may, 2009 csqcdii beijing 1 relativistic bcs-bec crossover at high density pengfei zhuang physics...

May, 2009 CSQCDII BeijingMay, 2009 CSQCDII Beijing 11

Relativistic BCS-BEC Crossover at High DensityRelativistic BCS-BEC Crossover at High Density Pengfei Zhuang Pengfei Zhuang

Physics Department, Tsinghua University, Beijing Physics Department, Tsinghua University, Beijing

100084100084 1) Introduction1) Introduction

2) Mean Field Theory 2) Mean Field Theory

3) Fluctuations 3) Fluctuations

4) Applications to Color Superconductivity and Pion 4) Applications to Color Superconductivity and Pion SuperfluiditySuperfluidity

5) Conclusions5) Conclusions

based on the works withbased on the works with Lianyi He, Xuguang Huang, Lianyi He, Xuguang Huang, Meng Jin, Shijun Mao, Chengfu Mu and Gaofeng Sun Meng Jin, Shijun Mao, Chengfu Mu and Gaofeng Sun


introduction: pairingintroduction: pairing

in BCS, Tin BCS, Tcc is determined by thermal excitation of fermions, is determined by thermal excitation of fermions,

in BEC, Tin BEC, Tcc is controlled by thermal excitation of collective modes is controlled by thermal excitation of collective modes

Tc in the BEC region is independent of the coupling Tc in the BEC region is independent of the coupling between fermions, since the coupling only affects the between fermions, since the coupling only affects the internal structure of the bosons.internal structure of the bosons.


introduction: BCS-BEC in QCDintroduction: BCS-BEC in QCD

pair dissociation line

BCSBEC

sQGP

QCD phase diagramQCD phase diagram

rich QCD phase structure at high density, natural attractive interaction rich QCD phase structure at high density, natural attractive interaction in QCD, possible BCS-BEC crossover ?in QCD, possible BCS-BEC crossover ?

new phenomena in BCS-BEC crossover of QCD:new phenomena in BCS-BEC crossover of QCD: relativistic systems, anti-fermion contribution, rich inner structure relativistic systems, anti-fermion contribution, rich inner structure (color, flavor), medium dependent mass, ……(color, flavor), medium dependent mass, ……

strongly coupled quark matter with both quarks and bosonsstrongly coupled quark matter with both quarks and bosons


introduction: theory of BCS-BEC introduction: theory of BCS-BEC

*) Leggett mean field theory *) Leggett mean field theory (Leggett, 1980)(Leggett, 1980) *)NSR scheme *)NSR scheme (Nozieres and Schmitt-Rink, 1985)(Nozieres and Schmitt-Rink, 1985) extension of of BCS-BEC crossover theory at T=0 to T extension of of BCS-BEC crossover theory at T=0 to T≠0 (above T≠0 (above Tcc

) ) Nishida and Abuki (2006,2007)Nishida and Abuki (2006,2007) extension of non-relativistic NSR theory to relativistic systems, extension of non-relativistic NSR theory to relativistic systems, BCS-NBEC-RBEC crossover BCS-NBEC-RBEC crossover

4

0 04

1ln ( ) ,

(2 )fl

d qq G G

G

*) *) GG00G scheme G scheme (Chen, Levin et al., 1998, 2000, 2005)(Chen, Levin et al., 1998, 2000, 2005)

asymmetric pair susceptibility asymmetric pair susceptibility 0 G G extension of non-relativistic extension of non-relativistic GG00G scheme to relativistic systems G scheme to relativistic systems

(He, Jin, PZ, 2006, 2007) (He, Jin, PZ, 2006, 2007)

*) Bose-fermion model *) Bose-fermion model (Friedderg, Lee, 1989, 1990)(Friedderg, Lee, 1989, 1990) extension to relativistic systems extension to relativistic systems (Deng, Wang, 2007)(Deng, Wang, 2007) Kitazawa, Rischke, Shovkovy, 2007,Kitazawa, Rischke, Shovkovy, 2007, NJL+phase diagram NJL+phase diagram Brauner, 2008,Brauner, 2008, collective excitations …… collective excitations ……


BCS limit

BEC limit

mean field: non-relativistic BCS-BEC at T=0mean field: non-relativistic BCS-BEC at T=0

