Thermal Quantum Field Theory 2014

First oder phase transition in QED3 withChern-Simon term
Yuichi Hoshino,Kushiro National College of Technology
4th September,2014,Okouchi Hall at Riken

Introduction!
Kinematics and D-S equation!
Numerical Results!
effects of phase fluctuation(super fluidity)!
T dependence of vev!
Finite temperature P-odd super conductivity!

First oder phase transition in QED3 with Chern-Simon term

2

1 Introduction
problems in Maxwell-Chern-Simon QED1(1L = (i$ ! D " m) " F() F() " *()+ A(F)+ " (@(A()2442aChern-Simon term breaks parity( ()+* A(F)+ :4(1)(2)

if m = 0,there is a U (2) symmetry generated by fI; $ 3; $ 5; [$ 3; $ 5]g for 4-spinor.
!
"53 0
Ordinary $ marices f$ 0; $ 1; $ 2g; f$ 3; $g = 0; f$ 5; $g = 0; 3 = [$ 3; $ 5]=2 =0 "5 3Chiral like symmetry breaking under ei$ 373 ; ei$ 575

3Parity! cos(27)! 3+ i sin(27) $ 3($ 5)

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(t; x; y) ! (t; "x; y) under P P "1 = i$ 1$ 5memo 3

Thermal Quantum Field Theory 2014

! me! "mo 3(5)First oder phase transition in QED3 with Chern-Simon term

32 Dyson-Schwinger analysis including C-S term& Kondo-Maris(1994),A.Raya(2010)
Solving Dyson-Schwinger eq.there may be 1-st order phase transition at critical value of topologicalmass (c:! for small (; me 6= 0; mo 6= 0,chiral symmetry and Parity symmetry are broken (( ( (c);forbroken me = 0; mo 6= 0 for ((c ( ():# large$ (;chiral#$symmetry restored# and$ Parity# symmery$!
6= 0; 3
6= 0 at ( ( (c:! 0; 3
6= 0 at ( ) (c:*topological mass weaken Coulomb force,"(g() " k(k) =k 2 " i(*()@k @=k 2)k( k)
D() (k) ="ak 2 " (2 + i*k4(6)*Parity violating mass! generation"I 0
In Euclid space 3 =;it is simple to separate upper and lower component byA+ = (1 + 3 )=20 "I
LF =SF (p; a) = "Thermal Quantum Field Theory 2014+ (i$ ! @ " m+ ) + +" (i$ ! @ " m" ) "A+(p; a)$ ! p + B+(p; a)A"(p; a)$ ! p + B"(p; a)A"A :A2+(p; a)p2 + B+2 (p; a) +A2"(p; a)p2 + B"2 (p; a) "(7)First oder phase transition in QED3 with Chern-Simon term

4q = k " p; + = e2=4F; a = 0Z
Z+(k!
q)(p!
q)
+((p ! q)
3S
A+(p) = 1 + 2 2 d3k5 V+ (k) 2 2,dk5(k);+F pq (q + (2)F 2 p2q 2(q 2 + (2)ZZ+1+((k ! q)3V
B+(p) = 2 d3k5 S+(k) 2,dk5(k):+Fq + (2F2q 2(q 2 + (2)(8)Parity violating mass is a second term in B+(p):It is proportional to A(p):In the high energy limit A(p) ! 1:High energy limit of mo(p) is not changed by higher order correction.Topological mass ( dependence of me by linear approximation to ladder Dyson-Schwinger
equation.
In the Landau gauge with A(p) = 1,it is written
Z 3 0dpm(p0)"((p0 ! (p " p0)=(p " p0)22m(p) = 2e:(9)(2F)3 (p " p0)2 + (2p0 2 + m2

e2 (e2(2mo("p )p!1 = p; mo("p )p!0 =:22Fm+j(j8 "p2(10)

In the linear approximation analytic solution has been known for ( = 0 case.

e2
m2
e2
m2 "mrme(p) =; m = m(0) = ; m(x) =e :4F p2 + (m + ()24F
8F
Thermal Quantum Field Theory 2014(11)First oder phase transition in QED3 with Chern-Simon term

5Here we assume this type of mass function me (p):#Z$d3 kB+(k)B+(k)= "2(+);(2F)3 A2+(k)k 2 + B+2 (k) A2+(k)k 2 + B+2 (k)Z#$d3 kB+(k)B"(k)
3= "2("):(2F)3 A2+(k)k 2 + B+2 (k) A2"(k)k 2 + B"2 (k)

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&Absence of screening correction in our approximation for small ( where B+; B")

0.

