Phase Transition in Anyon Superconductivity at Finite Temper

1、phase transition in anyon superconductivity at finite temper 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 magne

2、tic penetration, phasetransitioninanyonsuperconductivityatfinite temperature e.j.ferrer andv.delaincera dept.ofphysics,stateuniversityofnewyork,fredonia,ny14063,usa (suny-fre-98-04) themagneticresponseofthechargedanyon uidattemperatureslargerthanthefermionenery gap(t c)isinvestigatedintheself-consis

3、tent eldapproximation.inthistemperatureregiona newphase,characterizedbyaninhomogeneousmagneticpenetration,isfound.theinhomogeneity islinkedtotheexistenceofanimaginarymagneticmasswhichincreaseswiththetemperatureat t c.thesystemstabilityinthe(t c)-phaseisprovedbyinvestigatingtheelectromagnetic eldrest

4、-energyspectrum. arxiv:hep-th/9805039v1 7 may 1998i.introductioninrecentyearstherehasbeenmuchinterestininvestigating(2+1)-dimensionalgaugetheorieswithchern-simons(cs)interactions.thisinterestisdue,inpart,toavarietyofsigni cantphysicalapplicationsofthesetheoriesinqft,aswellasincondensedmatterphysics.

5、awellknownexampleinqftisthedeser,jackiwandtempletonparityanomaly1.thisresultshowsthatthe uctuationsofamassivefermi eldinducesacsterminthee ectiveactionofthegauge elds.inthiscontextthephotonacquiresatopologicalmass1,2.parityviolation3andvortexsolutionsinlowerdimensions4areamongtheconsequencesofthepre

6、senceofcsterms.ontheotherhand,ithasbeenshownthatwhendiracfermionsarecoupledtoamaxwell-chern-simons(mcs)gauge eld,thelorentzsymmetrycanbespontaneouslybrokenbythedynamicalgenerationofamagnetic eld5.thee ectsofcstermsinsupersymmetricmodelshavebeenalsoinvestigated6.incondensedmatter,csmodelshavebeencons

7、ideredinthestudyofdi erentphysicalapplications.cstheoriesinvolvingseveralvectorpotentialsareknowntobeparticularlyappropriatefordescribingthequantumhalle ect7,8,9.arecentmodelofthefractionalquantumhalle ectconsidersthattheelectronsaretransformedintocompositefermionsbyattachingtwoarti cialstatistical

8、uxquantatoeachelectron10.thegauge eldtheorydescribingthesecompositefermionsintroducesthestatisticalgauge uxviacs elds11.whenmatter eldiscoupledtothecsgauge eldsin(2+1)-dimensions,asuitabledescriptionforanyonsisobtained12,13.anyons14,15areparticleswithfractionalstatisticsin(2+1)-dimensions.theanyonde

9、scriptionwithinthecsgaugetheoryisequivalenttoattaching uxtubestothechargedfermions.theaharonov-bohmphasesresultingfromtheadiabatictransportoftwoanyonsisthesourceofthefractionalexchangestatistics15.ithasbeenarguedthatstronglycorrelatedelectronsystemsintwodimensionscanbedescribedbyane ective eldtheory

10、ofanyons16,17.anyonscanbealsoobtainedassolitonswhichfractionalspininelectronsystems.excitationswithfractionalspinintwodimensionalsystemsnecessarilyobeyfractionalstatistics18.animportant featureoftheanyontheoryisthatitviolatesparityandtime-reversalinvariances.althoughthereareclaimsthatanyonscouldplay

11、abasicroleinhigh-tcsuperconductivity17-20,atpresentnoexperimentalevidencesofpandtviolationinhigh-tcsuperconductivityhavebeencon rmed.itshouldbepointedout,nevertheless,thatitispossibletoconstructmoresophisticatedpandtinvariantanyonicmodels21. whetherlinkedornottohigh-tcsuperconductivity,anyonsupercon

12、ductivityisaninterestinge ectinitsownright,anddeservesadeeperstudy.asitisknown,anyonsuperconductivityhasanorigindi erentfromthenambu-goldstone-higgslikemechanism.thegenesisoftheanyonsuperconductivityisgivenbythespontaneouslyviolationofcommutativityoftranslationsinthefreeanyonsystem22.thisnewmechanis

13、mmayhavewiderapplicationsthantheoriginalphysicalproblemthatmotivateditsstudy. thesuperconductingbehaviorofanyonsystemsatt=0hasbeeninvestigatedbymanyauthors13,19-25.att=0,anyonsuperconductivityappearsduetotheexactcancellationbetweenthebareandinducedcstermsinthee ectiveactionofthetheory26.however,att=

14、0thiscancelationdoesnottakeplace27.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 penetrat

