Phase Transition in Anyon Superconductivity at Finite Temperature

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

PHASETRANSITIONINANYONSUPERCONDUCTIVITYATFINITE

TEMPERATURE

E.J.Ferrer andV.delaIncera

Dept.ofPhysics,StateUniversityofNewYork,Fredonia,NY14063,USA

(SUNY-FRE-98-04)

Themagneticresponseofthechargedanyon uidattemperatureslargerthanthefermionenery

gap(T ωc)isinvestigatedintheself-consistent eldapproximation.Inthistemperatureregiona

newphase,characterizedbyaninhomogeneousmagneticpenetration,isfound.Theinhomogeneity

islinkedtotheexistenceofanimaginarymagneticmasswhichincreaseswiththetemperatureat

T ωc.Thesystemstabilityinthe(T ωc)-phaseisprovedbyinvestigatingtheelectromagnetic

eldrest-energyspectrum.

arXiv:hep-th/9805039v1 7 May 1998I.INTRODUCTIONInrecentyearstherehasbeenmuchinterestininvestigating(2+1)-dimensionalgaugetheorieswithChern-Simons(CS)interactions.Thisinterestisdue,inpart,toavarietyofsigni cantphysicalapplicationsofthesetheoriesinQFT,aswellasincondensedmatterphysics.AwellknownexampleinQFTistheDeser,JackiwandTempletonParityAnomaly[1].Thisresultshowsthatthe uctuationsofamassiveFermi eldinducesaCSterminthee ectiveactionofthegauge elds.Inthiscontextthephotonacquiresatopologicalmass[1],[2].Parityviolation[3]andvortexsolutionsinlowerdimensions[4]areamongtheconsequencesofthepresenceofCSterms.Ontheotherhand,ithasbeenshownthatwhenDiracfermionsarecoupledtoaMaxwell-Chern-Simons(MCS)gauge eld,theLorentzsymmetrycanbespontaneouslybrokenbythedynamicalgenerationofamagnetic eld[5].Thee ectsofCStermsinsupersymmetricmodelshavebeenalsoinvestigated[6].Incondensedmatter,CSmodelshavebeenconsideredinthestudyofdi erentphysicalapplications.CStheoriesinvolvingseveralvectorpotentialsareknowntobeparticularlyappropriatefordescribingthequantumHalle ect[7],[8],[9].ArecentmodelofthefractionalquantumHalle ectconsidersthattheelectronsaretransformedintocompositefermionsbyattachingtwoarti cialstatistical uxquantatoeachelectron[10].Thegauge eldtheorydescribingthesecompositefermionsintroducesthestatisticalgauge uxviaCS elds[11].Whenmatter eldiscoupledtotheCSgauge eldsin(2+1)-dimensions,asuitabledescriptionforanyonsisobtained[12],[13].Anyons[14],[15]areparticleswithfractionalstatisticsin(2+1)-dimensions.TheanyondescriptionwithintheCSgaugetheoryisequivalenttoattaching uxtubestothechargedfermions.TheAharonov-Bohmphasesresultingfromtheadiabatictransportoftwoanyonsisthesourceofthefractionalexchangestatistics[15].Ithasbeenarguedthatstronglycorrelatedelectronsystemsintwodimensionscanbedescribedbyane ective eldtheoryofanyons[16],[17].Anyonscanbealsoobtainedassolitonswhichfractionalspininelectronsystems.Excitationswithfractionalspinintwodimensionalsystemsnecessarilyobeyfractionalstatistics[18].Animportant

featureoftheanyontheoryisthatitviolatesparityandtime-reversalinvariances.Althoughthereareclaimsthatanyonscouldplayabasicroleinhigh-TCsuperconductivity[17]-[20],atpresentnoexperimentalevidencesofPandTviolationinhigh-TCsuperconductivityhavebeencon rmed.Itshouldbepointedout,nevertheless,thatitispossibletoconstructmoresophisticatedPandTinvariantanyonicmodels[21].

Whetherlinkedornottohigh-TCsuperconductivity,anyonsuperconductivityisaninterestinge ectinitsownright,anddeservesadeeperstudy.Asitisknown,anyonsuperconductivityhasanorigindi erentfromtheNambu-Goldstone-Higgslikemechanism.Thegenesisoftheanyonsuperconductivityisgivenbythespontaneouslyviolationofcommutativityoftranslationsinthefreeanyonsystem[22].Thisnewmechanismmayhavewiderapplicationsthantheoriginalphysicalproblemthatmotivateditsstudy.

