1. Pengfei Zhuang Physics Department, Tsinghua University, Beijing 100084
Relativistic BCS-BEC Crossover at High Density
Pengfei Zhuang Physics Department, Tsinghua University, Beijing
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
May, CSQCDII Beijing

2. 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
May, CSQCDII Beijing

3. introduction: BCS-BEC in QCD
QCD phase diagram
pair dissociation line
BCS
BEC
sQGP
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, ……
May, CSQCDII Beijing

4. 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
*) G0G scheme (Chen, Levin et al., 1998, 2000, 2005) 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 ……
May, CSQCDII Beijing

5. universality behavior
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
BEC limit
BCS-BEC crossover
May, CSQCDII Beijing

6. mean field: broken universality in relativistic systems
Lianyi He, PZ, PRD75, (2007)
NJL-type model at moderate density
order parameter
mean field thermodynamic potential
fermion and anti-fermion contributions
gap equation and number equation:
broken universality extra density dependence
May, CSQCDII Beijing

7. ● ● ● ● QCD atom gas mean field: relativistic BCS-BEC
plays the role of non-relativistic chemical potential ●
BCS-NBEC crossover
fermion and anti-fermion degenerate, NBEC-RBEC 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
May, CSQCDII Beijing

8. fluctuations: scheme fermions and pairs are coupled to each other
Lianyi He, PZ, 2007
bare fermion propagator
mean field fermion propagator
pair propagator
pair feedback to the fermion self-energy
fermions and pairs are coupled to each other
approximation
the pseudogap is related to the uncondensed pairs, in G0G scheme the pseudogap does not change the symmetry structure
May, CSQCDII Beijing

9. ● ● fluctuations: BCS-NBEC-RBEC BCS: no pairs
NBEC: heavy pairs, no anti-pairs
RBEC: light pairs, almost the same number of pairs and anti-pairs
May, CSQCDII Beijing

10. applications: BCS-BEC in asymmetric nuclear matter
Shijun Mao, Xuguang Huang, PZ, PRC79, (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.
May, CSQCDII Beijing

11. applications: color superconductivity in NJL
Lianyi He, PZ, 2007
order parameters of spontaneous chiral and color symmetry breaking
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
diquark & meson polarizations
diquark & meson propagators at RPA
May, CSQCDII Beijing

12. applications: BCS-BEC and color neutrality
Lianyi He, PZ, PRD76, (2007)
gap equations for chiral and diquark condensates at T=0
to guarantee color neutrality, we introduce color chemical potential: ● ● ● ●
color neutrality speeds up the chiral restoration and reduces the BEC region
there exists a BCS-BEC crossover
going beyond MF, see the talk by He, May 24
May, CSQCDII Beijing

13. BCS BEC applications: BCS-BEC in pion superfluid
meson mass, Goldstone mode
Gaofeng Sun, Lianyi He, PZ, PRD75, (2007)
meson spectra function
BCS
BEC
going beyond MF, see the talk by Mu, May 24
May, CSQCDII Beijing

14. thanks for your patience
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
May, CSQCDII Beijing

15. backups
May, CSQCDII Beijing

16. vector meson coupling and magnetic instability
vector condensate
gap equation
vector meson coupling slows down the chiral symmetry restoration and enlarges the BEC region.
Meissner masses of some gluons are negative for the BCS Gapless CSC, but the magnetic instability is cured in BEC region.
May, CSQCDII Beijing

17. beyond mean field is determined by the coupling and chemical potential
● 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
May, CSQCDII Beijing

18. pion superfluid quark chemical potentials
NJL with isospin symmetry breaking
quark chemical potentials
chiral and pion condensates with finite pair momentum
quark propagator in MF
thermodynamic potential and gap equations:
May, CSQCDII Beijing

19. pole of the propagator determines meson masses
mesons in RPA
meson propagator at RPA
considering all possible channels in the bubble summation
meson polarization functions
pole of the propagator determines meson masses
mixing among normal in pion superfluid phase,
the new eigen modes are linear combinations of
May, CSQCDII Beijing

20. phase diagram of pion superfluid
chiral and pion condensates at 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:
pion superfluidity phase diagram in plane at T=0
: average Fermi surface
: Fermi surface mismatch
homogeneous (Sarma, ) and inhomogeneous pion superfluid (LOFF, )
magnetic instability of Sarma state at high average Fermi surface leads to the LOFF state
May, CSQCDII Beijing