2 /2

1 8ˆ, , 1

F s

ek a e

2

2

( ) /

16ˆ, ,

31

, 2

1( ) 0

1

bb

s

T

ma

n pe

BCS-BEC crossover0 0,

small large ,

0 0

A.J.Leggett, in Modern trends in the theory of condensed matter, A.J.Leggett, in Modern trends in the theory of condensed matter, Springer-Verlag (1980) Springer-Verlag (1980)

universality behavior


mean field: broken universality in relativistic systemsmean field: broken universality in relativistic systems

5 54T Tg

L i m i C iC

52TgiC

2 3

3(2 ) k k k k

d kE E

g

order parameterorder parameter

NJL-type model at moderate densityNJL-type model at moderate density

mean field thermodynamic potentialmean field thermodynamic potential

fermion and anti-fermion contributionsfermion and anti-fermion contributions

Lianyi He, PZ, PRD75, 096003(2007)Lianyi He, PZ, PRD75, 096003(2007)

22 2

0

2

0

1 1 1 1

2 2 2

21 1

3

1,

z

x x x x

zx x

x x

F

F s

dxxE E

dxxE E

k

k a m

gap equation and number equation:gap equation and number equation:

broken universality broken universality extra extra density dependencedensity dependence

( )2 2

2 2

k k

k

E

k m

x

x m

± ±

±

= +D

= + ±


●

●

mean field: relativistic BCS-BECmean field: relativistic BCS-BEC

mm- plays the role of non-relativistic chemical potentialplays the role of non-relativistic chemical potential

0 :mm- =fermion and anti-fermion degenerate, fermion and anti-fermion degenerate, NBEC-RBEC NBEC-RBEC crossovercrossover

0 :m=BCS-NBEC crossoverBCS-NBEC crossover ●

●in non-relativistic case, only one dimensionless variable , changing the density can not induce a BCS-BEC crossover. however, in relativistic case, the extra density dependence may induce a BCS-BEC.

QCD

atom gas

1/ F skh a=

/Fk mx=


0 G Gfluctuations: schemefluctuations: scheme

bare fermion propagatorbare fermion propagator

pair propagatorpair propagator

10 0 0( , ) ( )G k k k m

1 10( , ) ( , ) ( ) mfG k G k k

Lianyi He, PZ, 2007Lianyi He, PZ, 2007

mean field fermion propagatormean field fermion propagator

fermions and pairs are coupled to each otherfermions and pairs are coupled to each other

( ) ( ) ( )mf flk k k

pair feedback to the fermion self-energypair feedback to the fermion self-energy

approximationapproximation2

0( ) ( , )fl pgk G k the pseudogap is related to the uncondensed pairs, the pseudogap is related to the uncondensed pairs,

in G in G00G scheme the pseudogap does not change the symmetry structureG scheme the pseudogap does not change the symmetry structure


fluctuations: BCS-NBEC-RBEC fluctuations: BCS-NBEC-RBEC

*

*

*

: critical temperature

: pair dissociation temperature

: 0, 0,

condensed phase

: 0, 0,

normal phase with both fermions and pairs

:

c

c pg

c pg

T

T

T T

T T T

T T

0, 0

normal phase with only fermions

pg

BCS: no pairs BCS: no pairs

NBEC: heavy pairs, no anti-pairsNBEC: heavy pairs, no anti-pairs

RBEC: light pairs, almost the same RBEC: light pairs, almost the same number of pairs and anti-pairsnumber of pairs and anti-pairs

0, m

0 / , 0<Fm k m

/ , 0Fm k

●●

●●


applications: BCS-BEC in asymmetric nuclear matterapplications: BCS-BEC in asymmetric nuclear matter

Shijun Mao, Xuguang Huang, PZ, PRC79, 034304(2009)Shijun Mao, Xuguang Huang, PZ, PRC79, 034304(2009)

asymmetric nuclear matter with both asymmetric nuclear matter with both npnp and and nnnn and and pp pp pairingspairingsdensity-dependent contact interaction density-dependent contact interaction (Garrido et al, 1999)(Garrido et al, 1999)

and density-dependent nucleon mass and density-dependent nucleon mass (Berger, Girod, Gogny, 1991)(Berger, Girod, Gogny, 1991)

by calculating the three coupled by calculating the three coupled gap equations, there exists only gap equations, there exists only np pairing BEC state at low np pairing BEC state at low density and no nn and pp density and no nn and pp pairing BEC states.pairing BEC states.


