Phase Transition in Anyon Superconductivity at Finite Temper(10)

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,

Eq.(3.27)establishesaconnectionbetweenthelinearcombinationofthecoe cientsofthelong-rangemodesofthezerocomponentsofthegaugepotentials,(C6+eC7),andthecoe cientofthelong-rangemodeofthemagnetic eld,C5.NotethatiftheinducedCScoe cientΠ1,ortheDebye-screeningcoe cientΠ0werezero,therewouldbenolinkbetweenC5and(C6+eC7).Thisrelationbetweenthelong-rangemodesofB,A0anda0canbeinterpretedasasortofAharonov-Bohme ect,whichoccursinthissystemat nitetemperature[33].AtT=0,wehaveΠ0=0,andthise ectdisappears.

Afterusingtheboundaryconditions(3.23),itfollowsthattheyarenotsu cienttodeterminethecoe cientsC5andA.WeneedanotherphysicalconditionfromwhereC5andAcanbefound.Since,obviously,anymeaningfulsolutionhavetobestable,thenaturaladditionalconditiontobeconsideredisthestabilityequationderivedfromthesystemfreeenergy.Withthisgoalwestartfromthefreeenergyofthehalf-planesample

F=1

πa0b Π0(eA0+a0)2

Π0(eE+E) 2Π1(eA0+a0)(eB+b)+Π2(eB+b)′22

whereLandL′determinethetwosample’slengths.

In(3.28)wehavetosubstitutethe eldsolutions(3.16),(3.17),(3.24)and(3.25)togetherwiththesolutionsfortheCS elds(thatcanbefoundsubstituting(3.24)and(3.25)ineqs.(3.9)and(3.10)respectively)

b(x)=χξ1C1e xξ1 χ

ξ2

A,(A

C1by (3.28)ξ2+C1sinxξ2 C1cosxL′→∞)isgivenasafunctionofAand

f=1=LL′beingthesamplearea)inthesample’slengthlimit(L→∞,

γ22+G,2X2=gγ1+G,X3= 2gγ1

X6=g

+e2Π2

BBγ1,Π0

1N(3.33)G=γ22

Π1 χξ2

=ξ2 γ2e+χγ2+2 2Π2 2e+χγ2ξ2 e1

δA

δC1=1