueto anoma- lous velocity, ii) side jump, iii) skew scattering. Mechanism i) was suggested in 1950’s by Karpulus and Luttinger. In modern days, the anomalous velocity is understood by the Berry phase (v_a= ^e_¯h^E×^bBerry). Side jump mechanism is suggested by Berger inref.[40]whereheshowedthattheelectronvelocityisdeflected inopposite directionsby the opposite electricfields experienced upon approaching and leaving an impurity. The skew scattering wassuggestedbySmitin[41,42]wherehenoticedthatasymmet- ricscatteringfromimpuritiesis causedby thespin–orbitinterac- tion.Thefundamentalinteractionunderlyingallthesethreeisthe spin–orbitinteraction.

Ithas beenknown thatthere is apower lawrelationship be- tween the anomalous part (RS) of the Hall resistivity (

ρ

yx) and thelongitudinalresistivity(

ρ

xx):

R_s

∼ ρ

_xx^α

,

(19)

withtheanomalousHallcoefficient,RS,definedbytherelation

ρ

yx

=

^RHH

+

^RSM

,

(20) where RH is the usual Hall coefficient. The power

α

had been computedfor three scenarios to give

α

₌2 for i), ii)and

α

₌1 foriii).

From (14)and(18)our modeldescribesa ferromagneticcon- ductor, therefore it will be interesting to study AH effect. The resistivity matrix(

ρ

) can be computedby inverting theconduc- tivitymatrix (

σ

) in (16)i.e.

ρ

=

σ

⁻¹. RS is identified by

ρ

yx at H=^0:

ρ

yx

H=⁰

= θ

θ

²

+ (

1

+ μ

²

/β

²

)

²

=

^RSM0

.

(21) SinceM₀=¹₃^qθ,anomalousHallcoefficientisgivenby

Fig. 2.Relationbetweenρxxand ^q₃⁰Rs(q0≡μr0)atfixedqχ=^{1 with}μ/T=⁰.1 (red),2 (green),5 (blue)and10 (purple)inlog–logplot.Thearrowrepresentsthe directionofincreasingβ/μ. RS∼ρ²_xx insmallβ/μand RS∼ρxx inlargeβ/μ regime.(Forinterpretationofthereferences tocolourinthis figurelegend,the readerisreferredtothewebversionofthisarticle.)

R_S

=

³ r₀

μ

1

θ

²

+ (

1

+ μ

²

/β

²

)

²

.

(22)

ThelongitudinalresistivityatH=^{0 reads}

ρ

xx

=

¹

+ μ

²

/β

²

θ

²

+ (

1

+ μ

²

/β

²

)

²

.

(23)

Thescalingbehaviourscanbereadeasilyfortwolimits:

RS

∼ ρ

²_xx

,

or RS

∼ ρ

xx

,

(24)

depending on β/

μ

^{1 or} β/

μ

^{1. The same scaling relations} hold for θ^{1 or} θ^{1. Forgeneral value of} β/

μ

, the scaling behaviourisshowninFig. 2astheslope.Inthefigure,thearrow representsthe directionsofincreasing disorder parameterβ,and

μ

/T =⁰.1 (red), 2 (green), 5 (blue) and 10 (purple). Notice the transitionfrom

α

=^{2 to}

α

=^{1 is sharper forlarger}temperature.

Notice also that

α

₌2 for smallβ/

μ

whateveris the spin–orbit interactionstrengthθ.

Inref.[35],materialwithβ/

μ

^{1 wasidentifiedasacoherent} metalofwhichopticalconductivityhasawelldefinedDrudepeak asifithadquasiparticles.Forβ/

μ

^{1,thesystembehavesasan} incoherentmetalwithoutaDrudepeak.Interestingly,suchelectri- cally classifiedcoherent/incoherentmetalalsoshowsAHproperty withthecharacteristicpower

α

=²,1.Thisisconsistentwiththe interpretation ofβ asimpurity densitysince large β wouldhave much extrinsic disorder effects.Notice that we have two scaling regimesinonemodelwithinterpolatingparametersgivenby the disorder parameterβ orspin–orbitcoupling strengthθ,while in conventional methoddifferentscalings areassociatedwithdiffer- ent mechanisms. The fact that all three mechanisms are origi- nated fromspin–orbitinteraction isreflected toour resultwhich isaconsequenceofaddingjustoneinteractiontermrepresenting spin–orbitcoupling.

Method:DCconductivities fromblackholehorizon

Hereweexplain howtoobtaintheDCconductivities(16).The actionwithcountertermsaregivenby

2

κ

²S

=

M d⁴x

√

−

^g

⎧ ⎨

⎩

^R

+

⁶ L²

−

¹

4F²

−

I=¹,2

1 2

(∂ χ

I

)

²

⎫ ⎬

⎭

−

¹ 16

M

q_χ

(∂ χ

_I

)

²F

∧

^F

+

^Sc

,

S_c

= −

∂M d³x

− γ

⎛

⎝

2K

+

⁴

L

+

^R

[ γ ] −

I=¹,2

L

2

∇ χ

I

· ∇ χ

I

⎞

⎠ .

