カラー超伝導におけるゆらぎの効果

北沢正清

京大基研

Fluctuations in Color Superconductivity

基研研究会「熱場の量子論」

1, Introduction2, Gauge Fluctuations in Type I CSC3, Pair Fluctuations at lower densities4, Pseudogap of CSC

C O N T E N T S

1, Introduction1,1, Introduction

Color Superconductivity(CSC)Approaches to CSCNature of CSC at low and high densitiesTwo types of superconductor

Color SuperconductivityColor SuperconductivityColor Superconductivity

chiral symm. broken Color Superconductivity(CSC)

with attractive channel in one-gluon exchange interaction.

quark (fermion) systemDense Quark Matter:

Cooper instability at sufficiently low T

SU (3)c gauge symmetry is broken!

[３]c×[３]c＝[３]c＋[６]c

∆~100MeV at moderate density µq~ 400MeV

T

µ

confinement

Attractive!

Pairing patterns of CSCPairing patterns of CSCPairing patterns of CSC

ρ

u d su d s

ρ

Two Flavor Superconductor(2SC)

u d u d

s

µ<Ms µ>>Ms

Color-Flavor Locked (CFL)

5ij i jCiαβ α βψ γ ψ∆ =for JP=0+ pairing

ij ijk kαβ αβγ γε ε∆ = d

a,b : colori,j : flavor

attractive channel : color anti-symm.flavor anti-symm.

00

⎛ ⎞= ⎜ ⎟⎜ ⎟∆⎝ ⎠

d 12

3

∆⎛ ⎞= ∆⎜ ⎟⎜ ⎟∆⎝ ⎠

d

(3) (2)c cSU SU→ (3) (3) (3)(3)c L R

c L R

SU SU SUSU + +

× ×→

T

µ0

Approaches to CSCApproaches to CSCApproaches to CSC

first principle calculationeffective theoriesNJL-type 4-fermi model,random matrix model, etc..

observation ???

100MeV∆ ≈Asymptotic forms of

gap ∆,critical temperature Tc, gluon self energy,Ginzburg parameter, etc…

weak couplingstrong coupling

in compact stars and/orheavy ion collisions

due to asymptotic freedom

using one gluon exchange

low density high density

Structural Change of Cooper PairsStructural Change of Cooper PairsStructural Change of Cooper PairsMatsuzaki, PRD62,017501 (‘00) Abuki, Hatsuda, Itakura, PRD 65, 074014 (‘02)

Coherence length of Cooper pairsbecomes short as µ is lowered.

T

µ0

ξ – coherence lengthd – interquark distance

Bosonize?

µ[MeV]

ξ / d

weak coupling =validity of MFA

“Type” of CSC““TypeType”” of CSCof CSC

Fluctuations of SC pair field:gauge field:

– coherence length– penetration depth

ξλ

Ginzburg parameter: κ=λ/ξ

T

µ0

Matsuura, Iida, Hatsuda, Baym, PRD69,074012 (‘04)Giannakis, Ren, NP B669, 462 (‘03)

- Type I1κGauge fluctuations dominate

- Type II1κPair fluctuations dominate

:Type I CSC1κ:Type II CSC1κ >%

“Type” of SC““TypeType”” of SCof SC

Giannakis, Ren, NPB669, 462 (‘03)

H ∆

0

λ

ξx

λ ξ

H ∆

0

λ

ξx

– coherence length of ∆ 1/ 2/ 2 | |c aξ ε ε −≡

2 1/ 221/ 8ceλ ε −∆≡

CFL (weak limit)

metal SC and 2SC

22 4 21( , ) ( 2 ) ( )2 2bF a c AeA i Aε∆ ∆= + ∆− +∆ + ∇ ∇×

rr r rGL free energy:

– penetration depth of A

1/ 2 0.707cκ = ≅0.589cκ =

Type I : λ ξType II :

σ=0at κc

σ<0σ>0

Surface Energy σ C

C

T TT

ε −=

2, Gauge Fluctuations2,2, Gauge Fluctuations

First order transition in type I SCEstimation by Bailin & LoveRecent progress by Matsuura, et al., et al.

in Type I CSCin Type I CSC

First Order Transition in Type I SCFirst Order Transition in Type I SCFirst Order Transition in Type I SC