ThesuperconductingbehaviorofanyonsystemsatT=0hasbeeninvestigatedbymanyauthors[13],[19]-[25].AtT=0,anyonsuperconductivityappearsduetotheexactcancellationbetweenthebareandinducedCStermsinthee ectiveactionofthetheory[26].However,atT=0thiscancelationdoesnottakeplace[27].Hence,several

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

authors[27]-[31]haveadvocated

that

the

superconductingphaseevaporatesatany nitetemperature.Inref.[28],ithasbeenindependentlyclaimedthatthedestructionoftheanyonsuperconductingphaseat nitetemperatureisconnectedtotheexistenceofalong-rangemode, 1foundinsidethein nitebulkatT=0.Thislongrangemodeistheconsequenceoftheexistenceofapole～

λ～10 5cmforT～200Kandelectrondensitiescharacteristicofthehigh-Tcsuperconductors).

Ourmainconclusionwasthatthemagneticbehavioroftheanyon uidisnotjustdeterminedbyitsbulkproperties,butitisessentiallya ectedbythesampleboundaryconditions.Theimportanceoftheboundaryconditionsin(2+1)-dimensionalmodelshasbeenpreviouslystressedinref.[34].

Itisnaturaltoexpectthatattemperatureslargerthantheenergygapthissuperconductingbehaviorshouldnotexist.Atthosetemperaturestheelectronthermal uctuationsshouldmakeaccessiblethefreestatesexistingbeyondtheenergygap.Asaconsequence,thechargedanyon uidshouldnotbeaperfectconductoratT ωc.Asignalofsuchatransitionmaybefoundstudyingthesystemmagneticresponseatthosetemperatures.Themaingoalinthispaperistoinvestigatethecharacteristicsofthemagneticresponseinthishightemperaturephase.

InwhatfollowsweshowthatatT ωcanexternallyappliedconstantandhomogeneousmagnetic eldcanpenetratethesample,givingrisetoaperiodicinhomogeneousmagnetic eldwithinthebulk.Theinhomogeneityofthemagneticresponseincreaseswiththetemperature.Wealso ndthat,contrarytotheT ωccase,along-rangepenetration,associatedtoamasslessmodeoftheelectromagnetic eldwithintheanyon uid,canpenetratetheboundedsampleinthehightemperaturephase.Nevertheless,intherangeoftemperaturesconsideredthee ectofthishomogeneous eldpenetrationisnegligibleascomparedtotheinhomogeneouscomponent.

Theseresultscorroboratetheexistenceofaphasetransitioninthechargedanyon uidfromasuperconductingphase,atT ωc(seeRef.[33]),toanon-superconductingphase,atT ωc.Theinhomogeneouscharacterofthemagneticresponseinthehightemperaturephaseislinkedtotheinhomogeneityofthespatialdistributionoftheinducedmany-particlechargeandcurrentdensitiesatT ωc.

Wealsoprovethattheinhomogeneousmagneticpenetrationisassociatedtoanimaginarymagneticmassassociatedtooneoftheelectromagnetic eldmodeswithinthechargedanyon uid.TheappearanceofanimaginarymagneticmassatT ωcbynomeansindicatesthatthelinearapproximationusedinthesecalculationsisbrokenbythepresenceoftachyons.Tachyons,asitisknown,correspondtoimaginaryrest-energysolutions.TherestenergyandthemagneticmassspectrumoftheMCStheoryat nitedensityarenotthesameatT ωc,asprovedinSec.4.FortheMCStheoryat nitedensity,weshowthattherestenergiesoftheelectromagnetic eldmodesatT ωcarereal.

Theplanforthepaperisasfollows.InSec.2,forcompleteness,aswellasfortheconvenienceofthereader,wereviewthemany-particle(2+1)-dimensionalMCSmodelusedtodescribethechargedanyon uidintheself-consistent eldapproximation.Principalattentionisgiventothederivationofthehigh-temperatureMCSe ectiveactioninthelinearapproximation.InSec.3westudythemagneticresponseintheself-consistent eldapproximationofachargedanyon uidcon nedtoahalfplaneinthe(T ωc)-phase.We ndtheanalyticalsolutionoftheMCS eldequationsthatsatis esthecorrespondingboundaryconditionsandminimizesthesystemfreeenergy.Thepropagationmodesofthemagnetic eldwithintheanyon uidhavethreecontributions:onethatdecaysexponentiallyinspace,otherthataccountsforanhomogeneous eldpenetration,andathirdthatchangesperiodicallyinspacewithinthesample.Whenallthecoe cientsappearinginthemagnetic eldsolutionareevaluatedfortherangeoftemperaturesandothercharacteristicparametervalues,we ndthattheleadingterminthemagneticresponseisaninhomogeneousmagnetic eldwithacharacteristicwavelengththatdecreaseswiththetemperature.InSec.4weinvestigatethedispersionequationfortheMaxwell eldinthehigh-temperatureapproximation.WesolvethisequationinarbitrarycovariantgaugesfortheMaxwellandCS eldstoobtainthemagneticmassesofthechargedanyon uidinthehigh-temperaturephase.Asexpected,themagneticmassesturnouttobeequaltotheinverselengthscaleswhichcharacterizethemagneticresponseoftheanyon uidinthisphase.Wethenprovethattheexistenceofanimaginarymagneticmasscannotbeassociatedtoatachyonicmodeinthismany-particlesystemwithCSinteractions.Indoingthat,we ndtherestenergiesoftheelectromagnetic eldmodesatT ωc.Sec.6containsthesummaryanddiscussion.

