Field theory formulation of the Quantum Hall Effect

E.Abdalla∗,M.C.B.Abdalla†

arXiv:cond-mat/9406119v1 30 Jun 1994CERN,TheoryDivision,CH-1211Gen`eve23,Switzerland

Abstract

WeconsiderthequantumHalleffectintermsofaneffectivefieldtheoryformulationofthe

edgestates,providinganaturalcommonframeworkforthefractionalandintegraleffects.

PACS:73.40.Hm,11.10.Kk,72.15.Nj,11.30.Rd

CERN-TH7344June94

cond-mat/9406119

TherehasbeenalotofinterestinthelastyearsontheQuantumHallEffect(QHE)1.Theintegraleffect(IQHE)hasbeengivensimpleexplanation2,butthefractionaleffect(FQHE)ismoreinvolvedandrequirestheconceptofcollectivebehavioroftheelectrons3.Inparticular,Jain4usedaunifiedschemeinordertoexplainbotheffects,sincetheex-planationoftheintegraleffectwasbelievedtobedescribedintermsofnon-interactingelectrons1,whilethefractionaleffectincludestheabovementionedcollectivephenomenon.Theideaistointroducecopiesoftheelectron,andconsidercondensationsofsuchfields4.Furtherunderstandingcanbeachievedonceoneverifiesthatchiralityplaysanessen-tialroleintheexplanationoftheHalleffect,asstressedrecently5,6,7,8.OurpurposehereistoprovideafieldtheoreticalapproachtothequantumHalleffect,insuchawaythatthecollectiveelectronphenomenonisnaturallyaccomodated,beingjustaconsequenceofthedescriptionofelectronssqueezedinasmallspaceregion,andthechiralnatureoftheinteraction5−8,leadingtochiralanomaly,whichtogetherwiththesolitonbehaviorinher-itedfromthecollectiveeffect,naturallyaccomodatesthefractionaleffect,andgives,asabyproduct,theintegraleffectasaparticularcase.Werefrainfromreviewingthesubject,referingtothevastliterature1,9.However,wehavetoexplicitlyquotesomeresultswhichwillbeusedinthefollowing,aswellasrefertoasacheckofthecorrectednessoftheprocedure.

WestartoutoftheLagrangeandensityoffour-dimensionalQED,whichreads

L=

ΨiΓµAµΨi

,

i=1,...,N

,

(1)

wherethegammamatricesΓµandtheΨfieldarefour-dimensionalandread

󰀉0−1

10󰀌

󰀉0−σa

σ0

a

Γ0=

,

Γa=

󰀌

,

󰀉󰀌χ

Ψ=

η

,(2)

σaaretheusualPaulimatriceswitha=1,2,3andχandηarebispinors.InthisnotationtheLagrangianis

L=χ+∂0χ+η+∂0η−η+σa∂aη−χ+σa∂aη+χ+χA0+η+ηA0−e(χ+σaχ+η+σaη)Aa==

η∂η+(ηγµη)Aµ+χ+σM∂Mχ+η+σM∂Mχ

ψIγµψIAµ+M−derivatives,I=1,2andM=2,3.

(3)

Intheabovewehaveusedthefollowingnotation:ThespinorfieldψI=1standsforχ,thefirstbispinor,whileψI=2correspondstoγ1η(t,−󰀮x),andthetwo-dimensionalgamma

32

matricesareγ0=σ,γ1=σ,andγ5=γ0γ1;theslashderivativeis∂=γµ∂µ.Therefore,forΓµtheindexµgoesfrom0to3,whileforγµittakesthevalues0,1.Weneglectthederivativesinthe2and3directions(M),consideringonlythetwo-dimensionalproblem.