15、ion, authors27-31haveadvocated that the superconductingphaseevaporatesatany nitetemperature.inref.28,ithasbeenindependentlyclaimedthatthedestructionoftheanyonsuperconductingphaseat nitetemperatureisconnectedtotheexistenceofalong-rangemode, 1foundinsidethein nitebulkatt=0.thislongrangemodeistheconseq

16、uenceoftheexistenceofapole 10 5cmfort200kandelectrondensitiescharacteristicofthehigh-tcsuperconductors). ourmainconclusionwasthatthemagneticbehavioroftheanyon uidisnotjustdeterminedbyitsbulkproperties,butitisessentiallya ectedbythesampleboundaryconditions.theimportanceoftheboundaryconditionsin(2+1)-

17、dimensionalmodelshasbeenpreviouslystressedinref.34. itisnaturaltoexpectthatattemperatureslargerthantheenergygapthissuperconductingbehaviorshouldnotexist.atthosetemperaturestheelectronthermal uctuationsshouldmakeaccessiblethefreestatesexistingbeyondtheenergygap.asaconsequence,thechargedanyon uidshoul

18、dnotbeaperfectconductoratt c.asignalofsuchatransitionmaybefoundstudyingthesystemmagneticresponseatthosetemperatures.themaingoalinthispaperistoinvestigatethecharacteristicsofthemagneticresponseinthishightemperaturephase. inwhatfollowsweshowthatatt canexternallyappliedconstantandhomogeneousmagnetic el

19、dcanpenetratethesample,givingrisetoaperiodicinhomogeneousmagnetic eldwithinthebulk.theinhomogeneityofthemagneticresponseincreaseswiththetemperature.wealso ndthat,contrarytothet ccase,along-rangepenetration,associatedtoamasslessmodeoftheelectromagnetic eldwithintheanyon uid,canpenetratetheboundedsamp

20、leinthehightemperaturephase.nevertheless,intherangeoftemperaturesconsideredthee ectofthishomogeneous eldpenetrationisnegligibleascomparedtotheinhomogeneouscomponent. theseresultscorroboratetheexistenceofaphasetransitioninthechargedanyon uidfromasuperconductingphase,att c(seeref.33),toanon-supercondu

21、ctingphase,att c.theinhomogeneouscharacterofthemagneticresponseinthehightemperaturephaseislinkedtotheinhomogeneityofthespatialdistributionoftheinducedmany-particlechargeandcurrentdensitiesatt c. wealsoprovethattheinhomogeneousmagneticpenetrationisassociatedtoanimaginarymagneticmassassociatedtooneoft

22、heelectromagnetic eldmodeswithinthechargedanyon uid.theappearanceofanimaginarymagneticmassatt cbynomeansindicatesthatthelinearapproximationusedinthesecalculationsisbrokenbythepresenceoftachyons.tachyons,asitisknown,correspondtoimaginaryrest-energysolutions.therestenergyandthemagneticmassspectrumofth

23、emcstheoryat nitedensityarenotthesameatt c,asprovedinsec.4.forthemcstheoryat nitedensity,weshowthattherestenergiesoftheelectromagnetic eldmodesatt carereal. theplanforthepaperisasfollows.insec.2,forcompleteness,aswellasfortheconvenienceofthereader,wereviewthemany-particle(2+1)-dimensionalmcsmodeluse

24、dtodescribethechargedanyon uidintheself-consistent eldapproximation.principalattentionisgiventothederivationofthehigh-temperaturemcse ectiveactioninthelinearapproximation.insec.3westudythemagneticresponseintheself-consistent eldapproximationofachargedanyon uidcon nedtoahalfplaneinthe(t c)-phase.we n

25、dtheanalyticalsolutionofthemcs eldequationsthatsatis esthecorrespondingboundaryconditionsandminimizesthesystemfreeenergy.thepropagationmodesofthemagnetic eldwithintheanyon uidhavethreecontributions:onethatdecaysexponentiallyinspace,otherthataccountsforanhomogeneous eldpenetration,andathirdthatchange

26、speriodicallyinspacewithinthesample.whenallthecoe cientsappearinginthemagnetic eldsolutionareevaluatedfortherangeoftemperaturesandothercharacteristicparametervalues,we ndthattheleadingterminthemagneticresponseisaninhomogeneousmagnetic eldwithacharacteristicwavelengththatdecreaseswiththetemperature.i

27、nsec.4weinvestigatethedispersionequationforthemaxwell eldinthehigh-temperatureapproximation.wesolvethisequationinarbitrarycovariantgaugesforthemaxwellandcs eldstoobtainthemagneticmassesofthechargedanyon uidinthehigh-temperaturephase.asexpected,themagneticmassesturnouttobeequaltotheinverselengthscale