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

II.ANYONFLUIDATHIGHTEMPERATUREANDDENSITY

A.Many-particlemodelandenergygap

Thenon-relativisticchargedanyonsystemininteractionwithanelectromagnetic eldin2+1dimensionscanbemodeledbytheMCSLagrangiandensity

L= 1

4πεµνρaµ νaρ+eneA0+iψ D0ψ 1

4πεµνρfνρ= jµ (2.6)

(2.7) νFµν=e jµ eneδµ0

Toguaranteetheelectricneutralityofthesystemformedbytheelectron uidandthebackgroundchargeneweimposethecondition

0 j neδµ0=0,(2.8)

wherej0isthemany-particlesystemfermiondensity

0 j=

ip4+µ p2

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

Theexistenceofadi erentfromzerofermiondensity,asitisrequiredbytheneutralitycondition(2.8),generatesthrougheqs.(2.6)-(2.7)anontrivialbackgroundofCSmagnetic eld

f21=

b,

H0= 1a1)+22 22πne

In(2.13)weconsideredthebackgroundCSpotential,

ak= (2.13)

2 ωc

(2.16)

Ψnk=√

whereωc=√Lb(2.17)

b/2π.Thus,the llingfactor,de nedastheratiobetweenthedensityof

particlesneandthenumberofstatesperunitareaofafullLandaulevelnB,isequaltotheCScouplingconstantN.WhenthemagnitudeoftheCScouplingconstantNisconsideredasapositiveinteger,the

systemwillhaveNcompletely lledLandaulevels.Oncethisgroundstateisestablished,itcanbearguedimmediately[19],[20],[22],

[24],thatatT=0thesystemwillbecon nedtoa lledband,whichisseparatedbyanenergygap(ωc)fromthefreestates;therefore,itisnaturaltoexpectthatatT=0thesystemshouldsuperconduct.ThisresultisalreadyawellestablishedfactonthebasisofHartree-Fockanalysis[19]andRandomPhaseApproximation[20],[22].AtT=0itwasprovedinRefs.[33]thattheexistenceofanaturalscale(thecyclotronfrequencyωc)inthistheory,makespossibletherealizationofasuperconductingphaseatT ωc,forsystemscon nedtoaboundedregion.Itislogicaltoexpectthatthisenergyscale,ωc,separatestwodi erentphysicalphasesofthesystem.Aswewillprovebelow,thesuperconductingstate,foundatT ωc,disappearswhenthesystemreachestemperatureslargeenough(i.e.atT ωc)tomovetheelectronsbeyondtheenergygaptothefree-energyband.

B.E ectiveactioninthelinearapproximation

Thelinearresponseofthemediumcanbefoundundertheassumptionthatthequantum uctuationsofthegauge eldsabouttheground-statearesmall.Inthiscasetheone-loopfermioncontributiontothee ectiveaction,obtainedafterintegratingoutthefermion elds,canbeevaluateduptosecondorderinthegauge elds.Thee ectiveactionintermsofthequantum uctuationofthegauge eldswithinthelinearapproximation[28],[29]takestheform

1µνρΓeff(Aν,aν)=dx (2.18)εaµ νaρ+Γ(2)

4π

Γ(2)istheone-loopfermioncontributiontothee ectiveactioninthelinearapproximation

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

Γ(2)= dxdy[aµ(x)+eAµ(x)]Πµν(x,y)[aν(y)+eAν(y)].(2.19)

In(2.19)Πµνrepresentsthefermionone-looppolarizationoperatorinthepresenceoftheCSbackgroundmagnetic eld

k2+Π0′,A2=Π1,A3+A1=Π2(2.21)

From(2.20)and(2.21)wehavethatthenewindependentcoe cientscanbefoundfromthepolarizationoperatorcomponents

Π0Π00=

+Π0

′

k2 k0k2 iΠ1k1(2.23)

2Π22=Π2k1 Π0

b

[28],[29],[36]

G(p4,p)=∞ dρ

n=0 ∞0∞ ∞ dx2bexp (ip2x2)exp ip4+µ 2m ρ n(ξ) n(ξ′)tn(2.25)

where

t=expmρ,ξ=p1

2

,ξ′=p12(2.26)

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

IntheLandaugauge,theΠµνEuclideancomponents:Π00,Π02andΠ22aregivenby[29],

Π00k,µ,

b=

Πjkk,µ,i(2π)2β dpp0 +G(p)·DjG(p k)+DjG(p)·G(p k),(2.28)4m2β dpp0

2m

wherethenotationΠ4,(2.29)

±DjG(p)=ipj εjk pkG(p),2 ±DjG(p k)=i(pj kj) εjk pkG(p k),2 (2.30)

hasbeenused.Theindependentcoe cients:Π0,Π0′,Π1andΠ2,foundfrom(2.22)-(2.24),arefunctionsofk2,βand

b

π

8π n(2n+1)Θn (2.34)

Π1=1b

πm

where n(2n+1) n β32πm2 n(2n+1)2Θn (2.36)

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

Θn=sech2β( n

2, n=(eβ( n

2π tanhβµ

48πsech2 βµb

12m2Π0(2.38)

Intheseexpressionsm=2me(meistheelectronmass).