22 43 1( , ) ( 2 )2 4 lm lmbF A d r a c ieA F Fε⎡ ⎤= + + ∇ − +∆ ∆ ∆ ∆⎢ ⎥⎣ ⎦∫

r r r%

GL free energy functional

integrate out A ( ) ( , ) V AF Fd Ae eβ β− ∆ − ∆= ∫%

rr

( )3

2 4 2 2 23

1( ) ln( ) ln2 (2 ) A

k

d kF at b T k m kπ<Λ

∆ ∆ ∆= + + + −∫2 3

2

12 6A AT m mπ πΛ⎛ ⎞= −⎜ ⎟

⎝ ⎠

Negative 3rd order term induces thefirst order transition.

1Am λ−= ∆

T =TcT=Tc

*T >Tc

T <Tc

( )F ∆

∆

( )F ∆

∆

though, too weak to observe..

Halperin, Lubensky, Ma, PRL32,292(’74)

Gauss approx.

First Order Transition in CSCFirst Order Transition in CSCFirst Order Transition in CSCBailin, Love, Phys. Rep. 107, 325 (‘84)

( )3

2 2 2 2 33 2

1( ) ln( ) ln 8(2 ) 2 6A c A A A

k

d kF T k m k T m mπ π π<Λ

Λ∆ ⎛ ⎞≡ + − = −⎜ ⎟

⎝ ⎠∫

23* 2'' 4

c c

c c

T T gT T

µπ

⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

renormalize Tc Tc’

leads to 1st order Tc*

232

5

1~ gcT eg

π

µ

−

* ''

c c

c

T TT−

→ ∞ as g 0

( ') / ( 0) 2.8cT T T∆ = ∆ = = for moderate values of g, Tc, µ

calculation for 2SC pairing

ReconsiderationsReconsiderationsReconsiderationsMatsuura, Iida, Hatsuda, Baym, PRD 69, 074012 (‘04)

( )3

2 2 23

22 3 4

2 2

( ) ln( ) ln(2 )

4 4 23

c

A c Ak T

c cA A A

d kF T k m k

T Tm m m

π

π π π

<

≡ + −

= − +

∆ ∫

*

0c c

c

T T gT−

→ as

*2~ 0c c

c

T T gT−

− →

as

Giannakis, Hou, Ren, Rischke, hep-ph/0406031

introduced the momentum cutoff Λ = Tc~1/ξ0

0g →

0g →momentum dependence of mA (k)

(assume mA<<Tc)

CJT effective action

3, Pair Fluctuations3,3, Pair Fluctuations

Precursory phenomena / “sQGP”Response functionSpecific heatTime Dependent GL equation

at Lower Densitiesat Lower Densities

Pair Fluctuation in Type II SCPair Fluctuation in Type II SCPair Fluctuation in Type II SCelectric conductivity

ε

ε ~10-3

enhancementabove Tc

Precursory Phenomena in Alloys•Electric Conductivity•Specific Heat•etc…

Thouless, 1960Aslamasov, Larkin, 1968Maki, 1968, …

High-Tc Superconductor(HTSC)

large fluctuations induced bystrong coupling and quasi-two dimensionality

pseudogap

1986~in quasi-two-dimensional cuprates

the “sQGP”the the ““sQGPsQGP””

CSC

Hadrons

Success of hydrodynamicsat RHIC energy

J/ψ peak above Tc on lattice

= strongly coupled QGP= = sstrongly coupled trongly coupled QGPQGP

T

ω

jet quenching, etc…

elliptic flow v2

Quark matter is strongly interacting!!“Strongly interacting CSC” is also expected.