1

IntegratingoverthegaugefieldAµ(µ=0,···,3)wegetL=

ψjγµψjDµν

ψjψjD33

ψjγ5ψjD22

ψji∂ψj+geff

󰀂

ψkγµψk+geff(

ψjγ5ψj)2

󰀄

,(5)

wheretheeffectivecurrentvertexcorrespondingtoµ=2intwodimensionsis

ψψ.WefindalsoatrivialThirringinteraction,whichisneglected

inthefollowing.Theelectromagneticfieldisresponsiblefortheinternalinteractionsinthecristal,andfortheformationofthetwo-dimensionallayer.Nowtheeffectivetwo-dimensionalinteractionisthechiralGross-Neveu(CGN)theory,whichhasbeenmuchstudiedintheliterature(see[10]andreferencestherein).Sincetheelectromagneticfieldhasbeenusedinordertoformthesystem,weshallforthetimebeingdisregarditsinteraction.Laterwehavetoreconsiderit,inordertoknowtheinteractionofthesamplewiththeexternalfieldsresponsablefortheHalleffect.AsfarastheCGNinteractionisconcerned,werecallthefollowingfacts.Thesolutionofsuchatheoryisgivenintermsof

ˆj,whichsatisfythenon-linearrelation10,11solitonsψ

󰀁j···ψ󰀁j󰀁+∼ǫjj···jψψ12N−112N−1j

,

(6)

whereontherighthandsideasuitableredefinitionoftheKlein10factorandthenormal

productprescriptionarerequired.

Suchsolitonsarechargelessfields,sincetheU(1)-chargehasbeenseparatedinorderthattheaboverelationbevalid.Infact,thegaugeinteractionhasbeenusedupinthequantumgaugedegreesoffreedom,inorderthatthetwo-dimensionalinfra-redbehaviordoesnotcauseproblems.AsshownbyWen12,theexcitationsresponsablefortheQHEcontainseveralbranches.Infact,theexcitationsineachbranchcarryafractionalcharge.SuchedgeexcitationsformtheFermiliquid.However,asstressedbyWen,thefermionsintheFermiliquidarenottheelectrons,butdescribetheedgeexcitation,apossibilityopenedbytheboson/fermiondualityintwodimensions.ThismeansthattheFermiliquidisbuiltofsolitons,describedbygeneralizedstatistics13,14.

Thenextproceduretobefollowedconcernstheinteractionofthesolitonswiththeexternalgaugefields.Thegaugeinteractionofthedegeneratesolitonsdisplayinganap-parentSU(2N−1)symmetry,withthegaugefieldisfundamentallyimportantinordertoexplaintheeffect.Thediscussionofgaugeinvarianceinthiscontextisgivenin[15].Fromthefactthatwehaveaboundstatestructure,orcollectiveapproachofthetypepicturedbyeq.(6),wecandescribethephysicalelectronfieldbythel.h.s.of(6)times

󰀁j),sinceitissuchfieldwhichhasphysicalinterpretation.ThistheU(1)factor(ψj=eiχψ

meansthatwecandistributethechargeequallyamongthesolitonfields,andtrytofindan

2

effectiveLagrangeanwhichdisplaysthedesiredfeaturesoftheHalleffect.Asapuretwo-dimensionaltheory,thechiralGrossNeveumodelpresentsfieldswithgeneralizedstatistics.Asafour-dimensionaltheoryweinterpreteitsmeaningasthefactthatonlyboundstatesoftheform(6)describethephysicalelectron.Therefore,wesupposethatthephysicalelectronisoftheaboveform,multipliedbytheconvenientexponentialrepresentingtheU(1)charge,thatis

󰀇ˆeψ

−iχ/n

󰀈+

=e

iχ

󰀇󰀈󰀇󰀈

−iχ/n−iχ/nˆjeˆjeǫjj1···jnψ···ψ1n

e

,(7)

andeachofthesolitonscarriesacharge

ψji∂ψj+eˆ

4π

AµAµ+(ψ−selfinteractions)

,(8)

wherewefollowed[5,6]andintroducedtheinteractionoftheelectromagneticfieldwith

ψjγµ1−γ5thecurrentofdefinitechirality,thatis

17

,andaˆ=(2N−1)arepresentstheregularizationambiguity2N−1

(seebelow).IntheQHEchiralityplaysafundamentalrole.Itiswellknownthatchiralgaugeinteractionsareanomalous10,17,18,andtwo-dimensionalchiralQEDwithanomalousbreakdownofgaugeinvariance17admitsexactsolutionsinapositivemetricHilbertspacerespectingunitarity,providedtheparametera,definedabove(see[17])isrestrictedtoa≥1.Infact,theJR17termae2A2µtakesaccountoftheambiguityintheregularizationprocedureresultingfromthelackofgaugeinvariance.ItmayalsobeobtainedasthemasslesslimitoftheProcatheory10.

