Transcript Relativistic BCS-BEC Crossover at High Density Pengfei Zhuang Physics Department, Tsinghua University, Beijing 100084 1) Introduction 2) Mean Field Theory 3) Fluctuations 4) Applications to Color.

Relativistic BCS-BEC Crossover at High Density
Pengfei Zhuang
Physics Department, Tsinghua University, Beijing 100084
1) Introduction
2) Mean Field Theory
3) Fluctuations
4) Applications to Color Superconductivity and Pion
Superfluidity
5) Conclusions
based on the works with Lianyi He, Xuguang Huang,
Meng Jin, Shijun Mao, Chengfu Mu and Gaofeng Sun
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introduction: pairing
Tc in the BEC region is independent of the coupling
between fermions, since the coupling only affects the
internal structure of the bosons.
in BCS, Tc is determined by thermal excitation of fermions,
in BEC, Tc is controlled by thermal excitation of collective modes
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introduction: BCS-BEC in QCD
QCD phase diagram
pair dissociation line
sQGP
BEC
BCS
strongly coupled quark matter with both quarks and bosons
rich QCD phase structure at high density, natural attractive interaction
in QCD, possible BCS-BEC crossover ?
new phenomena in BCS-BEC crossover of QCD:
relativistic systems, anti-fermion contribution, rich inner structure (color,
flavor), medium dependent mass, ……
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introduction: theory of BCS-BEC
*) Leggett mean field theory (Leggett, 1980)
*)NSR scheme (Nozieres and Schmitt-Rink, 1985)
extension of of BCS-BEC crossover theory at T=0 to T≠0 (above Tc )
Nishida and Abuki (2006,2007)
extension of non-relativistic NSR theory to relativistic systems,
BCS-NBEC-RBEC crossover
d 4q
1

 fl  
ln


(
q
)
4
 G
,
(2 )

G0G0
*) G0G scheme (Chen, Levin et al., 1998, 2000, 2005)

G0G
asymmetric pair susceptibility
extension of non-relativistic G0G scheme to relativistic systems
(He, Jin, PZ, 2006, 2007)
*) Bose-fermion model (Friedderg, Lee, 1989, 1990)
extension to relativistic systems (Deng, Wang, 2007)
Kitazawa, Rischke, Shovkovy, 2007, NJL+phase diagram
Brauner, 2008, collective excitations
……
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mean field: non-relativistic BCS-BEC at T=0
A.J.Leggett, in Modern trends in the theory of condensed matter,
Springer-Verlag (1980)
BCS limit
universality behavior

1
 ,
kF as

8 2 / 
e
,
e2
ˆ  1
BEC limit
  ,

b
2
n( p ) 
16
,
3
1
b 
mas2
ˆ   2

,
1
e
(   ) / T
1

0
BCS-BEC crossover
  0    0,
small   large ,
 0   0
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mean field: broken universality in relativistic systems
NJL-type model at moderate density Lianyi He, PZ, PRD75, 096003(2007)
g

L   i    m    i 5C T  T iC 5
4
order parameter
g
T



2



 iC 5
mean field thermodynamic potential
2
d 3k








E

E




k
k
k
k
3 
g
(2 )
±
k
±
k
2
E =
(x )
xk± =
k 2 + m2 ± m
+ D2
fermion and anti-fermion contributions
gap equation and number equation:
z
 

  1

1
1
2  1
  
    dxx   
2 
2  
2
E


2

E


2

x
x
  x


0
 x

z

 x  
 x  
2

2 
  dxx 1      1    

3 0
Ex  
Ex  


broken universality
kF
1

,
 
extra density dependence
k F as
m
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mean field: relativistic BCS-BEC
m- m
plays the role of non-relativistic chemical potential
m- m = 0 :
BCS-NBEC crossover
●
m= 0 : fermion and anti-fermion degenerate,
NBEC-RBEC crossover ●
●
●
in non-relativistic case, only one dimensionless
variable h = 1/ kF a s, changing the density can
not induce a BCS-BEC crossover.
however, in relativistic case, the extra density
dependence x = kF / m may induce a BCS-BEC.
QCD
atom gas
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fluctuations:
G0G scheme
Lianyi He, PZ, 2007
bare fermion propagator
G01 (k, )  (k0  ) 0    k  m
mean field fermion propagator
G1 (k, )  G01 (k, ) mf (k )
pair propagator
pair feedback to the fermion self-energy
(k )  mf (k )   fl (k ) 
fermions and pairs are coupled to each other
approximation
 fl (k ) 2pg G0 (k, )
the pseudogap is related to the uncondensed pairs,
in G0G scheme the pseudogap does not change the symmetry structure
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fluctuations: BCS-NBEC-RBEC
Tc :
critical temperature
T * : pair dissociation temperature
T  Tc :   0,  pg  0,
condensed phase
●
Tc  T  T * :   0,  pg  0,
normal phase with both fermions and pairs
T  T* :
  0,  pg  0
normal phase with only fermions
●
BCS:
  0,   m
no pairs
NBEC: 0    m / kF , 0<  m heavy pairs, no anti-pairs
RBEC:   m / kF ,  0 light pairs, almost the same
number of pairs and anti-pairs
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applications: BCS-BEC in asymmetric nuclear matter
Shijun Mao, Xuguang Huang, PZ, PRC79, 034304(2009)
asymmetric nuclear matter with both np and nn and pp pairings
density-dependent contact interaction (Garrido et al, 1999)
and density-dependent nucleon mass (Berger, Girod, Gogny, 1991)
by calculating the three coupled
gap equations, there exists only
np pairing BEC state at low
density and no nn and pp pairing
BEC states.
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applications: color superconductivity in NJL
Lianyi He, PZ, 2007
order parameters of spontaneous chiral and color symmetry breaking
  
