Phase Transition in Anyon Superconductivity at Finite Temper(8)

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,