(25)

Theequationsofmotionare

∇

²

χ

I

+

^qχ 8

∇

M

1

√ −

g

^{P Q R S}FP QFR S

∇

^M

χ

I

=

⁰

,

∇

MF^{M N}

+

^qχ 4

∇

M

(∂ χ

I

)

²

√

¹

−

g

^{M N P Q}FP Q

=

0

,

RM N

−

¹

2gM N

R

+

6

−

¹ 4F²

−

¹

2

(∂ χ

I

)

²

−

¹

2F_{M P}F_N^P

−

¹

2

∂

M

χ

I

∂

N

χ

I

−

^qχ

16

∂

M

χ

I

∂

N

χ

I

√

1

−

g

^{P Q R S}FP QFR S

=

0

,

(26)

where ⁰¹²³=

^txyr=^1.

Oncewegetabackgroundsolutionoftheseequations,wecan computetheDCtransportcoefficientsfromtheblackholehorizon data.Letusstartbydefiningausefulquantity:

F^{M N}

≡ √

−

^{g FM N}

+

^qχ

4

(∂ χ )

²

^{M N P Q}F_{P Q}

.

(27)

ThentheMaxwellequationandtheboundarycurrentcanbewrit- tenas

∂

_MF^{M N}

=

⁰

,

J^μ_tot

=

^lim

r→∞F^μr

.

(28)

Ifweassumeallfieldsdependont andr,onecanobtainanother expressionforthetotalboundarycurrentusingtheMaxwellequa- tion

J^μ_tot

=

^lim

r→^r0F^μr

+

∞ r₀

dr

∂

tF^tμ

.

(29)

Now we want to consider fluctuations corresponding to the boundary DC electric field E_i and the DC temperature gradient ζi=^∇_T^{iT.Following[11],wemayconsider}fluctuationsaroundthe backgroundasfollows:

δ

A_i

= − (

E_i

− ζ

_ia

(

r

))

t

+ δ

a_i

(

r

) , δ

g_ti

= −

^U

(

r

) ζ

_it

+

^r2

δ

h_ti

, δ

g_ri

=

^r2

δ

h_ri

,

δ χ

i

= δ χ

i

(

r

) .

(30)

In the linear level the time dependent part of the equations of motiondropsout by theabove choiceoffluctuations. Thus these fluctuationsarestablestaticfluctuationstotheDCsources,(E_i,ζi). Tobe aphysicalfluctuation inthe blackholebackground,the fluctuation should satisfy the in-falling boundary condition as it approachesthehorizon.Thiscondition canbe describedinterms of the Eddington–Finkelstein coordinates v =^t+ _4π¹_T^ln(r−^r0), and the in-falling fluctuation should depend on v near horizon.

Therefore,theregularityatthehorizonimplies

δ

h_ti

∼ − ζ

i

4

π

T log

(

r

−

^r0

)

^U

(

r

)

r²

+

^h(_ti⁰⁾

+

O

((

r

−

^r0

)) , δ

h_ri

∼ H

ri

r²U

(

r

) +

^h(_ri⁰⁾

+

O

((

r

−

^r0

)) , δ

a_i

∼ −

^Ei

− ζ

ia

(

r

)

4

π

T log

(

r

−

^r0

) +

^a(_i⁰⁾

+

O

((

r

−

^r0

)) , δ χ

i

∼ δ χ

_i⁽⁰⁾

+

O

((

r

−

r0

)) .

(31)

Byexpanding(r,i)componentand(t,i)componentoftheEinstein equations, it turns out that Hri=^r20h⁽_ti⁰⁾ and h_ti⁽⁰⁾ can be deter- mined.⁵

Nowwe arereadytoconsiderthetotalcurrent(29)inthelin- ear level.The current can be calculated by plugging (4)and the fluctuation(30).Firstly,thesecondpartis

∞ r₀

dr

∂

tF^ti

=

i j

r₀

−

^H

+

¹ 3

θ

q

−

¹

5

θ

²H

ζ

j

=

_{i j}

r₀

(−

^H

+

^M

) ζ

j

.

(32)

The second termis thecontributionofthemagnetisation current Jmag [43].Thus therelevant partofthecurrentis thefirst term, whichcanbewrittenintermsofthehorizondata:

J^μ

=

^lim

r→^r0F^μr

=

^Ei

− (

q

− θ

H

)

h⁽_ti⁰⁾

−

^H

r²₀

i j

H

r j

+

i j

θ

E_j

−

i j

H

r₀

ζ

j

.

(33) Finally, the expressions h⁽_ti⁰⁾ and Hri obtained from the Einstein equationgive ustheelectricconductivitiesandthethermoelectric coefficients(16)usingformula

σ

i j=_∂^∂_E^Ji_j ^and

α

i j= ¹_T^∂_∂ζ^Ji_j^{.The de-} tailsofh⁽_ti⁰⁾andHriwillbegiveninalongerversionofthiswork.

