Phase Transition in Anyon Superconductivity at Finite Temper(9)

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