28、swhichcharacterizethemagneticresponseoftheanyon uidinthisphase.wethenprovethattheexistenceofanimaginarymagneticmasscannotbeassociatedtoatachyonicmodeinthismany-particlesystemwithcsinteractions.indoingthat,we ndtherestenergiesoftheelectromagnetic eldmodesatt c.sec.6containsthesummaryanddiscussion. th

29、e 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.man

30、y-particlemodelandenergygap thenon-relativisticchargedanyonsystemininteractionwithanelectromagnetic eldin2+1dimensionscanbemodeledbythemcslagrangiandensity l= 1 4a a+enea0+i d0 1 4f= j (2.6) (2.7) f=e j ene0 toguaranteetheelectricneutralityofthesystemformedbytheelectron uidandthebackgroundchargenewe

31、imposethecondition 0 j ne0=0,(2.8) wherej0isthemany-particlesystemfermiondensity 0 j= ip4+ 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, charac

32、terized by an inhomogeneous magnetic penetration, theexistenceofadi erentfromzerofermiondensity,asitisrequiredbytheneutralitycondition(2.8),generatesthrougheqs.(2.6)-(2.7)anontrivialbackgroundofcsmagnetic eld f21= b, h0= 1a1)+22 22ne in(2.13)weconsideredthebackgroundcspotential, ak= (2.13) 2 c (2.16

33、) nk= wherec=lb(2.17) b/2.thus,the llingfactor,de nedastheratiobetweenthedensityof particlesneandthenumberofstatesperunitareaofafulllandaulevelnb,isequaltothecscouplingconstantn.whenthemagnitudeofthecscouplingconstantnisconsideredasapositiveinteger,the systemwillhavencompletely lledlandaulevels.once

34、thisgroundstateisestablished,itcanbearguedimmediately19,20,22, 24,thatatt=0thesystemwillbecon nedtoa lledband,whichisseparatedbyanenergygap(c)fromthefreestates;therefore,itisnaturaltoexpectthatatt=0thesystemshouldsuperconduct.thisresultisalreadyawellestablishedfactonthebasisofhartree-fockanalysis19a

35、ndrandomphaseapproximation20,22.att=0itwasprovedinrefs.33thattheexistenceofanaturalscale(thecyclotronfrequencyc)inthistheory,makespossibletherealizationofasuperconductingphaseatt c,forsystemscon nedtoaboundedregion.itislogicaltoexpectthatthisenergyscale,c,separatestwodi erentphysicalphasesofthesyste

36、m.aswewillprovebelow,thesuperconductingstate,foundatt c,disappearswhenthesystemreachestemperatureslargeenough(i.e.att c)tomovetheelectronsbeyondtheenergygaptothefree-energyband. b.e ectiveactioninthelinearapproximation thelinearresponseofthemediumcanbefoundundertheassumptionthatthequantum uctuations

37、ofthegauge eldsabouttheground-statearesmall.inthiscasetheone-loopfermioncontributiontothee ectiveaction,obtainedafterintegratingoutthefermion elds,canbeevaluateduptosecondorderinthegauge elds.thee ectiveactionintermsofthequantum uctuationofthegauge eldswithinthelinearapproximation28,29takestheform 1

38、eff(a,a)=dx (2.18)a a+(2) 4 (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

39、phase, characterized by an inhomogeneous magnetic penetration, (2)= dxdya(x)+ea(x)(x,y)a(y)+ea(y).(2.19) in(2.19)representsthefermionone-looppolarizationoperatorinthepresenceofthecsbackgroundmagnetic eld k2+0,a2=1,a3+a1=2(2.21) from(2.20)and(2.21)wehavethatthenewindependentcoe cientscanbefoundfromth

40、epolarizationoperatorcomponents 000= +0 k2 k0k2 i1k1(2.23) 222=2k1 0 b 28,29,36 g(p4,p)= d n=0 0 dx2bexp (ip2x2)exp ip4+ 2m n() n()tn(2.25) where t=expm,=p1 2 ,=p12(2.26) the magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-c

41、onsistent field approximation. in this temperature region a new phase, characterized by an inhomogeneous magnetic penetration, inthelandaugauge,theeuclideancomponents:00,02and22aregivenby29, 00k, b= jkk,i(2)2 dpp0 +g(p)djg(p k)+djg(p)g(p k),(2.28)4m2 dpp0 2m wherethenotation4,(2.29) djg(p)=ipj jk pk

42、g(p),2 djg(p k)=i(pj kj) jk pkg(p k),2 (2.30) hasbeenused.theindependentcoe cients:0,0,1and2,foundfrom(2.22)-(2.24),arefunctionsofk2,and b 8 n(2n+1)n (2.34) 1=1b m where n(2n+1) n 32m2 n(2n+1)2n (2.36) the magnetic response of the charged anyon fluid at temperatures larger than the fermion energy ga