&For)

0.

large

(whereB+B"(0,2() (k) induces

Chern-Simon

correction

3(&In weak coupling (c ' :01e2 for quenched case.Chiral symmetry is restored for the same region of( in unquenched case with screening correction.(cmay be very small.for unqenched case as 10"6e2.
Infrared mass vanishes at (c:zero momentum mass is sensitive by solving Dyson equation.
&if (=me'0:1,me!
0 by iteration.or (effective potential for me and mo)
Shortdistance origin :zero anomalous dimension for ( ) (c:Anomalous dimension D = e2=8Fm " e2=32F( = 0 for 2-component fermion.by deriving electron spectral function.e2=8Fm = 1; e2=32F( = 1 ! (c = e2=32F = :01e2:Thermal Quantum Field Theory 2014

vev..008+(h-1).0005

B(+) at critical myu

B(-)

Bo

First oder phase transition in QED3 with Chern-Simon term

63 Phase ?uctuation by massless Aeld(Goldstone)
uniform vev (order parameter)!distribution by super\uidty(r) =0ZeiJ(r)

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;

1
d2rK0jrJj2;
2
vs _ rJ
F =DiJ(x) "iJ(y)Sik ! "i 0jT i(x)ee
DEiJ(x) "iJ(y)= Sik 0jT eej0(15)k (y)j0E

We expand the exponentail factor in powers of J as far as quadratic terms:
DEiOJ(x) "iOJ(y)0jT eej0$1#2
2' " 0jOJ(x) + OJ(y) " 2T OJ(x)OJ(y)j02

Thermal Quantum Field Theory 2014

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First oder phase transition in QED3 with Chern-Simon term

7

Propagator changes as the gauge transformation

S ! S + OS; OS = S(x " y)(D(x " y) " D(0))
S(x " y)0 = S(x " y)0 exp(D(x " y) " D(0));"D(x) = 0:2@(r) = j 0j

2= j 0jDe*i(J(x)"J(0))

exp("i= j 0j2 exp["XY model:

EXk(eik!r " 1)JkX kB TkK0 k 2+f1 " cos(k ! r)g]kB Tr@(x) = j 0j2 exp("ln( ))2FK0 ar "P_ ( )aThermal Quantum Field Theory 2014

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First oder phase transition in QED3 with Chern-Simon term

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3.1 T dependence of vev
2-dimension: vanishing vev by infrared divergence
2j 0j2 ! j 0j2e"D(0)=c(23)

D

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D(x) is a massless propagator
vev =Ej0 =1exp("D(0));
0c2$0jJ(x)2j0Z 31dk=(on " shell exp)(2F)32! kD(0) =#00jeiJ(x)=cD(0) =Z

D(0) =ZAt high Tvev =Thermal Quantum Field Theory 2014dn"1k
1(2F)n"12! k exp(U! k ) " 1d2 kkB T
k2k T =ln( 2 )2 2k 2 B(2F)
8F((kB Tk2ln( )) ! 00 exp("8Fc2 (2(25)

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First oder phase transition in QED3 with Chern-Simon term

At low T

D(0) =vev =9Z

d2 kkkB Texp(")=(2F)22kkB T
8F
kB T
exp(")
08Fc2(29)(30)

4 Finite temperature superconductivity
At _nite temperature me disappear without chiral symmetry breaking.
*mo may survives at _nite temperature.Phonon attraction and weak Coulomb repulsive force may beattractive for two electron system.

5 SummaryFor one particle problem as mass and condensation effects of Chern-Simon term was examined.
There is a _rst order phase transion at critical value of topological mass (c above which chiral orderparameter vanishes and parity odd mass survives.At high temperature chiral order parameter vanishes
by phase \ucutuation.
Future problem:Parity odd mass may survive at high temperature and candidate for super conductor.

Thermal Quantum Field Theory 2014