BCS-BEC crossover in relativistic superfluid

1. BCS-BEC crossover in relativistic superfluid Yusuke Nishida(University of Tokyo)with Hiroaki Abuki (Yukawa Institute) hep-ph/0504083 Seminar@KEK 3 October, 2005

2. New fermion superfluid in 40K, 6Li Regal et al., PRL 92 (2004) 040403 Bartenstein et al., PRL 92 (2004) 120401 Zwierlein et al., PRL 92 (2004) 120403 Kinast et al., PRL 92 (2004) 150402 Bourdel et al., cond-mat/0403091 and more… Interaction is arbitrarily tunable using Feshbach resonance weak strong • Weak coupling : • BCS by Cooper pairs • Strong coupling : • BEC by molecules

3. Idea of BCS-BEC crossover Eagles, Phys. Rev. 186 (1969) 456; Leggett, J. Phys. 41 (1980) C7-19 Nozi`eres and Schmitt-Rink, J. Low Temp. Phys. 59 (1985) 195 BCS : large size pairing BEC : small size pairing d bound state x Strongerattractive int. x/d~104-5 Bose-Einstein Condensationof bound bosons Condensation ofCooper pairs

4. T m BCS-BEC crossover in QCD ? Abuki, Hatsuda and Itakura, PRD 65 (2002) 074014 Perturbative QCD at high density => BCS instability in color 3, flavor 1 and JP=0+ diquark channel QGP ? Color superconductivity Hadron phase larger g • Possible realization of BEC in low or intermediate density region of CSC

5. Contents • Introduction • BCS-BEC in non-relativistic system • BEC in relativistic system • Crossovers in relativistic system • BCS-BEC-RBEC phases • Phase diagram • Summary and implication for QCD

6. strength of attraction: Result in non-relativistic system S’a de Melo et al., PRL 71 (1993) 3202 Critical temperature Tc with fixed density BCS behavior Fermi gas Bose gas BEC behavior Tc

7. Why Tc in BEC is constant ? • Bound boson’s mass : • density : • Tc of ideal BEC : independent of coupling Non-relativistic superfluids : • liquid 3He (Leggett, J. Phys. 41 (1980) C7-19) • trapped fermionic alkali atoms (Ohashi and Griffin, PRL (’02)) • nuclear matter (Lombardo et al., PRC 64 (2001) 064314) In relativistic system, binding effect on the Boson mass appears : mB decreases as increasing the coupling

8. Ideal BEC in relativistic system Kapsta, Finite Temperature Field Theory (Cambridge, 1989) boson density anti-boson density At T=Tc of ideal BEC => mB=mB Non-relativistic limit(mB3>>NB) Relativistic limit(mB3<<NB) Anti-boson density appears Anti-boson density is negligible Relativistic BEC usual BEC state

9. Ideal BEC in relativistic system boson density TRL Tc TNR LARGE mB small mB anti-boson density RBEC BEC “Crossover” fromBECto RBECas decreasing the boson mass

10. BCS-BEC in relativistic system ? What we expect is … TRL Tc Tc TNR BEC RBEC BCS BEC G 2 crossovers : BCS->BECandBEC-> RBEC as increasing the coupling Our analysis

11. _ NF+NF : (anti-)fermion density Our formulation hep-ph/0504083 2-body contact interaction with massive fermion fermion mass fermion chemical potential Attraction in JP=0+ channel Fermion pair correlation in normal phase <Mean field+Gauss fluctuation> Nozi`eres and Schmitt-Rink (’85); S’a de Melo et al. (’93) • Fermion number density Ntotal NB : pair correlation : phase shift

12. Stable (anti-)boson density NB : bosonic contribution to the density If attraction G is strong enough, bound state poles appear in • Bound boson density internal structure of boson • Bound anti-boson density • Unstable boson density

13. Numerical calculations • Fixed number density Ntotal • Critical temperature Tc by Thouless criterion Pair fluctuation diverges : pair susceptibility Tc and mas functions of coupling G with fixed number density (2m=mBon T=Tc) Parameter set : scaled by ultraviolet cutoff L

14. Critical temperature vs. coupling Chemical potential and densities on Tc line Normal phase Tc Superfluid phase intermediate strong weak BEC RBEC BCS

15. BCS phase Weak coupling region Mean field result • Exponentially increasing Tc • Mean field result is valid • m ~EF • Fermion density is dominant

16. BEC phase Intermediate coupling BEC TNR with mB=2m TNR with mB=2m • Slowly increasing Tc • Well described by BEC • m <m • Bound boson density is dominant

17. Relativistic BEC Strong coupling 1/T ~ (NF)-1/3 BEC TRL with mB=2m • Very large Tc~(NF)1/3 • Small interparticle distance • m ~0 • Anti-particles are available

18. Entropy vs. coupling At T=Tc Stotal=SF+SB : total : fermion : boson BCS>BEC<< RBEC F _ F

19. Dissociation in (R)BEC phases Bound boson poles Bound bosons melt at T=Tdiss > Tc (G/G0=1) : T=Tc : Tc<T<Tdiss : T=Tdiss 2m-2m Tdiss Bound anti-boson poles Tc • Normal phase without • stable bosons (T>Tdiss) • Preformed boson phase • (Tc<T<Tdiss) Superfluid phase _ Cf. q-q bound state above Tc by Asakawa and Hatsuda

20. Phase diagram in m-G plane Half of boson mass in the vacuum (T=m=0) • Crossover regions • BCS->BEC: m ~m • BEC-> RBEC : m ~0 mB/2 BEC Approximated by points where the stable bosonis formed / becomesmassless in the vacuum RBEC BCS

21. Summary 2 crossovers as increasing the coupling Tdiss Dissociation ofbound bosons above Tc =preformed boson phase(Tc<T<Tdiss) Tc Fermi gas Bose gas Cf. pseudogap phase discussed by Kitazawa et al., PRD 70 (2004) 056003 Superfluid phase

22. T m Significance of (R)BEC in QCD • BEC criterion : m<m Speculative QCD phase diagram Hard dense loop gives fermion mass Cf. diquark bound state above Tc by Shuryak and Zahed RBEC Preformedboson phase BEC criterion : Probable realization of BEC in non-perturbativeregion of CSC BEC(g>1) BCS • RBEC (large Tc) is realized when large g & small m • Future work : competition b/w (R)BEC and c-phase : realistic treatment of plasmino mass

23. Backup Slides

24. Hints from lattice simulations Nakamura and Saito, PTP111 (2004) 733; PTP 112 (2004) 183 Q-Q potentials Q-Q potentials _

25. Size of pairing Engelbrecht et al., PRB 55 (1997) 15153

26. Idea of BCS-BEC crossover Eagles, Phys. Rev. 186 (1969) 456; Leggett, J. Phys. 41 (1980) C7-19 Nozi`eres and Schmitt-Rink, J. Low Temp. Phys. 59 (1985) 195 BCS : pairing in k-space BEC : pairing in x-space kz bound state kx Strongerattractive int. ky Bose-Einstein Condensationof bound bosons Condensation ofCooper pairs