Phase Transition in Anyon Superconductivity at Finite Temper(7)

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,

Θn=sech2β( n

2, n=(eβ( n

2π tanhβµ

48πsech2 βµb

12m2Π0(2.38)

Intheseexpressionsm=2me(meistheelectronmass).

III.MAGNETICRESPONSEINTHEHIGH-TEMPERATUREAPPROXIMATION

Inthehightemperatureregion,fortemperaturesabovetheenergygap(T ωc),theelectronswillbeenergizedenoughtoreachtheemptyLandaulevelbands,aswementionedabove.Consequently,theelectroncon nementintoacompletely lledbandislost,beingpossiblefortheelectronstochangetheirinitialstates.Hence,itisnaturaltoexpect,fromaheuristicpointofview,thatinthishightemperaturephasethesystemcannotbehaveasasuperconductor.

IfthesystemisnotasuperconductoratT ωc,thenithastoallowthepenetrationofanexternallyappliedconstantmagnetic eld.Inotherwords,noMeissnere ectcantakeplace.Inthissectionwewillshowthatthisisindeedthecase.

A.Many-ParticleSystemLinearEquations

Toobtaintheextremumequationscorrespondingtotheanyonmany-particlesystemwehavetoconsiderthevariationalproblemderivedfrom

thee ectiveaction(2.18).Thisformulationisknownintheliteratureastheself-consistent eldapproximation[29].

Incomponentform,the eldequationsinthepresenceoftheinducedcurrentsandtheCS eldsare

·E=eJ0

0Ek+εkl lB=eJk

eN(3.1)(3.2)

2πf0k=εkl 0El+ kB(3.4)

µwherefµνistheCSgauge eldstrengthtensor,de nedasfµν= µaν νaµ,andJindisthecurrentdensityinduced

bythemany-particlesystem.

Insolvingeqs.(3.1)-(3.4)wecon neouranalysistogauge eldcon gurationswhicharestaticanduniforminthey-direction.WithinthisrestrictionwetakeagaugeinwhichA1=a1=0.Then,thedi erentcurrentdensitycomponentsare

0(x)=Π0[a0(x)+eA0(x)]+Π0′ x(E+eE)+Π1(b+eB)Jind

1Jind(x)=0,2Jind(x)=Π1(E+eE)+Π2 x(b+eB)(3.5)(3.6)

Intheaboveexpressionsweusedthefollowingnotation:E=f01,E=F01,b=f12andB=F12.Eqs.(3.5)-(3.6)playtheroleintheanyon uidoftheLondonequationsinBCSsuperconductivity.Whentheinducedcurrents(3.5)-(3.6)aresubstitutedineqs.(3.1)-(3.2)we nd,aftersomemanipulation,asetofindependentdi erentialequationsdependingonthe eldsBandE,alongwiththezerocomponentsofthegaugepotentials,A0anda0,