ThisisourproposalfortheeffectivefieldtheoryLagrangeantodescribetheQHE.ThereforewearriveatthechiralQED2interaction,togetherwithchiralGross-Neveutypeself-interactionfortheFermifields,aswellastheconstraint(6).SincetheessenceofthechiralGross-Neveuself-interactionistoimplytherelation(6)characteristicofthetwo-dimensionaltheory,whiletheHalleffectitselfdependsontherelationbetweenthecurrentandtheexternalgaugefield,wesimplifymatters,consideringchiralQED2asthemodelfortheQHE,togetherwithrelation(6).(Infact,inthepresenceofimpuritiesthefreeelectronpictureholdstrue13,14).

Aneffectivebosonictheoryisobtainedbymeansofthecomputationofthefermionicdeterminantintwodimensions,throughthePolyakov-Wiegmanformula19.Insuchacase,

3

integrationoverthefermionsleadstoanexternalgaugefielddependentpartitionfunctionoftheform10󰀋2a

ˆ2LidxˆeiW[A]

e=e

4π

wheretheWZWfunctionalisgivenbytheexpression

Γ[h]=

1

4π

i

󰀆

−1

󰀅µ)h,d2xAµh(∂µ−∂

(10b)

trǫµν

󰀆

1

dr

0

󰀆

−1−1−1

d2x󰀅h∂r󰀅h󰀅h∂µ󰀅h󰀅h∂ν󰀅h.(10c)

IntheabovewechoseA−=

2

󰀃eˆ

(∂µφj)2+

j

2

󰀃

j

(∂µφj)−

2

1

2π

A

µ

󰀃

j

󰀅µ)φj+(∂µ−∂

aˆeˆ2

2π

󰀃

i

ˆeˆ󰀅ν)φi+a(∂ν−∂

2π

(13)

.

Considernowthe(classicallyconserved)currents

µ

=JL

󰀅µ)Aµ=0(∂µ−∂

Aµ

.,

eˆ

2π

(14)

µ

=−(gµν+ǫµν)(∂νφi−eˆAν)JiR

µ

,asexpected,Inthequantumtheorywehaveconservationoftheright-movingcurrentJiR

µ

=0∂µJiR

,(15)

buttheleftcurrentisanomalous!Indeed,wehave

µ

∂µJL

=(2N−1)

eˆ2

2π(2N−

=

e2

1)2

󰀊󰀍

󰀅µAµ(a−1)∂µ+∂

,

,wereadtheHallconductivityfromtheanomaly

coefficient.Thefirstterm,namely(a−1)∂µAµisapuretwo-dimensionaleffect,anddoes

󰀮,B󰀮usualsystem.Therefore,thenotappeariftheexternalgaugefielddescribestheE

(minimal)Hallconductivityisgivenbythecoefficientoftheanomaly22,

σH=

e2

2ǫµνF

µν

2π5

eˆ2δijδ′(x−y),

(19)

whichshouldbecomparedto[12].Moreover,theW∞algebraobtainedforthenon-relativisticelectrongasdescription23canbeunderstoodasthealgebraofhigherspinchiralcurrentstudiedis[27]fortheabeliancase,andin[28]inthenon-abeliancase.Suchhigher-dimensionalalgebrasunderlinethemodelsconsidered21.

Althoughwedidnotdiscusstheissueofthehierarchyofthedifferentfillingfactors,itshouldbeclearatthispointthatoncewehavegotten(18),andconsequentlythesimplestfillingfactorν=1/3(forN=2),theargumentsusedin[29]nowapply.

ThisworkwaspartiallysupportedbyCAPES(E.A.),Brazil,undercontractNo.1526/93-4andbyCNPq(M.C.B.A.),Brazil,undercontractNo.204220/77-7.

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