  3   iC  ij  3i 5 j
color breaking from SU(3) to SU(2)
quarks at mean field and mesons and diquarks at RPA
quark propagator in 12D Nambu-Gorkov space

 u1 
   u1  dC2  d 2  uC1  d 1  uC2  u 2  dC1
 C 
 d 2 
 SA
 

 d2 
SB

 uC1 

SC


 d 1  S  
SD

 C 
u2


 
SE
 u 2 


SF
 C 


d
1


 
 uC3  diquark & meson propagators at RPA
 u 3 



 d3 
 C 
 d3 
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 u 3  uC3  d 3  dC3 
 GI  I 

I  A, B, C, D, E, F
SI   
 

 I GI 



M q  m0  2GS
Ek  k 2  M q2


2
2
E   Ek      2GD  



 diquark & meson polarizations
D
M
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applications: BCS-BEC and color neutrality
Lianyi He, PZ, PRD76, 056003(2007)
gap equations for chiral and diquark condensates at T=0


d 3k 1  Ek   B / 3 Ek  B / 3
m

m

8
G
m



E


/
3
 k B 

0
s 

3


(2

)
E
E
E
p 





d 3k  1
1 



8
G


to guarantee color neutrality, we introduce
d 
3  
 

(2

)
E
E
 
 

color chemical potential:
r  g  B / 3  8 / 3,
b  B / 3  28 / 3
●
r  m
●
●●
m

B / 3  m
  Gd / Gs
there exists a BCS-BEC crossover
color neutrality speeds up the chiral
restoration and reduces the BEC region
going beyond MF, see the talk by He, May 24
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applications: BCS-BEC in pion superfluid
meson mass, Goldstone mode
Gaofeng Sun, Lianyi He, PZ, PRD75, 096004(2007)
meson spectra function
 (, k )  2Im D(, k )
BCS
BEC
going beyond MF, see the talk by Mu, May 24
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conclusions
*
BCS-BEC crossover is a general phenomena from cold atom gas
to quark matter.
*
BCS-BEC crossover is closely related to QCD key problems:
vacuum, color symmetry, chiral symmetry, isospin symmetry ……
*
BCS-BEC crossover in color superconductivity and pion
superfluid is not induced by simply increasing the coupling
constant of the attractive interaction, but by changing the
corresponding charge number.
*
there are potential applications in heavy ion collisions (at
CSR/Lanzhou, FAIR/GSI and RHIC/BNL) and compact stars.
thanks for your patience
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backups
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vector meson coupling and magnetic instability
vector-meson coupling
2
2
LV  GV       5  


vector condensate
V  2GV  0
gap equation

d 3k  Ek  B / 3 Ek   B / 3
V  8GV 




E


/
3


k
B


(2 )3 
E
E

 1
vector meson coupling slows down
the chiral symmetry restoration and
enlarges the BEC region.
r  m
Meissner masses of some gluons
are negative for the BCS Gapless
CSC, but the magnetic instability
is cured in BEC region.
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beyond mean field
0  (T  0) is determined by the coupling and chemical potential
0  100  200 MeV  q  300  500 MeV
● going beyond mean field reduces the critical temperature of color
superconductivity
● pairing effect is important around the critical temperature and dominates
the symmetry restored phase
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pion superfluid
NJL with isospin symmetry breaking

LNJL    i     m0   0   G     i i 5 
2
2

quark chemical potentials

  u
 0
0   B / 3  I / 2
0


d  
0
 B / 3   I / 2 
chiral and pion condensates with finite pair momentum
     u   d ,  u  uu ,  d  dd
   2 ui 5 d 
quark propagator in MF

2
e2iqx ,    2 di 5u 
   p    q   u  0  m

2iG 5
S ( p, q )  



2
iG


k



q




m
5

d
0


1
thermodynamic potential and gap equations:
T
  G ( 2   2 )  Tr Ln S 1
V

 2

 2

 2
 0,
 0,
 0,
 0,
 0,
 0,
 u
 u2
 d
 d2

 2
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
2
e2iqx
m  m0  2G

 2
 0,
0
q
q 2
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mesons in RPA
meson propagator
D at RPA
considering all possible channels in the bubble summation
 1, m  
meson polarization functions
i  , m  
4
  5

d p
*



m
 mn (k )  i
Tr m S ( p  k )n S ( p)
i  5 , m   
(2 )4
 i 3 5 , m   0
pole of the propagator determines meson masses M m



2G  (k )
2G 0 (k ) 
1  2G (k ) 2G  (k )



2
G

(
k
)
1

2
G

(
k
)

2
G

(
k
)

2
G

(
k
)
 
  
  
  0


det 
0
2G  (k ) 2G   (k ) 1  2G    (k ) 2G   0 (k ) 


 2G  (k ) 2G  (k )
2G 0  (k ) 1  2G 0 0 (k ) 
0
0 

k0  M m , k  0
mixing among normal
the new eigen modes
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 ,   ,   in pion superfluid phase,
 ,   ,   are linear combinations of  ,   ,  
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phase diagram of pion superfluid
chiral and pion condensates at T  B  q  0
in NJL, Linear Sigma Model and Chiral
Perturbation Theory, there is no remarkable
difference around the critical point.
analytic result:
critical isospin chemical potential for pion
superfluidity is exactly the pion mass in the
vacuum: c
I  m
pion superfluidity phase diagram in I  B
plane at T=0
I : average Fermi surface
B (nB ): Fermi surface mismatch
homogeneous (Sarma, q  0 ) and
inhomogeneous pion superfluid (LOFF, q  0 )
magnetic instability of Sarma state at high
average Fermi surface leads to the LOFF state
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