III.MAGNETICRESPONSEINTHEHIGH-TEMPERATUREAPPROXIMATION

Inthehightemperatureregion,fortemperaturesabovetheenergygap(T ωc),theelectronswillbeenergizedenoughtoreachtheemptyLandaulevelbands,aswementionedabove.Consequently,theelectroncon nementintoacompletely lledbandislost,beingpossiblefortheelectronstochangetheirinitialstates.Hence,itisnaturaltoexpect,fromaheuristicpointofview,thatinthishightemperaturephasethesystemcannotbehaveasasuperconductor.

IfthesystemisnotasuperconductoratT ωc,thenithastoallowthepenetrationofanexternallyappliedconstantmagnetic eld.Inotherwords,noMeissnere ectcantakeplace.Inthissectionwewillshowthatthisisindeedthecase.

A.Many-ParticleSystemLinearEquations

Toobtaintheextremumequationscorrespondingtotheanyonmany-particlesystemwehavetoconsiderthevariationalproblemderivedfrom

thee ectiveaction(2.18).Thisformulationisknownintheliteratureastheself-consistent eldapproximation[29].

Incomponentform,the eldequationsinthepresenceoftheinducedcurrentsandtheCS eldsare

·E=eJ0

0Ek+εkl lB=eJk

eN(3.1)(3.2)

2πf0k=εkl 0El+ kB(3.4)

µwherefµνistheCSgauge eldstrengthtensor,de nedasfµν= µaν νaµ,andJindisthecurrentdensityinduced

bythemany-particlesystem.

Insolvingeqs.(3.1)-(3.4)wecon neouranalysistogauge eldcon gurationswhicharestaticanduniforminthey-direction.WithinthisrestrictionwetakeagaugeinwhichA1=a1=0.Then,thedi erentcurrentdensitycomponentsare

0(x)=Π0[a0(x)+eA0(x)]+Π0′ x(E+eE)+Π1(b+eB)Jind

1Jind(x)=0,2Jind(x)=Π1(E+eE)+Π2 x(b+eB)(3.5)(3.6)

Intheaboveexpressionsweusedthefollowingnotation:E=f01,E=F01,b=f12andB=F12.Eqs.(3.5)-(3.6)playtheroleintheanyon uidoftheLondonequationsinBCSsuperconductivity.Whentheinducedcurrents(3.5)-(3.6)aresubstitutedineqs.(3.1)-(3.2)we nd,aftersomemanipulation,asetofindependentdi erentialequationsdependingonthe eldsBandE,alongwiththezerocomponentsofthegaugepotentials,A0anda0,

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

2ω xB+αB=γ[ xE σA0]+τa0,

2 xB=κ xE+ηE,(3.7)(3.8)

Inobtainingtheseequationswehaveusedtheeqs.(3.3)-(3.4)toeliminatetheCSmagneticandelectric eldsintermsoftheMaxwell elds

b= χ xE

E= χ xB(3.9)(3.10)

Thecoe cientsappearinginthesedi erentialequationsdependonthecomponentsofthepolarizationoperatorsthroughtherelations

ω=2π

eN

γ=1+e2Π0′ 2π

N,σ= e2δ2πΠ1,Π1,κ=

A0,ThesolutionsforB,a0andA0,canbeobtainedusingeqs.(3.8),(3.9),(3.13)andthede nitionofEintermsof

B(x)= γ1C1e xξ1 C2exξ1 γ2C3e xξ2 C4exξ2+C5(3.15)

(3.16) 1d2 4ac2a(3.14) a0(x)=χγ1C2exξ1 C1e xξ1+χγ2C4exξ2 C3e xξ2+C6

A0(x)=1

ξ2 C3e xξ2 C4exξ2+C7(3.17)

2 2 Intheaboveformulasweintroducedthenotationγ1=ξ1κ+η/ξ1,γ2=ξ2κ+η/ξ2.

Ascanbeseenfromthemagnetic eldsolution(3.15),therealcharacteroftheinverselengthscales(3.14)iscrucialfortherealizationoftheMeissnere ect.Attemperaturesmuchlowerthantheenergygap(T ωc)thisisindeedthecase,asitwasshowninourpreviousworks(seeref.[33]).