1 2 2 1 1 2 2 1 3 3 3 3 C C C C C Cu d d u d u u d u u d d 1

2

2

1

1

2

2

1

3

3

3

3

u

Cd

d

Cu

d

Cu

u

Cd

u

Cu

d

Cd

3 3

5C iji ji

order parameters of spontaneous chiral and color symmetry breakingorder parameters of spontaneous chiral and color symmetry breaking

quark propagator in 12D Nambu-Gorkov spacequark propagator in 12D Nambu-Gorkov space

A

B

C

D

E

F

S

S

SS

S

S

S

I II

I I

GS

G

0 2q SM m G

, , , , ,I A B C D E F

2 2

2 22

k q

k D

E k M

E E G

applications: color superconductivity in NJLapplications: color superconductivity in NJL

color breaking from SU(3) to SU(2)

diquark & meson polarizationsdiquark & meson polarizations

M D

diquark & meson propagators at RPAdiquark & meson propagators at RPA

quarks at mean field and mesons and diquarks at RPAquarks at mean field and mesons and diquarks at RPA

Lianyi He, PZ, 2007Lianyi He, PZ, 2007


applications: BCS-BEC and color neutralityapplications: BCS-BEC and color neutrality

gap equations for chiral and diquark condensates at T=0gap equations for chiral and diquark condensates at T=0

3

0 3

3

3

/ 3 / 318 / 3

(2 )

1 18

(2 )

k B k Bs k B

p

d

E Ed km m G m E

E E E

d kG

E E

/ 3 /B d sm m G G

●● ●●

there exists a BCS-BEC crossoverthere exists a BCS-BEC crossover

to guarantee color neutrality, we introduce to guarantee color neutrality, we introduce color chemical potential: color chemical potential:

8 8/ 3 / 3, / 3 2 / 3r g B b B

color neutrality speeds up the chiral color neutrality speeds up the chiral restoration and reduces the BEC regionrestoration and reduces the BEC region

●● ●●

r m

Lianyi He, PZ, PRD76, 056003(2007)Lianyi He, PZ, PRD76, 056003(2007)

going beyond MF, see the talk by He, May 24going beyond MF, see the talk by He, May 24


applications: BCS-BEC in pion superfluidapplications: BCS-BEC in pion superfluid

( , ) 2 Im ( , )k D k meson spectra functionmeson spectra function

meson mass, Goldstone modemeson mass, Goldstone mode

BEC

BCS

Gaofeng Sun, Lianyi He, PZ, PRD75, 096004(2007)Gaofeng Sun, Lianyi He, PZ, PRD75, 096004(2007)

going beyond MF, see the talk by Mu, May 24going beyond MF, see the talk by Mu, May 24


conclusionsconclusions

* * BCS-BEC crossover is a general phenomena from cold atom gas BCS-BEC crossover is a general phenomena from cold atom gas to quark matter.to quark matter.

* * BCS-BEC crossover is closely related to QCD key problems: BCS-BEC crossover is closely related to QCD key problems: vacuum, color symmetry, chiral symmetry, isospin symmetry ……vacuum, color symmetry, chiral symmetry, isospin symmetry ……

* BCS-BEC crossover in color superconductivity and pion * BCS-BEC crossover in color superconductivity and pion superfluid is not induced by simply increasing the coupling superfluid is not induced by simply increasing the coupling constant of the attractive interaction, but by changing the constant of the attractive interaction, but by changing the corresponding charge number.corresponding charge number.