5 10 [GeV]Asakawa, Hatsuda (‘04)

( )5 2 2† h.c.ex

CexH d iψ γ τ λ ψ= +∆∫ x

5 2 22 ( )( ( ))indC

C exG x i xx ψ γ τ λ ψ= −∆

Apply an external pair field ∆ex

( , )R ωΞ =k + + ⋅⋅⋅

Q =

RPA approx.: ( ) 11 ( , )C nG Q ω−−= − + k

Response Function of Pair FieldResponse Function of Pair FieldResponse Function of Pair Field

Pair field ∆ind is dynamically induced

Linear Response

total pair field: ( , ) (( , ) , )tot ind ex exRω ωω∆ = ∆ + ∆Ξ∆ = kk k

1+

Thouless Criterion

2

2

( 0)∂ Ω ∆ ==

∂∆1 1 ( ,0)CG Q−Ξ = + 0

ThermodynamicPotential

CT T=

∆Ω(∆

)

2

2

( 0)∂ Ω ∆ =∂∆

ΞR(0,0) diverges at Tc - for any second order transitions

D.J. Thouless, AoP 10,553(‘60)

11( ) Im ( )ρ ω ωπ

−= − Ξk k

Spectral FunctionSpectral FunctionSpectral Function M.K., T.Koide, T.Kunihiro, Y.Nemoto, PRD 65, 091504 (2002)

int 5 2 5 2( )( )CC

C A AL G i iψ γ τ λ ψ ψ γ τ λ ψ=Nambu-Jona-Lasinio model:

ε→０（Ｔ→ＴＣ）

for k=0

As T Tc, the peak becomes sharp.The peak diverges at Tcowing to the Thouless criterion.

C

C

T TT

ε −=

The peak survives up to ~ 0.2 electric SC： ~ 10-3

summation of connected diagrams

:free fermionsfreeΩ =

:collective modescol.Ω =

3

3 ln ( , )(2 ) n

n

d kT G k iωπ

= ∑∫

+L+ + +L

free col.Ω Ω Ω= +

3 3

3 32 2( , ) ( , )13

(2 ) 2 (2 )C n Cn n

nd k d kG Q k i G Q kT T iω ωπ π

⎛ ⎞= − − +⎜ ⎟

⎝ ⎠∑ ∑∫ ∫ L

( )3

31 1

3

33 ln 3 ln(2 ) (2 )

( , ) ( , )n

C nn

nGd k d kQ Ti iT k kπ

ωπ

ω− −= − =+ Ξ∑ ∑∫ ∫

# of possible collective excitation in color space.

Thermodynamic PotentialThermodynamic PotentialThermodynamic Potential2

2VdC TdT

−Ω

=

Specific Heat

We use an approximation.

CV

/107

anomalous enhancement of cV above Tc.The enhancement is clearly seen from ε~0.1 (T~1.1Tc).

Cfree

Ccol

Tc ε

ε

CV

/107

free col.C C+

Tc

free (BCS approx.)from collecitve mode

Specific HeatSpecific HeatSpecific Heat2

2VdC TdT

Ω= −

2 2

2 2. .

/

/free free

col col

C Td dT

C Td dT

= − Ω

= − Ω

D.N. Voskresensky, PRC69,065209(‘04)

Cfree~ Ccol. at ε~1

M.K., et al., hep-ph/0403019

Inconsistency:

2110( ,, ) )( CG C CQ Aωω ω ε−− = + ≅ +Ξ +k kk

Approximation for ΞApproximation for Approximation for ΞΞ

( , ) ( ,( ), )tot exω ωωΞ∆ = ∆kk k

1 ( 0( ), ,) totω ω− ∆ =Ξ kk effective equation to describethe pair field without external field

with (0,0) /cT T

T QA T=

= ∂ ∂ 2(0,0) /cT T

QC=

= ∂ ∂ k

0 (0,0) /cT T

QC ω=

= ∂ ∂

C

C

T TT

ε −=

( )20 / ( ) 0i t xtC C Aε∂ ∂ − ∇ + ∆ =

3

col. 313 ln (

(2,

))n

n

d kT k iωπ

−ΞΩ = ∑∫Thermodynamic pot. :

M.K., Kunihiro, in preparation

Linear response:

Notice: C0 takes a complex (not pure imaginary) value.

linear part of time dependentGL equation

Time-Dependent GL equationTimeTime--Dependent GL equationDependent GL equation

( )2 2 32 1/ ( / ) 0i t i t c a bκ κ ε∂ ∂ ∆ + ∂ ∂ ∆ − ∇ ∆ + ∆ + ∆ =

diffusion equationκ2 ~ 0

wave equationκ1 ~ 0

Abrahams, Tsuneto, Phys.Rev.152,416(‘66)