43、p is investigated in the self-consistent field approximation. in this temperature region a new phase, characterized by an inhomogeneous magnetic penetration, n=sech2( n 2, n=(e( n 2 tanh 48sech2 b 12m20(2.38) intheseexpressionsm=2me(meistheelectronmass). iii.magneticresponseinthehigh-temperatureappr

44、oximation inthehightemperatureregion,fortemperaturesabovetheenergygap(t c),theelectronswillbeenergizedenoughtoreachtheemptylandaulevelbands,aswementionedabove.consequently,theelectroncon nementintoacompletely lledbandislost,beingpossiblefortheelectronstochangetheirinitialstates.hence,itisnaturaltoex

45、pect,fromaheuristicpointofview,thatinthishightemperaturephasethesystemcannotbehaveasasuperconductor. ifthesystemisnotasuperconductoratt c,thenithastoallowthepenetrationofanexternallyappliedconstantmagnetic eld.inotherwords,nomeissnere ectcantakeplace.inthissectionwewillshowthatthisisindeedthecase. a

46、.many-particlesystemlinearequations toobtaintheextremumequationscorrespondingtotheanyonmany-particlesystemwehavetoconsiderthevariationalproblemderivedfrom thee ectiveaction(2.18).thisformulationisknownintheliteratureastheself-consistent eldapproximation29. incomponentform,the eldequationsinthepresen

47、ceoftheinducedcurrentsandthecs eldsare e=ej0 0ek+kl lb=ejk en(3.1)(3.2) 2f0k=kl 0el+ kb(3.4) wherefisthecsgauge eldstrengthtensor,de nedasf= a a,andjindisthecurrentdensityinduced bythemany-particlesystem. insolvingeqs.(3.1)-(3.4)wecon neouranalysistogauge eldcon gurationswhicharestaticanduniforminth

48、ey-direction.withinthisrestrictionwetakeagaugeinwhicha1=a1=0.then,thedi erentcurrentdensitycomponentsare 0(x)=0a0(x)+ea0(x)+0 x(e+ee)+1(b+eb)jind 1jind(x)=0,2jind(x)=1(e+ee)+2 x(b+eb)(3.5)(3.6) intheaboveexpressionsweusedthefollowingnotation:e=f01,e=f01,b=f12andb=f12.eqs.(3.5)-(3.6)playtheroleinthea

49、nyon 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 an

50、yon 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, 2 xb+b= xe a0+a0, 2 xb= xe+e,(3.7)(3.8) inobtainingtheseequationswehaveusedtheeqs.(

51、3.3)-(3.4)toeliminatethecsmagneticandelectric eldsintermsofthemaxwell elds b= xe e= xb(3.9)(3.10) thecoe cientsappearinginthesedi erentialequationsdependonthecomponentsofthepolarizationoperatorsthroughtherelations =2 en =1+e20 2 n,= e221,1,= a0,thesolutionsforb,a0anda0,canbeobtainedusingeqs.(3.8),(3

52、.9),(3.13)andthede nitionofeintermsof b(x)= 1c1e x1 c2ex1 2c3e x2 c4ex2+c5(3.15) (3.16) 1d2 4ac2a(3.14) a0(x)=1c2ex1 c1e x1+2c4ex2 c3e x2+c6 a0(x)=1 2 c3e x2 c4ex2+c7(3.17) 2 2 intheaboveformulasweintroducedthenotation1=1+/1,2=2+/2. ascanbeseenfromthemagnetic eldsolution(3.15),therealcharacterofthei

53、nverselengthscales(3.14)iscrucialfortherealizationofthemeissnere ect.attemperaturesmuchlowerthantheenergygap(t c)thisisindeedthecase,asitwasshowninourpreviousworks(seeref.33). inthehightemperatureregion(t c)thepolarizationoperatorcoe cientsaregivenbyeqs.(2.38).usingthisapproximation,andassumingthatn

54、=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

55、investigated in the self-consistent field approximation. in this temperature region a new phase, characterized by an inhomogeneous magnetic penetration, a 202 c e20 d 1(3.18)(3.19)(3.20) substitutingwith(3.18)-(3-20)ineq.(3.14),weobtainfortheinverselengthscalesinthehigh-temperaturelimit 1 e 1 m/2tan

56、h2m 22(3.21) 2 tanh2 (3.22) theimaginaryvalueoftheinverselength2is duetothefactthatatt c,20and00(seeeq. (2.38).animaginary2impliesthattheterm2c3e x2 c4ex2,inthemagnetic eldsolution(3.15),doesnothaveadampingbehavior,butanoscillatingone.ontheotherhand,thepresenceoftheconstantcoe cientc5inthemagnetic eldsolution(3.15)meansthatthereexistsamagneticlong-rangemode.insec.4wewillseehowthislong-rangemodeisassociatedtotheexistenceofamodewithzeromagneticmass,m1=0. nevertheless,tocompletelydeter