* * there are potential applications in heavy ion collisions (at there are potential applications in heavy ion collisions (at CSR/Lanzhou, FAIR/GSI and RHIC/BNL) and compact stars.CSR/Lanzhou, FAIR/GSI and RHIC/BNL) and compact stars.

thanks for your patience thanks for your patience


backupsbackups


vector meson coupling and magnetic instabilityvector meson coupling and magnetic instability

vector-meson couplingvector-meson coupling 2 2

5V VL G

gap equationgap equation

02V VG

vector meson coupling slows down vector meson coupling slows down the chiral symmetry restoration and the chiral symmetry restoration and enlarges the BEC region. enlarges the BEC region.

vector condensatevector condensate

3

3

/ 3 / 38 / 3

(2 )k B k B

V V k B

E Ed kG E

E E

1

Meissner masses of some gluons Meissner masses of some gluons are negative for the BCS Gapless are negative for the BCS Gapless CSC, CSC, but the magnetic instability but the magnetic instability is cured in BEC region.is cured in BEC region.

r m


beyond mean fieldbeyond mean field

0 100 200 MeV 300 500 MeVq 0 ( 0)T is determined by the coupling and chemical potentialis determined by the coupling and chemical potential

● going beyond mean field reduces the critical temperature of color going beyond mean field reduces the critical temperature of color superconductivitysuperconductivity ● ● pairing effect is important around the critical temperature and dominates the pairing effect is important around the critical temperature and dominates the symmetry restored phasesymmetry restored phase


NJL with isospin symmetry breakingNJL with isospin symmetry breaking

2 2 1

2 2 2 2

2 2 2 2

( ) Tr Ln

0, 0, 0, 0, 0, 0, 0, 0u u d d

TG S

V

q q

2 2

0 0 5NJL iL i m G i

, , u d u duu dd chiral and pion condensates with finite pair momentumchiral and pion condensates with finite pair momentum

quark propagator in MFquark propagator in MF 0 51

5 0

2( , )

2u

d

p q m iGS p q

iG k q m

0 2m m G

2 25 52 , 2

2 2iq x iq xui d e di u e

quark chemical potentialsquark chemical potentials

0 / 3 / 2 0

0 0 / 3 / 2u B I

d B I

thermodynamic potential and gap equations:thermodynamic potential and gap equations:

pion superfluidpion superfluid


5

5

3 5 0

1,

,

,

,

m

m

i m

i m

i m

4

*4

( ) Tr ( ) ( )(2 )mn m n

d pk i S p k S p

meson polarization functionsmeson polarization functions

meson propagator at RPAmeson propagator at RPA

pole of the propagator determines meson masses pole of the propagator determines meson masses

mixing among normal in pion superfluid phase, mixing among normal in pion superfluid phase,

the new eigen modes are linear combinations of the new eigen modes are linear combinations of

considering all possible channels in the bubble summationconsidering all possible channels in the bubble summation

0

0

0

0 0 0 0 00 , 0

1 2 ( ) 2 ( ) 2 ( ) 2 ( )

2 ( ) 1 2 ( ) 2 ( ) 2 ( )det

2 ( ) 2 ( ) 1 2 ( ) 2 ( )

2 ( ) 2 ( ) 2 ( ) 1 2 ( )mk M k

G k G k G k G k

G k G k G k G k

G k G k G k G k

G k G k G k G k

0

, ,

mM

, ,

D

mesons in RPAmesons in RPA

, ,


I B

chiral and pion condensates at chiral and pion condensates at in NJL, Linear Sigma Model and Chiral in NJL, Linear Sigma Model and Chiral Perturbation Theory, there is no remarkable Perturbation Theory, there is no remarkable difference around the critical point.difference around the critical point.

analytic result: analytic result: critical isospin chemical potential for pion critical isospin chemical potential for pion superfluidity is exactly the pion mass in the superfluidity is exactly the pion mass in the vacuum: vacuum:

0BT q

cI m

( )B BnI

pion superfluidity phase diagram in pion superfluidity phase diagram in plane at T=0plane at T=0

: average Fermi surface: average Fermi surface

: Fermi surface mismatch : Fermi surface mismatch

homogeneous (Sarma, ) and homogeneous (Sarma, ) and inhomogeneous pion superfluid (LOFF, )inhomogeneous pion superfluid (LOFF, )

magnetic instability of Sarma state at high magnetic instability of Sarma state at high average Fermi surface leads to the LOFF stateaverage Fermi surface leads to the LOFF state

0q 0q

phase diagram of pion superfluidphase diagram of pion superfluid

may, 2009 csqcdii beijing 1 relativistic bcs-bec crossover at high density pengfei zhuang physics...

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