Summaryanddiscussion: We view the holographic principle as a set of axioms to calculate strongly interacting systems. For reader’sconveniencewefirstlistthembelow[9].

1. For a strongly interacting system with conformal symmetry at UV, there is a dual gravity with asymptotic AdS bound- ary. Non-AdS geometry maybe regardedas an IR partofan asymptoticAdSgeometry.

2. To calculate the correlation function of an operator O with dimension and spin p, we introduce a source field φ0(x) with spin p and dimension . Extend φ0(x) into one higher dimensional space φ (x,r) such that φ (x,r= ∞)∼ φ0(x)/r^d−2p−. Identify the generating function of confor- mal field theory Z[φ0] ^{with that of} gravitational system exp(−^S[φ0]).

3. Foraglobalsymmetryatthebulk,wehavealocalgaugesym- metryattheboundary.

4. ForEuclideanGreenfunctions,Dirichletboundaryvalueatin- finity isenough.Forcausalgreenfunction,assign thebound- arycondition(BC)attheIRregioninadditiontotheDirichlet BCattheinfinity.

5. Temperatureandchemical potential are provided by regular- ity of metric, gauge fields at the horizon or its replacing IR geometry.

6. Characterise the system by lifting the least irrelevant inter- actions at the boundary to the bulk. For strongly correlated electron system, the electron–electron interaction is counted by the gravity, but electron–lattice interactions should be takenintoaccountexplicitly.Thegaugefielddualtothecon- servedU(1)currentcanhaveinteractiontermstotakecareof electron–latticeinteractions.

The last item is what we added in thispaper. In this paper,we consideredthemagneto-electricphenomena inducedbythespin–

orbit coupling interaction as an example of dimensional lifting inholographic theory.Whenelectronspinsarecorrelated,adding chargecarrierchanges themagneticpropertyaswellasthecharge

5 Sincetheexpressionislengthybutnotveryilluminatingwedon’tpresenthere.

transport.Ineffectivefieldtheoryapproach,suchelectric–magnetic effectcanbe implied byaddingthe Chern–Simonsterm∼^A∧^F in 2+1 dimension. It act as a crossing source of electricity and magnetism.Withsuch aterm,thesystemcanpickupamagnetic sourcewhenweprovideelectricchargeandviceversa.

We work atfinite temperature, chemical potential, andmag- netic fields.The metric, gauge and axion fields (gμν,Aμ,

χ

I) are playingthe roleofcoupled orderparameters.We havefoundthe exactandanalytic solutionofsucha complicatedcoupledsystem withanon-trivialinteraction,whichmadeitpossibletogetanex- plicitandanalyticDCconductivityformulas.

Whenwesplitthecurrentintoorbitalandmagnetisation parts, itdependsonthedefinitionofthemagnetisation,namely,whether we use M or M.˜ The electric conductivity does not change but thermo-electricconductivitychanges.The

α

forMandthatforM˜ arerelatedinsimple manner:

σ

i j and

α

xx arenot changedwhile

α

_xy^M₌

α

_xy^M˜ ₋H/r0. Itturns outthat Onsagerrelations arevalidin bothcases,

α

= ¯

α

,aremarkablefact.

Ourresultsonthe Hallresistivity showsanon-lineardiamag- netic response is similar to that of graphite system and also to highTc superconductor Bi2Sr2CaCu2O8.Alsowe haveshownthat theanomalousHallcoefficientsinourmodelinterpolatebetween thelinearandquadraticregimeontheresistivitydependenceasa functionofdisorderparameterandspin–orbitinteractioncoupling.

Itisparticularlyinterestingtoseethatelectricalcoherent/incoher- entmetalhasmagneticbehaviour withquadratic/linearresistivity dependence.Experimentallydiversematerialswere studied.Some shows

α

near 2 and the other 1, sometimes old and new data crashes.So, detailed data mining is postponed to future investi- gation.

Ourmodel doesnot include paramagnetic behaviour because weintegratedout fermionandthereforespin degreesoffreedom is not included explicitly. Also we could have chosen the scalar field

χ

itselfinsteadofitskinetictermasascalarpartnerofF∧^F term.Wewillreportontheseissueselsewhere.

Acknowledgements

TheauthorswanttothankJunghoonHanforsuggestingtolook graphite data for diamagnetism. SS appreciates discussions with YunkyuBang,YongbaekKim,KwonParkandespeciallyKiseokKim on manyrelated issues. YS wantto thankKIAS for the financial support during his visit. The work of SS and YS was supported byMid-career ResearcherProgramthrough theNationalResearch FoundationofKorea (NRF)grantNo.NRF-2013R1A2A2A05004846.

The work of KKY and KKK was supported by Basic Science Re- searchProgramthroughtheNationalResearchFoundationofKorea (NRF)fundedbythe MinistryofScience, ICTandFuturePlanning (NRF-2014R1A1A1003220)andthe 2015GISTGrant fortheFARE Project (Further Advancement of Researchand Education atGIST College).

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