Inthehightemperatureregion(T ωc)thepolarizationoperatorcoe cientsaregivenbyeqs.(2.38).Usingthisapproximation,andassumingthatN=2,togetherwiththeassumptionne m2(thisapproximationisinagreementwiththetypicalvaluesfoundinhigh-TCsuperconductivity),wecancalculatethecoe cientsa,canddthatde nethebehavioroftheinverselengthscales,

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

a π2Π0′Π2

c e2Π0

d 1(3.18)(3.19)(3.20)

Substitutingwith(3.18)-(3-20)ineq.(3.14),weobtainfortheinverselengthscalesinthehigh-temperaturelimit

ξ1 e

1 βµm/2πtanh2m

22(3.21) ξ2 βµtanh2

(3.22)

Theimaginaryvalueoftheinverselengthξ2is duetothefactthatatT ωc,Π2>0andΠ0′<0(seeeq. (2.38)).Animaginaryξ2impliesthatthetermγ2C3e xξ2 C4exξ2,inthemagnetic eldsolution(3.15),doesnothaveadampingbehavior,butanoscillatingone.Ontheotherhand,thepresenceoftheconstantcoe cientC5inthemagnetic eldsolution(3.15)meansthatthereexistsamagneticlong-rangemode.InSec.4wewillseehowthislong-rangemodeisassociatedtotheexistenceofamodewithzeromagneticmass,M1=0.

Nevertheless,tocompletelydeterminethecharacteristicsofthemagneticresponseinthiscase,itisneededto ndthevaluesoftheC′sunknowncoe cientswhichareinagreementwiththeproblemboundaryconditionsandtheminimizationofthesystemfree-energydensity.

C.BoundaryConditionsandStabilityConditionfortheSemi-in niteSample

Inordertodeterminetheunknowncoe cients,weneedtousetheboundaryconditions.Henceforthweconsiderthattheanyon uidiscon nedtoahalfplane ∞<y<∞withboundaryatx=0.Theexternalmagnetic eldisappliedfromthevacuum( ∞<x<0).Theboundaryconditionsforthemagnetic eldareB(x=0)=Bconstant),andB(x→∞) nite.Sinceineqs.(3.7)-(3.8)themagnetic eldistangledtotheelectric eldE,theboundaryvaluesofEhavetobetakenintoaccountindeterminingtheunknowncoe cients.Becausenoexternalelectric eldisapplied,theboundaryconditionsforthis eldare,E(x=0)=0,E(x→∞) nite.Afterusingtheseconditionsitisfoundthat,

C2=0, C1=C3+C4,C1=C5+(C3 C4)γ2 γ1(3.23)

Introducingthenewnotationξ2=i

ξ2+Asinx

γ2Acosx

A=i(C4 C3), ξ22κ η/ ξ2+C5(3.25)where

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

Eq.(3.27)establishesaconnectionbetweenthelinearcombinationofthecoe cientsofthelong-rangemodesofthezerocomponentsofthegaugepotentials,(C6+eC7),andthecoe cientofthelong-rangemodeofthemagnetic eld,C5.NotethatiftheinducedCScoe cientΠ1,ortheDebye-screeningcoe cientΠ0werezero,therewouldbenolinkbetweenC5and(C6+eC7).Thisrelationbetweenthelong-rangemodesofB,A0anda0canbeinterpretedasasortofAharonov-Bohme ect,whichoccursinthissystemat nitetemperature[33].AtT=0,wehaveΠ0=0,andthise ectdisappears.

Afterusingtheboundaryconditions(3.23),itfollowsthattheyarenotsu cienttodeterminethecoe cientsC5andA.WeneedanotherphysicalconditionfromwhereC5andAcanbefound.Since,obviously,anymeaningfulsolutionhavetobestable,thenaturaladditionalconditiontobeconsideredisthestabilityequationderivedfromthesystemfreeenergy.Withthisgoalwestartfromthefreeenergyofthehalf-planesample

F=1

πa0b Π0(eA0+a0)2

Π0(eE+E) 2Π1(eA0+a0)(eB+b)+Π2(eB+b)′22

whereLandL′determinethetwosample’slengths.

In(3.28)wehavetosubstitutethe eldsolutions(3.16),(3.17),(3.24)and(3.25)togetherwiththesolutionsfortheCS elds(thatcanbefoundsubstituting(3.24)and(3.25)ineqs.(3.9)and(3.10)respectively)

b(x)=χξ1C1e xξ1 χ

ξ2

A,(A

C1by (3.28)ξ2+C1sinxξ2 C1cosxL′→∞)isgivenasafunctionofAand

f=1=LL′beingthesamplearea)inthesample’slengthlimit(L→∞,

γ22+G,2X2=gγ1+G,X3= 2gγ1

X6=g

+e2Π2

BBγ1,Π0

1N(3.33)G=γ22

Π1 χξ2

=ξ2 γ2e+χγ2+2 2Π2 2e+χγ2ξ2 e1

δA

δC1=1

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

tobe

A=2+γ1B(3.37)

C1= 3gγ1

γ22

γ2A+

From(3.39)weseethatforT ωctheelectromagnetic eldlong-rangemodepropagatesintothesample,producinganhomogeneousmagneticpenetration.Thisresultisdi erentfromtheoneobtainedinthelow-temperaturelimit(T ωc)[33].InthatlimititwasfoundthatC5=0,whichimpliesthatthelong-rangemodecannotpropagatewithinthesamplewhenauniformandconstantmagnetic eldisperpendicularlyappliedatthesample’sboundaries.

