京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in...
-
Upload
nguyenduong -
Category
Documents
-
view
213 -
download
0
Transcript of 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in...
![Page 1: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/1.jpg)
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
![Page 2: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/2.jpg)
1, Introduction1,1, Introduction
Color Superconductivity(CSC)Approaches to CSCNature of CSC at low and high densitiesTwo types of superconductor
![Page 3: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/3.jpg)
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!
![Page 4: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/4.jpg)
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 + +
× ×→
![Page 5: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/5.jpg)
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
![Page 6: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/6.jpg)
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
![Page 7: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/7.jpg)
“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 >%
![Page 8: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/8.jpg)
“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 −=
![Page 9: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/9.jpg)
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
![Page 10: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/10.jpg)
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.
![Page 11: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/11.jpg)
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
![Page 12: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/12.jpg)
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
![Page 13: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/13.jpg)
3, Pair Fluctuations3,3, Pair Fluctuations
Precursory phenomena / “sQGP”Response functionSpecific heatTime Dependent GL equation
at Lower Densitiesat Lower Densities
![Page 14: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/14.jpg)
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
![Page 15: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/15.jpg)
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)
![Page 16: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/16.jpg)
( )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)
![Page 17: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/17.jpg)
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
![Page 18: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/18.jpg)
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.
![Page 19: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/19.jpg)
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:
![Page 20: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/20.jpg)
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
![Page 21: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/21.jpg)
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
![Page 22: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/22.jpg)
4, Pseudogap of CSC4,4, Pseudogap of CSCM.K., T.Koide, T.Kunihiro, Y.Nemoto,hep-ph/0309026, to appear in PRD
![Page 23: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/23.jpg)
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
![Page 24: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/24.jpg)
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)
![Page 25: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/25.jpg)
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!
![Page 26: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/26.jpg)
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
![Page 27: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/27.jpg)
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
![Page 28: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/28.jpg)
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−
![Page 29: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/29.jpg)
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
![Page 30: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/30.jpg)
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!
![Page 31: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/31.jpg)
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
![Page 32: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/32.jpg)
( )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
![Page 33: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/33.jpg)
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:
![Page 34: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/34.jpg)
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
![Page 35: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/35.jpg)
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
![Page 36: 京大基研 北沢正清 - riise.hiroshima-u.ac.jp · 1, Introduction 2, Gauge Fluctuations in Type I CSC ... gauge field: – coherence length ... zEstimation by Bailin & Love zRecent](https://reader031.fdocuments.net/reader031/viewer/2022022605/5b77edfd7f8b9ad3338e310d/html5/thumbnails/36.jpg)
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 −=