盛岡研究会、つなぎ温泉、2009年6/26
カラー超伝導
北沢 正清
大阪大学
Contents:
(1) クォーク
(2) 低～中間密度領域のカラー超伝導
(3) 冷却原子系からの情報
Phase Diagram of QCD
LHC
•success of ideal hydro. models
•early thermalization
strongly coupled QGP near Tc
RHIC
T
FAIR@GSI
Confined
Color SC
0
m
Quark “Quasi-particles”
in the Deconfined Phase
Is There Quark Quasi-Particles in QGP?
Yes, at asymptotically high T.
Quark quasi-particles:
• width G~g2T
 / mT
•2 collective excitations
having a “thermal mass”
mT~ gT
normal
“plasmino”
gT
mT 
6
(, p) 
p / mT
G
~g
mT
The decay width grows as T is lowered.
NOT clear, near Tc.
Lattice QCD Simulation for Quarks
Karsch, MK, 2007
Imaginary-time quark correlator in Landau gauge
in quenched approx., 643x16
C (t , p  0)
T = 3Tc
2-pole ansatz for
quark spectral function:
  ( )  Z n (  En ) :normal
 Z p (  E p ) :plasmino
tT
projection by   (1   0 ) / 2
•Similar result is obtained even with 1283x16! MK et al. in preparation.
•Result is insensitive to # of data points used in the analysis.
Quark excitations would have small decay rate even near Tc.
Quark Dispersion
Karsch, MK, arXiv:0906.3941.
in quenched approx., 643x16
HTL(1-loop)
(plasmino)
p/T
•Lattice results behave reasonably as functions of p.
•Quarks have a thermal mass mT ~ 0.8T. (1.25<T/Tc<3)
Notice: Similar result is obtained even with 1283x16!
Decay width may be small even for V∞.
Phase Diagram
0th approximation: (quasi-)fermions + interaction (gluon-ex.)
T
0
analogy to condensed matter phys.
•Polarized gas
•BCS-BEC crossover
•strongly correlated system
m
Is thermal mass mT~0.8T not negligible?  See, a trial in Hidaka, MK 2007
Phase Diagram
0th approximation: (quasi-)fermions + interaction (gluon-ex.)
T
•crossover transition
analogy to condensed matter phys.
•Polarized gas
•BCS-BEC crossover
•strongly correlated system
•quarkyonic region McLerran, Pisarski, 2007
chirally restored but confined
0
m
Is thermal mass mT~0.8T not negligible?  See, a trial in Hidaka, MK 2007
Phase Diagram
0th approximation: (quasi-)fermions + interaction (gluon-ex.)
T
•crossover transition
analogy to condensed matter phys.
•Polarized gas
•BCS-BEC crossover
•strongly correlated system
•quarkyonic region McLerran, Pisarski, 2007
chirally restored but confined
0
m
•Are 2 phases connected
continuously at lower T?
1st CP : Asakawa, Yazaki, 1989.
2nd CP : MK, et al., 2002
(See, also Yamamoto, et al., 2006).
3rd CP : Zhang, et al.,2009.
Color Superconductivity
at intermediate densities
Color Superconductivity
At extremely dense matter,
(    )
%
quark (fermion) system
attractive channel in
one-gluon exchange interaction.
[３]c×[３]c＝[３]c＋[６]c
Cooper instability at sufficiently low T
ud
•pairing in scalar (JP=0+) channel
qiqj     I ijI I
u
d
I
color,flavor anti-symmetric
us
s
ds
T
m
Various Phases of Color Superconductivity
T
m
m  ms
%
u
us
ud
ud
u
d
s
ds
2-flavor SuperCondoctor (2SC)
SU ( )L  SU ( )R  SU ( )c  U ()B
 SU ( )L  SU ( )R  SU ( )c  U%()B
us
ms = m
d
ds
s
Color-Flavor Locking (CFL)
SU ( )L  SU ( )R  SU ( )c  U ()B
 SU ( )L R c  U%()B
analogy with B-phase
in 3He superfluid
Color Superconductivity in Compact Stars
ud
•effect of strange quark mass ms
•neutrality and -equilibrium conditions
u
us
Mismatch of densities
(1) strong coupling!
(2) mismatched Fermi surfaces
pF ~
d
s
ds
m  m 
(1) weak coupling
(2) common Fermi surface
T
m
Various Phases of Color Superconductivity
ud
3 order parameters ud, us, ds
 2*2*2=8 possibilities of distinct phases
CFL Alford, et al. ‘98
ud=us=ds >0
ud>0, us=ds =0
2SC
Bailin, Love ‘84
ud>0, us>0, ds =0 uSC
ud>0, ds>0, us =0 dSC
Ruster, et al. ‘03
u
us
d
s
ds
Matsuura, et al., ‘04
cf.) Neumann, Buballa, Oertel ’03
+ chiral symmetry restoration
many phases at intermediate densities
T
Abuki, Kunihiro, 2005; Ruster et al.,2005; Fukushima, 2005
m
Various Phases of Color Superconductivity
ud
3 order parameters ud, us, ds
 2*2*2=8 possibilities of distinct phases
CFL Alford, et al. ‘98
ud=us=ds >0
ud>0, us=ds =0
2SC
Bailin, Love ‘84
ud>0, us>0, ds =0 uSC
ud>0, ds>0, us =0 dSC
Ruster, et al. ‘03
u
us
d
s
ds
Matsuura, et al., ‘04
cf.) Neumann, Buballa, Oertel ’03
+ chiral symmetry restoration
many phases at intermediate densities
T
Abuki, Kunihiro, 2005; Ruster et al.,2005, Fukushima, 2005
m
Sarma Instability
The gapless SC is realized
only as the maximum of the effective potential.
V ( )
gapless
Sarma instability