D.InhomogeneousMagneticResponse 2+G)γ2+(gγ11B(3.39)

Asithasbeenpreviouslyestablished,inthehigh-temperaturelimitthecoe cientsA,C1andC5arealldi erentfromzero,i.e.,inthe(T ωc)-phasethemagneticresponseofthechargedanyon uidhasanexponentialdecayingcomponent,aswellas,bothhomogeneousandinhomogeneouspenetrations..Tocompleteourstudyofthemagneticresponseathightemperaturewestillneedtoestimatethecorrespondingvaluesofeachcomponentfortherangeofparametersandtemperatureshereconsidered.

Atthedensitiesunderconsideration,ne m2,theestimatedvaluesofthecoe cientsA,C1andC5inthehigh-temperatureapproximation(T ωc)are

A≈103B,C5≈10 4

λ x(3.41)

B(x)=

where

E0(T)=12√λ x 1(3.42)

2+1

B.Moreover,theinhomogeneousmagnetic eldpenetration(3.42)is

characterizedbyawavelengthλ,whichisproportionaltotheinverseofthelengthscalemagnitude

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

AtT ωc,usingthatµ πne

ξ2increaseswiththetemperature(seeeq.(3.22)),wehave,thatthewavelengthdecreases

withT.Thehigh-temperatureleadingbehaviorforλisgivenby

λ≈π1

B,andthetemperature.TheE’sinhomogeneity

alsoincreaseswiththetemperaturethroughλ.

Fromtheobtainedresultsweconcludethatfortemperatureslargerthantheenergygap,thechargedanyon uidisinanewphaseonwhichthesuperconductivityislost(nonMeissnere ectisfoundinthisphase).

Theinductionofinhomogeneouselectricandmagnetic eldswithinthechargedanyon uidathightemperature,indicatesthatsomeredistributionoftheinducedchargeandcurrentsoccursatT ωc.

Toverifythis,letuscalculatetheinducedelectricchargedensityofthechargedmediuminthehigh-temperatureapproximation.Consideringeq.(3.5)inthehigh-temperaturelimit,we ndthattheinducedelectricchargedensitypresentsaninhomogeneousspatialdistributionwithhigh-temperatureleadingcontributiongivenby

√βµeJ0(x)=24Btanhx(3.46)λ

Asdiscussedabove,inthehigh-temperatureregimeλ～

2 Bcosh βµ

2 +1 1 x(3.47)λ

Obviously,thecurrentdensity(3.47)isnotasupercurrentcon nedtothesample’sboundary.

IV.MAGNETICMASSANDRESTENERGYINTHECHARGEDANYONFLUIDATT ωC

Wehaveseenthattheinverselengthscales,ξ1andξ2,arebasicelementsinthedeterminationofthemagneticresponseofthechargedanyon uid.InthisSec.weshallgoonestepforwardinclarifyingthephysicalinterpretationoftheseparameters.Wewillshowthattheinverselengthscales(3.21),(3.22)canbeidenti edwiththemagneticmassesoftheelectromagnetic eldwithinthe uidatT ωc.AparticularlyimportantpointinthisSec.isourproofthattheexistenceathightemperatureofanimaginarymagneticmass(correspondingtotheinverselength(3.22))isnotlinkedtothepresenceoftachyonsinthetheory,ortothebreakingofthelinearapproximation,aswassuggestedinref.[28].Asshownbelow,themagneticmassandtherestenergyarenotthesameintheMCStheory(contrarytowhathappensinaKlein-Gordon-liketheory).

Toinvestigatethemagneticmassesandtherestenergiesassociatedwiththeelectromagneticmodes,weneedtostudytheelectromagnetic elddispersionequation.Withthisaimwestartfromthee ectiveaction(2.18)takeninthecovariantgaugesfortheMaxwellandCS elds

1

α2 µaµ=0(4.1)

α1andα2beingtwoindependentgaugeparameters.

