Phase Transition in Anyon Superconductivity at Finite Temper(15)

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,

σ1=2

4,σ3=2σ1

3

with

A=σ1,y2,3= A+B21/3√3√(4.26)6(1 B), A,B=33 3i

σ1

Inthehigh-temperaturelimit A,B 1,sowecantaketheexpansions,

(1 A,B)1/3≈1 1

9 2A,B(4.27)(4.28)

Substitutingwith(4.27)and(4.28)in(4.26)weobtaintheleadinghigh-temperatureapproximationfortherootsofeq.(4.25),

y1= e2Π0,y2,3= 1

π2(Π2Π0′) 1(4.33)

Weshouldnotethatthemagneticmasses(4.32)-(4.33)aregaugeindependent.Thatis,theydonotdependonthegaugeparametersα1andα2.Wecanseethattheequation(4.23),fromwherethemagneticmassesarefound,isindependentofthegaugeparameterα2,sinceα2Ddoesnotdependonα2;andα1appearsonlyasamultiplicativefactor.

From(4.31)-(4.33)wehavethatoneoftheinfraredmodesoftheelectromagnetic eldintheanyon uidismassless,2M1=0,whileM22>0andM3<0.Thesignsobtainedforthesquareofthemasses(4.32),(4.33),areaconsequenceofthefactthatinthehigh-temperaturelimit,Π0>0,Π2>0andΠ0′<0(eq.(2.38)).

Comparingeqs.(4.32),(4.33)witheqs.(3.21),(3.22)respectively,onecanseethat

ξ1=M2,ξ2=M3(4.34)

Hence,themagneticmassescoincidewiththeinverselengthscales,ξ1andξ2,whichdeterminethemagneticresponseofthemedium.Thisispreciselythephysicalmeaningoftheseinfraredmasses(Mi).Theycannotbeinterpreted,otherwise,astherestenergiesoftheelectromagnetic eldmodes,aswewillseebelow.Thezeromode(4.31)islinkedtothelong-rangecomponentC5appearinginthemagneticresponse(3.25).