Phase Transition in Anyon Superconductivity at Finite Temper(16)

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)