Phase Transition in Anyon Superconductivity at Finite Temper(4)

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