( )20 / ( ) 0i t xtC C Aε∂ ∂ − ∇ + ∆ =

( )2 20 ( / ) ( ) 0i t xtc c aε∂ ∂ − ∇ + ∆ =

Our result

Voskresenskysecond time derivative

cT T>%

cT T

complex C0 owing to the particle hole asymmetryleads to a damped oscillation of pair field

4, Pseudogap of CSC4,4, Pseudogap of CSCM.K., T.Koide, T.Kunihiro, Y.Nemoto,hep-ph/0309026, to appear in PRD

µ

ω

k

2 2sgn( ) ( )k kω µ µ= − − + ∆

2 2( )d kdk k

ε µµ−

=− + ∆

Quasi-particle energy:

2( ) dkN kd

ωω

( )N ω

ωµ

2∆

∆∆

Density of State:

The gap on the Fermi surface becomes smalleras T is increased, and it closes at Tc.

Density of State in BCS theoryDensity of State in BCS theoryDensity of State in BCS theory

The origin of the pseudogap in HTSC is still controversial.

:Anomalous depression of the density of state near the Fermi surface in the normal phase.

Pseudogap

Conceptual phase diagram of HTSC cuprates

Renner et al.(‘96)

Yanase,Yamada(‘01), …

x: doping

T-matrix approximation

Approaches for the pseudogap in HTSC.

Analogy in BKT transition

It is, however, widely believed thatlarge fluctuation of pair-field causes the pseudogap.

Loktev et al.(‘01), …Different Origin??

Pseudogap in low density nuclear matterA.Schnell G.Roepke, P.SchuckPRL83 1926(1999)

013

ρ ρ= TC=4.34MeV

4.35 1.00254.34C

TT

= ≈

Pseudogap manifests itself!

30

3( ) ( , )(2 )dN ω ρ ωπ

= ∫k k 0 01( , ) Tr Im ( , )

4RGρ ω γ ω⎡ ⎤= ⎣ ⎦k k

Density of State N(ω) 3 0N d x ψγ ψ= ∫

0

1( , )( , ) ( , )n

n n

GG i iω ω

ω =− Σ k

kk

( , )nωΣ =k + + + ⋅⋅⋅3

03

q ( , ) ( , )(2 ) n m m

m

dT Gω ω ωπ

= Ξ + +∑∫ k q q≡, n mi iω ω+ +k q

, miωq

=Σ

10 0( , ) ( )n nG i iω ω µ γ γ−

⎡ ⎤= + − ⋅ =⎣ ⎦k k v:free progagator

T-matrix Approximation

The pseudogap survives up to ε =0.05~0.1 ( 5~10% above TC ).

Numerical Result : Density of StateNumerical Result : Density of StateNumerical Result : Density of State

( )( )free

NN

ωω

Dispersion RelationDispersion RelationDispersion Relation µ= 400 MeV, ε=0.01

pω∂

∂:increases ( ) pN ω

ω∂∂

:decrease

ω

p

pω

∂∂

ω [MeV]

kF=400MeV

affects the dispersion relation ω =ω−(p).

Rapid increase around ω =0

Re ( , ) 0ω µ ω−− + − Σ =p psolution

Re Σ−

quasi-particle peak of anti-particle, ω = −k−µ

quasi-particle peak,ω =ω−(k)~ k−µ

Fermi energy

Quasi-particle peak has a depression around the Fermi energy.

µ= 400 MeVε=0.01

ω−µ =−400MeV 0

kF=400MeVk

position of peaks

quasi-particle peaksat ω =ω−(k)~ k−µ and ω =−k−µ.

kω

1-Particle Spectral Function11--Particle Spectral FunctionParticle Spectral Function

,00

,k µ− −k

: collective mode

: on-shell

SummarySummarySummaryFluctuations of Color Superconductivity

high density type Igauge field fluctuations dominatemake the phase transition first order

More observables !?

See You Again in KOCHI !

low density type IIpair fluctuations dominatecause various precursory phenomena above Tc

recently reexamined

They can be experimental observables!

TDGL eq. for metallic SC

2 3 0'c

c

T Tit T

c b a a−∂Ψ − ∇ Ψ + Ψ + Ψ =

∂

2c

c

T TT

A BC C

ω −= − − k

2c

c

T TT

a bc c

ω −= − − k

Particle-hole asymmetry in CSC caused finite real part of ω.

