1/23 BCS-BEC crossover in relativistic superfluid Yusuke Nishida (University of Tokyo) with Hiroaki Abuki (Yukawa Institute) ECT*19 May, 2005 hep-ph/

2/23 weakstrong Interaction is arbitrarily tunable using Feshbach resonance New fermion superfluid in 40 K, 6 Li Regal et al., PRL 92 (2004) Bartenstein et al., PRL 92 (2004) Zwierlein et al., PRL 92 (2004) Weak coupling : BCS by Cooper pairs Strong coupling : BEC by molecules Kinast et al., PRL 92 (2004) Bourdel et al., cond-mat/ and more…

3/23 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 kxkx kyky kzkz BEC : pairing in x-space bound state Condensation of Cooper pairs Bose-Einstein Condensation of bound bosons Stronger attractive int.

4/23 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 bound state Condensation of Cooper pairs Bose-Einstein Condensation of bound bosons Stronger attractive int. BCS : large size pairingBEC : small size pairing  d  /d~10 4-5

5/23 T  BCS-BEC crossover in QCD ? Perturbative QCD at high density => BCS instability in color 3, flavor 1 and J P =0 + diquark channel Possible realization of BEC in low or intermediate density region of CSC QGP Color superconductivity ? Hadron phase larger g Abuki, Hatsuda and Itakura, PRD 65 (2002)

6/23 Contents 1. Introduction 2. BCS-BEC in non-relativistic system 3. BEC in relativistic system 4. Crossovers in relativistic system BCS-BEC-RBEC phases Phase diagram 5. Summary and implication for QCD

7/23 strength of attraction: Result in non-relativistic system Critical temperature T c with fixed density S’a de Melo et al., PRL 71 (1993) 3202 Bose gas BEC behavior BCS behavior Fermi gas TcTc

8/23 Why T c in BEC is constant ? Bound boson’s mass : density : T c of ideal BEC : independent of coupling m B decreases as increasing the coupling trapped fermionic alkali atoms (Ohashi and Griffin, PRL (’02)) liquid 3 He (Leggett, J. Phys. 41 (1980) C7-19) nuclear matter (Lombardo et al., PRC 64 (2001) ) Non-relativistic superfluids : In relativistic system, binding effect on the Boson mass appears :

9/23 Ideal BEC in relativistic system Kapsta, Finite Temperature Field Theory (Cambridge, 1989) boson densityanti-boson density Non-relativistic limit (m B 3 >>N B ) Relativistic limit (m B 3 <<N B ) Anti-boson density is negligible Anti-boson density appears At T=T c of ideal BEC =>   B = m B usual BEC state Relativistic BEC

10/23 Ideal BEC in relativistic system BEC RBEC TcTc boson density anti-boson density T NR T RL “Crossover” from BEC to RBEC as decreasing the boson mass LARGE m B small m B

11/23 BCS-BEC in relativistic system ? BECRBEC BECBCS 2 crossovers : BCS - > BEC and BEC - > RBEC as increasing the coupling G What we expect is … TcTc T NR T RL TcTc Our analysis

12/23 Our formulation hep-ph/ body contact interaction with massive fermion fermion mass fermion chemical potential Attraction in J P =0 + channel Fermion pair correlation in normal phase Nozi`eres and Schmitt-Rink (’85); S’a de Melo et al. (’93) Fermion number density N total N B : pair correlation : phase shift N F + N F : (anti-)fermion density _

13/23 Stable (anti-)boson density N B : bosonic contribution to the density If attraction G is strong enough, bound state poles appear in Bound boson density Bound anti-boson density Unstable boson density internal structure of boson

14/23 Fixed number density N total Numerical calculations Critical temperature T c by Thouless criterion Pair fluctuation diverges : pair susceptibility T c and  as functions of coupling G with fixed number density Parameter set : scaled by ultraviolet cutoff  (2  = m B on T=T c )

15/23 Critical temperature vs. coupling weak intermediatestrong BCS BECRBEC Superfluid phase TcTc Chemical potential and densities on T c line Normal phase

16/23 Weak coupling region Exponentially increasing T c Mean field result is valid  ~ E F Fermion density is dominant BCS phase Mean field result

17/23 Intermediate coupling BEC phase Slowly increasing T c Well described by BEC  < m Bound boson density is dominant BEC T NR with m B =2  T NR with m B =2m

18/23 Strong coupling Relativistic BEC Very large T c ~ (N F ) 1/3 Small interparticle distance  ~ 0 Anti-particles are available 1/T ~ (N F ) -1/3 BEC T RL with m B =2 

19/23 Entropy vs. coupling : total : fermion : boson BCS > BEC << RBEC F F _ S total = S F + S B At T=T c

20/23 Dissociation in (R)BEC phases TcTc T diss Superfluid phase Bound bosons melt at T = T diss > T c : T=T c : T c <T<T diss : T=T diss Bound anti-boson poles 2m-2  Bound boson poles Normal phase without stable bosons (T>T diss ) Preformed boson phase (T c <T<T diss ) (G/G 0 =1) Cf. q-q bound state above T c by Asakawa and Hatsuda _

21/23 Phase diagram in m-G plane RBEC BCS BEC Crossover regions BCS - > BEC :  ~ m BEC - > RBEC :  ~ 0 Approximated by points where the stable boson is formed / becomes massless in the vacuum Half of boson mass in the vacuum (T=   =0) mB/ 2mB/ 2

22/23 Summary 2 crossovers as increasing the coupling TcTc exponentially↑slowly↑very large  ~ EF~ EF < m< m~ 0~ 0 N fermions are dominant bound bosons are dominant anti-particles are available Dissociation of bound bosons above T c = preformed boson phase (T c <T<T diss ) Cf. pseudogap phase discussed by Kitazawa et al., PRD 70 (2004) Fermi gasBose gas BCS phaseBEC phase RBEC phase TcTc T diss Superfluid phase

23/23 T  Significance of (R)BEC in QCD BEC criterion :  < m Hard dense loop gives fermion mass Probable realization of BEC in non-perturbative region of CSC BEC criterion : RBEC BEC (g>1) Speculative QCD phase diagram RBEC (large T c ) is realized when large g & small  Future work : competition b/w (R)BEC and  -phase : realistic treatment of plasmino mass Preformed boson phase Cf. diquark bound state above T c by Shuryak and Zahed BCS

24/23 Backup Slides

25/23 Hints from lattice simulations Nakamura and Saito, PTP111 (2004) 733; PTP 112 (2004) 183 Q-Q potentials _

26/23 Size of pairing Engelbrecht et al., PRB 55 (1997) 15153