Thecorrespondinge ectiveLagrangiandensityfortheMaxwellandCS eldcon gurationscanberepresentedas

Leff= 1

2 1aµ( k)Dµν(k)aν(k) eAµ( k)Πµν(k)aν(k)(4.2)

1 1wherethematrices µνandDµνaregivenby

12 µν(k)=kgµν 1 1

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

1Dµν(k)=iN

α2kµkν+Πµν(k)(4.4)

Πµνistheone-loopfermionpolarizationoperatorgivenfrom(2.20)-(2.21)by A1k2A1kω iΠ1k iΠ1ωΠµν= A1kω A1ω2

22iΠ1k iΠ1ω A1ω+Π2k(4.5)

In(4.5)weareconsideringtheframek=(k,0),andthenotationω=k0hasbeenused.

Thee ectivetheoryfortheelectromagnetic eldinthechargedanyon uid,isfoundintegratingtheCS eldsinthepartitionfunction(2.3)withLagrangiandensity(4.2).Thenewe ectiveLagrangiandensitysoobtainedis

L′eff= 1

2± 2

Furthermore,inthepresenceofabackgroundmagnetic eld,thedistinctionbetweenmassandrest

energybecomesessential.Forinstance,inthecontextofstringtheoryinabackgroundmagnetic eld,ithasbeenshown[37]thatthemass(de nedinagreementwithWigner’sde nition)ofhigherspin(s≥1)chargedbosonparticlesdoesnotcoincidewiththeirrestenergy.Thisisduetothemodi cationofthealgebraoftheglobalsymmetriesbythebackground eld.

Fromtheabovediscussion,itisalsoclearthatincalculatingthemagneticmass(eq.(4.10))andtherestenergy(eq.(4.11))at nitetemperature,wehavetotakethepolarizationoperatorscoe cientsinthestaticlimit(ω=0,k～0)andintheplasmonlimit(k=0,ω～0)respectively.Now,becauseofthelackofanalyticityoftheGreen’sfunctionaboutkµ=0atT=0,itisknownthatinQFTtheselimitsdonotcommute[35],[38].InanyontheoryatT=0onefacesasimilarsituation,asitwasshowninref.[31].Then,ineachcasewehavetoconsiderthepolarizationoperatorcoe cientsevaluatedinthecorrespondinglimit.

InanyontheorytheCSinteractiongivesrisetoadispersionequationwithastructuremorecomplicatedthanthatcorrespondingtoaKlein-Gordon-liketheory.Asshownbelow,inthiscasethemagneticmassesandtherestenergiesoftheelectromagneticmodesaredi erent.

2+M2(4.13)

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

A.Electromagnetic eldmagneticmassesathightemperatures

To ndtheelectromagnetic eldmagneticmasseswemustsolvethedispersionequation(4.8)atω=0.Sinceweareinterestedinthemagneticmassesattemperatureshigherthantheenergygap(T ωc),weshouldconsiderinsolving(4.8)thepolarizationoperatorcoe cientsinthehigh-temperatureapproximation(2.38).

Letusdetermine rsttheexpressionofthematrixNµν(k)(eq.(4.7))atω=0.InthiscasethematrixDλρ(k)appearingineq.(4.7)takestheform

Dλρ(k)=k2

π+Π1(4.16)

Thus,aftertakingthematrixproductsindicatedin(4.7)withthepolarizationoperator(4.5)evaluatedatω=0,we nd,

Nµν(k)=e2

α1

with 2k+e2A1G1k21+e2G1Π2+e4G22=0

G1=B(4.19)

α2D+Π1(4.21)

ForN=2,takingintoaccountthatinnaturalunitse2～105cm 1,andconsideringthecharacteristicvaluesne=2×1014cm 2andme=2.6×1010cm 1,wecanestimatetherelativeordersbetweenthepolarizationoperatorcoe cients(2.38)attemperatureslargerthanωcas

Π0≈ 108e2Π1≈ 107e4Π0′≈1020e4Π2

is

det[ (k) N(k)] (Π2Π0′)2(4.22)Takingintoaccounttherelations(4.22),thehigh-temperatureleadingcontributiontothedispersionequation(4.19)

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

σ1=2

4,σ3=2σ1

3

with

A=σ1,y2,3= A+B21/3√3√(4.26)6(1 B), A,B=33 3i

σ1

Inthehigh-temperaturelimit A,B 1,sowecantaketheexpansions,

(1 A,B)1/3≈1 1

9 2A,B(4.27)(4.28)

Substitutingwith(4.27)and(4.28)in(4.26)weobtaintheleadinghigh-temperatureapproximationfortherootsofeq.(4.25),

y1= e2Π0,y2,3= 1

π2(Π2Π0′) 1(4.33)

Weshouldnotethatthemagneticmasses(4.32)-(4.33)aregaugeindependent.Thatis,theydonotdependonthegaugeparametersα1andα2.Wecanseethattheequation(4.23),fromwherethemagneticmassesarefound,isindependentofthegaugeparameterα2,sinceα2Ddoesnotdependonα2;andα1appearsonlyasamultiplicativefactor.