Damping Behavior of Collective ModeColor Superconductivity

2 0c

c

T TC B AT

ω −+ + =k

C :complex c :pure imaginary

is NOT pure imaginary.

Damped Oscillation

is pure imaginary.

Overdampingmode

ω0 ω0

( )5 2 2† h.c.ex

CexH d iψ γ τ λ ψ= +∆∫ x

expectation value of induced pair field:external field:

[ ]0

5 2 2( ) ( ) ( ), ( , )tC

exex tx i x i ds H s O tψ γ τ λ ψ = ∫ x

5 2 22 ( )( ) ( '( ,( )) ' )'indC

C x

Rexe

G x i x Dd x x xtx dψ γ τ λ ψ∆ ∆= − = ∫ ∫ x

Linear Response

5 2 2 5 2 22 ( ) ( ), (0) (0) (, ) )( CR CCG x i xD t i tψ γ τ λ ψ ψ γ τ λ ψ θ⎡ ⎤= − ⎣ ⎦x

Retarded Green function

† †(( ) , ) ( )ind n ex nnDω ωω∆ ∆= kk kFourier transformationwith Matsubara formalism

( , )nD ω =k + + ⋅⋅⋅( , )nQ ω =k

RPA approx.:

=( , )

1 ( , )C n

C n

G QG Q

ωω

−+

kk

with

Response Function of Pair Field

Model

2 25

5 2 5 2

( ) ( )

( )( )

S

CCC A A

L i G i

G i i

ψ γ ψ ψψ ψ γ ψ

ψ γ τ λ ψ ψ γ τ λ ψ

⎡ ⎤= ⋅∂ + +⎣ ⎦

+

τ

Nambu-Jona-Lasinio model (2-flavor,chiral limit)：

τ：SU(2)F Pauli matricesλ：SU(3)C Gell-Mann matricesC :charge conjugation operator

Aλ AλIH =

3( 250MeV) , 93MeVfπψψ = − =so as to reproduce

25.01GeV650MeV

/ 0.62

S

C S

G

G G

−=Λ =

=

Parameters:

Klevansky(1992), T.M.Schwarz et al.(1999)

M.K. et al., (2002)

2SC (not CFL) is expected at low µ and near Tc.We neglect the gluon degree of freedom.

Notice:

Numerical Check for µ=400MeV (Tc=40.04MeV)

Our effective equation well reproduces the full calculation1.3 , 100MeVcT T k≈ ≈up to

covers the region where clear collective mode appears.

Re ω(k) Im ω(k)

2A BC C

εω = − − k

1 ( , ) 0CG Q ω− + =k C

C

T TT

ε −=

ω

effective equationfull calculation :ω

0 0 0 0 0 0ˆ( , ) ( , ) ( , ) ( , )V Sp p p pγ γΣ = Σ − Σ ⋅ + Σp p p p prΣ has spinor indices:

0 00

0 0 0

1( , )G pp p p

γ γµγ µ µ

− +

− +

= = ++ − Σ + − − Σ + Σ

Λ Λ+ −

pp p

012

pγ γΛ

± ⋅=m

v v

0 VΣ = Σ Σm m

Decomposition of G

:projection op.

=0 in chiral limit

where,

( )

( )

030 0

3 0

( , )1( , ) tanh coth2 (2 ) 2 2

Rni qd dp

p i T Tµωω ωγ γ

π π ω µ η⎡ − ⎤Ξ + +

Σ = − − ⋅ −⎢ ⎥− − + − ⎣ ⎦+ → −

∫ ∫qk qqp q q

q

q q

r

( )0 0 0( , ) ( , )p pγ −− ++Λ + Σ Λ= Σ p p

:self-energy for the positve and negative energy particles.

positive energy part

Im Σ− (ω,k)

quasi-particle peak = –kpeak of Im =k–

,00

,k µ− −k

: collective mode

: on-shell

ω

|Im Σ−| has peaks around ω =µ−k, which is found to be the hole energy.

|Im Σ-|

k

coincide at fermi surface. R

e Σ −

(ω,k

)

= –k

µ

-µ0

ω

kPeak of |Im Σ− |

kF

C

C

T TT

ε −=