1. 第9届QCD相变和重离子碰撞物理研讨会，杭州
BEC-BCS crossover and the liquid-gas phase transition in hot and dense nuclear matter 金猛
华中师范大学
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2. outline Background 2.Pair correlations: BEC and BCS
3.The method beyond mean field
4.Summary and perspectives

3. The study on the properties of hot and dense nuclear matter is a very important aspect of nuclear physics
The equation of states (EoS), pairing instability, nucleon correlations
Low energy heavy-ion collision
Neutron star

4. Cooper pair or bound state
Interacting Fermion system
Interaction strength
Strong: bound state
Bose-Einstein condensation
(BEC)
Weak: Cooper pair
Superconductivity, superfluidity
(BCS)
Interacting Fermions
Quarks
Color superconductivity
Nucleons
Nucleon superfluidity
Cold atoms
Superfluidity
Electrons
Superconductivity

5. Nucleon interaction
nonrelativistic, effective interaction, Gogny force
the single particle properties, low T nuclear matter

6. Mean field theory
Hartree, Hartree-Fock, Hartree-Fock-Bogliubov

7. Beyond mean field approximation
1. Brueckner-Bethe-Goldstone (G-matrix)
can give a better saturation point, but not satisfy Hugenholtz van Hove
(HvH) theorem (including the rearrangement term can improve the result)
2. Generalized Beth-Uhlenbeck approach (T-matrix)
can study the different components of nuclear density, Mott effect to
the production of deuteron.
M.Schmidt, G.Roepke and H.Schulz, Ann. Phys. 202(90)57
3. Nozières-Schmitt-Rink (NSR) approach (T-matrix)
P. Nozeieres and S.Schmitt-Rink, J. low temp. phys. 59(84)195
4. Selfconsistent Green’s function (SCGF) method (T-matrix)
introducing the spectral function and calculating the correlation effect
selfconsistently. The thermodynamic consistency is also fulfilled in this
approach
A.Rios et al, Phys. Rev. C 78(08)044314;
V.Soma and P.Bozek, Phys.Rev.C 80(09)025803

8. Nozieres-Schmitt-Rink (NSR)
J. low temp. phys. 59(84)195

9. BEC-BCS crossover
For the separable interaction, the T-matrix can be solved analytically
Thouless criterion: Ann. Phys.10(60)553
The threshold for superfluidity defined as the pole of the two-particle T-matrix
In the weak coupling limit, this gives the same result for Tc as the BCS theory

10. T.Alm, A. Schnell, G. Roepke and H. Stein, ZPA 351(1995)295
Tc=7MeV
1、At low density (the strong coupling region), the critical temperature coincides
with that for the BEC in Boson gas,
2、at high density (the weak coupling region), the critical temperature coincides
with that for the BCS instability in Fermion gas,
3、In the middle density region, the critical temperature is determined from the
Thouless criterion

11. NSR approach based on the Hartree-Fock

12. Introducing the phase shift,
M. Jin, M.Urban,P.Schuck, PRC 82(2010)024911
Introducing the phase shift,
The result is same as that obtained from the optical theorem
in the electron-hole system,
R.Zimmermann and H. Stolz, Phys. Stat. Sol(b) 131(1985)151

13. Equation of state
The liquid-gas phase transition still exist with the same
critical temperature as mean field result. Tc=15.9MeV

14. The contribution of the nucleon correlation
The nucleon correlation is important when the density is
below the saturation density of nuclear matter

15. BEC-BCS crossover within MNSR
Tc=4.5MeV
Compared to the NSR result, the critical temperature for the
BEC-BCS crossover becomes lower

16. Liquid-gas phase transition
The boundary of the liquid-gas phase transition can be determined by the equilibrium condition

17. Phase diagram
The S-type region is covered by the unstable region of the liquid-gas
phase transition
2. When we decrease the T below the Tc of the BEC-BCS crossover there is a
pole in the T-matrix.
We need to introduce the pairing gap if we want to go inside the phase
boundary of the BEC-BCS crossover

18. Summary and perspectives
With the NSR, one can treat the bound states and the Cooper pairing together in nuclear matter.
We develop the Nozieres Schmitt-Rink approach based on the Hartree-Fock approximation, from which we can consider the correlation between nucleons.
We need to introduce the pairing gap to go down below the BEC-BCS crossover boundary.
We plan to apply our method to study the n-p, p-p and n-n pairing in the asymmetric nuclear matter, pure neutron matter etc..

19. Thank you !