BCS
Gapless state is unstable against the phase separation.
n
p
m
unlocking
region
p
What is the True Ground State?
gapless phases at T=0 have imaginary
color Meissner masses mM2<0. Huang, Shovkovy,2003
Chromo-magnetic instability
There is more stable state.
Candidates of true ground state:
•LOFF
•gluonic phase
•crystalline CSC
•spin-one superconductivity
•CSC + kaon condensation
high 
density
 low
Color Superconductivity in Compact Stars
ud
•effect of strange quark mass ms
•neutrality and -equilibrium conditions
u
us
Mismatch of densities
(1) strong coupling!
(2) mismatched Fermi surfaces
pF ~
d
s
ds
m  m 
(1) weak coupling
(2) common Fermi surface
T
m
Structual Change of Cooper Pairs
 ~ 100MeV
 / EF ~ 0.1
in electric SC
 / EF ~ 0.0001
m
x/d
T
m[MeV]
x – coherence length
d – interquark distance
Matsuzaki, 2000
Abuki, Hatsuda, Itakura, 2002
Phase Diagram
3-flavor NJL model
w/ slightly strong coupling GD/GS=0.75
mu,d=5MeV
ms = 80MeV
bound diquarks
for us, ds pairs
MK, Rischke, Shovokovy,2008
• m > m  superfluidity
• m < m  vacuum: No BEC region.
•Nevertheless, bound diquarks exist in the phase diagram.
Phase Diagram at Strong Coupling
GD/GS=1.1
BEC
MK, Rischke, Shovokovy,2008
•BEC manifests itself.
•Bound diquarks would exist in the deconfined phase.
Conceptual Phase Diagram
Conceptual phase diagram
T
Tdiss
preformed
stable bosons
Tc
BEC
strong coupling
lower 
large m
superfluidity
m~m
hidden by mass discontinuity
at 1st order transition
BCS
weak coupling
higher 
Conceptual Phase Diagram
Conceptual phase diagram
T
Tdiss
preformed
stable bosons
Tc
BEC
strong coupling
lower 
large m
superfluidity
m~m
BCS
weak coupling
higher 
Pseudogap in HTSC
Depression of the DoS around the Fermi surface above Tc
Pseudogap



m
N ( )
k
2
m

Pseudogap Region
2-flavor NJL; GD/GS = 0.61
N ( )
N free ( )
pseudogap region
The pseudogap survives up to  =0.05~0.1 ( 5~10% above TC ).
MK, et al., 2005
Conceptual Phase Diagram
Conceptual phase diagram
T
Tdiss
preformed
stable bosons
Tc
BEC
strong coupling
lower 
large m
Pseudogap
T*
(pre-critical)
region
superfluidity
m~m
BCS
weak coupling
higher 
Some Progress in Cold Atom
Crossover in Polarized Fermi gas
Pao, Wu, Yip, cond-mat/0506437
Son, Stephanov, cond-mat/0507586
Strong coupling limit
homogeneous
•mixture of fermions
and bound bosons
Weak coupling limit
spatially inhomogeneous
•LOFF
•phase separation
Question: How is the intermediate region between
two limits in the polarized Fermi gas?
Various Efforts in Cold Atom Society
•Monte Carlo simulation
•Renormailzation group method
•etc…
•Experimental result at unitarity
in the trapped gas
—no polarized SC at unitarity
T/TF
Shin, et al., Nature451,689(2008)
polarization
Cold Fermions with N Species
•Trapped potential + optical trap
•Select N hyperfine states w/ magnetic trap
N-”flavor” attractive fermion system
1
for N=3,
2
3
E
3
2
1
Cold Fermions with N Species
•Trapped potential + optical trap
•Select N hyperfine states w/ magnetic trap
N-”flavor” attractive fermion system
E
1
for N=3,
2
3
2
1
3
strong coupling:
Fermi-liquid of “trions”
weak coupling:
“2SC” BCS state
phase transition
Trion-BCS Transition
Rapp, et al., PRL99,130406(2007).
Rapp, et al., PRB77,144520(2008).
3-component Hubbard model:
Gutzwiller ansatz:
•MFA for g and BCS
•large d limit
strong
weak
Fermion-boson mixture : Another interesting system
Maeda, et al., 0904.4372
fermion
attraction
boson
Strong coupling
superfluid molecules
Weak coupling
BEC of bosons
Summary
•QCD相転移の向こう側ではクォーク物質（もしくはQGP状態）が
実現しており、低温高密度の基底状態はカラー超伝導である。
•低密度領域のカラー超伝導は強結合系であり、かつフェルミ面が
不揃いな超伝導状態である。
•冷却原子系から得られる情報は極めて興味深く、QCD相図およ
び、ハドロン化のメカニズムを理解する上でも有用な可能性があ
る。
Summary
Conceptual phase diagram
T
Tdiss
Bound diquark would
exist in sQGP.
preformed
stable bosons
Tc
BEC
strong coupling
lower 
large m
Pseudogap
T*
(pre-critical)
region
RHIC; hadronization, etc.
measurement on lattice QCD
Large fluctuations
affect various observables.
superfluidity
m~m
FAIR@GSI?
BCS
weak coupling
higher 

Wiedemann Effect (G. H. Wiedemann 1822

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