From(4.31)-(4.33)wehavethatoneoftheinfraredmodesoftheelectromagnetic eldintheanyon uidismassless,2M1=0,whileM22>0andM3<0.Thesignsobtainedforthesquareofthemasses(4.32),(4.33),areaconsequenceofthefactthatinthehigh-temperaturelimit,Π0>0,Π2>0andΠ0′<0(eq.(2.38)).

Comparingeqs.(4.32),(4.33)witheqs.(3.21),(3.22)respectively,onecanseethat

ξ1=M2,ξ2=M3(4.34)

Hence,themagneticmassescoincidewiththeinverselengthscales,ξ1andξ2,whichdeterminethemagneticresponseofthemedium.Thisispreciselythephysicalmeaningoftheseinfraredmasses(Mi).Theycannotbeinterpreted,otherwise,astherestenergiesoftheelectromagnetic eldmodes,aswewillseebelow.Thezeromode(4.31)islinkedtothelong-rangecomponentC5appearinginthemagneticresponse(3.25).

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

B.Electromagnetic eldrestenergiesathightemperatures

Therestenergiesoftheelectromagneticmodesarefoundbysolvingthedispersionequation(4.8)forωatk=0.Werecallthatthepolarizationoperatorcoe cientshavetobetakennowintheplasmonlimit(k=0,ω～0).Usingthislimit,we ndthatΠ0(k=0,ω～0)=0,whiletherestofthecoe cientsmaintainthesamefunctionalbehavior(2.38)obtainedinthestaticlimit[31].Thenthepolarizationoperatorisgivenby(4.5)withΠ0=0andk=0.InthiscasethematrixDλρ(ω)appearingineq.(4.7)takestheform D00ω21 ′ω4λρ0 Π(4.35)D(ω)=0α2

ω30iHα2

Inwriting(4.35)thefollowingnotationwasused

D= 1detDµν(k)=ω4

with

M= ωΠ0

56′

Using(4.3)and(4.37)inthedispersionequation(4.8)weobtainthegeneralexpressionfortherest-energyequation

2 1 ω2

M+det µν(ω) Nµν(ω)=α2D

ieΠ1ω 2 2′22N=iω(2Π1 H)(Π0)ω H(Π1) ′22(Π0)ω+(Π1 2H)Π1,000 0MN α2D0 NM (4.37)(4.38)(4.39)e2

α1(α2D)

where

θ1= 2H2

42 6ω+θ1ω4+θ2ω2+θ3=0θ3= 1(4.41)2θ1,

2πΠ0′

2From(4.43)-(4.45)wehavethatω2,3>0;therefore,atT ωcthereisnonegativesquaredrestenergy,which

meansthatthehightemperaturephaseisstable.Hence,theexistenceofanegativesquaredmagneticmass,M23,simplyindicatesthatthereisaninhomogeneousmagneticpenetrationinthechargedanyon uidatT ωc.Finally,weshouldpointoutthattherestenergiesω2,3arealsogaugeindependent(theydonotdependonα1andα2),andtheyaredeterminedbytheexplicit(proportionaltoN)andinduced(proportionaltoΠ1)CScontributions. 2(4.45)

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

V.CONCLUDINGREMARKS

Theparticleenergyspectrumoftheanyontheoryexhibitsabandstructuregivenbydi erentLandaulevelsseparatedbyanenergygapωc.TheenergygapisproportionaltothebackgroundCSmagnetic eld

bwhatdeterminestheenergygap

(ωc=

Tinitshigh-

temperatureleadingorder.Thatis,thespatialinhomogeneityofthemany-particlemagneticresponsewillincreasewiththetemperatureinthisnewphase.

TheabsenceoftheMeissnere ectatT ωcisrelatedtotheappearingofanimaginarymagneticmassM3inthisphase(eq.(4.33)).Theexistenceofanimaginarymagneticmasscannotbeassociatedwithatachyonicmodeinthismany-particlesystemwithCSinteractions.Thereasonisthatthemagneticmassesandtherestenergiesoftheelectromagnetic eldmodesarenotthesamewithinthechargedanyon uid.

Itisimportanttonotethatinobtainingtheinhomogeneousmagneticresponseathightemperatures,itwascrucialthatthepolarizationoperatorcoe cientΠ0′changeditssignfromapositivevalueatT ωc[28],[31],[33],toanegativevalueatT ωc(eq.(2.38)).Thatis,becauseΠ0′changesitssign,whileΠ2continuespositive,wehavethatM3isimaginaryineq.(4.33).

Finally,weshouldstatethattheresultsobtainedinRef.[33]andinthispaperforthesuperconductingpropertiesofthechargedanyon uidhavebeenderivedonthebaseofalinearapproximation.Ifnonlineare ects,asforinstancevortices,areconsidered,itispossiblethataricherscenarioforthesuperconductingphasesoftheanyonsystemwillappear.

Acknowledgments

ThisresearchhasbeensupportedinpartbytheNationalScienceFoundationunderGrantNo.PHY-9722059.

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

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