1/23

BCS-BEC crossover in relativistic superfluid

Yusuke Nishida (University of Tokyo) with Hiroaki Abuki (Yukawa Institute)

Seminar @ ECT* 19 May, 2005

hep-ph/0504083

2/23

weak strong

Interaction is arbitrarily tunable using Feshbach resonance

New fermion superfluid in 40K, 6LiRegal et al., PRL 92 (2004) 040403

Bartenstein et al., PRL 92 (2004) 120401Zwierlein et al., PRL 92 (2004) 120403

• Weak coupling : BCS by Cooper pairs

• Strong coupling : BEC by molecules

Kinast et al., PRL 92 (2004) 150402Bourdel et al., cond-mat/0403091

and more…

3/23Idea of BCS-BEC crossoverEagles, 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

kx

ky

kz

BEC : pairing in x-space

bound state

Condensation ofCooper pairs

Bose-Einstein Condensationof bound bosons

Strongerattractive int.

4/23Idea of BCS-BEC crossoverEagles, 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 ofCooper pairs

Bose-Einstein Condensationof bound bosons

Strongerattractive int.

BCS : large size pairing BEC : small size pairing

d

/d~104-5

5/23

T

BCS-BEC crossover in QCD ?

Perturbative QCD at high density => BCS instability in color 3, flavor 1

and JP=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) 074014

6/23Contents

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 Tc with fixed density

S’a de Melo et al., PRL 71 (1993) 3202

Bose gas BEC behavior

BCS behavior Fermi gas

Tc

8/23Why Tc in BEC is constant ? Bound boson’s mass : density :

Tc of ideal BEC : independent of coupling

mB decreases as increasing the coupling

• trapped fermionic alkali atoms (Ohashi and Griffin, PRL (’02))• liquid 3He (Leggett, J. Phys. 41 (1980) C7-19)• nuclear matter (Lombardo et al., PRC 64 (2001) 064314)

Non-relativistic superfluids :

In relativistic system, binding effect on the Boson mass appears :

9/23Ideal BEC in relativistic systemKapsta, Finite Temperature Field Theory (Cambridge, 1989)

boson density anti-boson density

Non-relativistic limit(mB

3>>NB)Relativistic limit

(mB3<<NB)

Anti-boson density is negligible Anti-boson density appears

At T=Tc of ideal BEC => B = mB

usual BEC state Relativistic BEC

10/23Ideal BEC in relativistic system

BEC RBEC

Tc

boson density

anti-boson density

TNR

TRL

“Crossover” from BEC to RBECas decreasing the boson mass

LARGE mB small mB

11/23BCS-BEC in relativistic system ?

BEC RBECBECBCS

2 crossovers : BCS -> BEC and BEC -> RBEC as increasing the coupling

G

What we expect is …

Tc

TNR

TRL

Tc

Our analysis

12/23Our 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

NF + NF : (anti-)fermion density_

13/23Stable (anti-)boson density

NB : 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 Ntotal

Numerical calculations

Critical temperature Tc by Thouless criterion

Pair fluctuation diverges :

pair susceptibility

Tc and as functions of coupling Gwith fixed number density

Parameter set : scaled by ultraviolet cutoff

(2 = mB on T=Tc)

15/23Critical temperature vs. coupling

weak intermediate strong

BCS BEC RBEC

Superfluid phase

Tc

Chemical potential and densities on Tc line

Normal phase

16/23Weak coupling region

• Exponentially increasing Tc

• Mean field result is valid• ~ EF

• Fermion density is dominant

BCS phase

Mean field result

17/23Intermediate coupling BEC phase

• Slowly increasing Tc

• Well described by BEC • < m• Bound boson density is dominant

BEC TNR with mB=2

TNR with mB=2m

18/23Strong coupling Relativistic BEC

• Very large Tc ~ (NF)1/3

• Small interparticle distance• ~ 0• Anti-particles are available

1/T ~ (NF)-1/3

BEC TRL with mB=2

19/23Entropy vs. coupling

: total

: fermion

: boson

BCS > BEC << RBEC

FF_

Stotal = SF + SB

At T=Tc

20/23Dissociation in (R)BEC phases

TcTdiss

Superfluid phase

Bound bosons melt at T = Tdiss > Tc

: T=Tc

: Tc<T<Tdiss

: T=Tdiss

Bound anti-boson poles

2m-2

Bound boson poles

• Normal phase without stable bosons (T>Tdiss)

• Preformed boson phase(Tc<T<Tdiss)

(G/G0=1)

Cf. q-q bound state above Tc by Asakawa and Hatsuda

_

21/23Phase diagram in m-G plane

RBECBCS

BEC

Crossover regions• BCS -> BEC : ~ m• BEC -> RBEC : ~ 0

Approximated by points where the stable bosonis formed / becomesmassless in the vacuum

Half of boson mass in the vacuum (T==0)

mB/ 2

22/23Summary2 crossovers as increasing the coupling

Tc exponentially↑ slowly↑ very large

~ EF < m ~ 0

Nfermions are

dominantbound bosons are dominant

anti-particlesare available

Dissociation ofbound bosons above Tc

= preformed boson phase(Tc<T<Tdiss)

Cf. pseudogap phase discussed by Kitazawa et al.,

PRD 70 (2004) 056003

Fermi gas Bose gas

BCS phase BEC phase RBEC phase

Tc

Tdiss

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 Tc) is realized when large g & small • Future work : competition b/w (R)BEC and -phase

: realistic treatment of plasmino mass

Preformedboson phase

Cf. diquark bound state above Tc

by Shuryak and Zahed

BCS

24/23

Backup Slides

25/23Hints from lattice simulationsNakamura and Saito, PTP111 (2004) 733; PTP 112 (2004) 183

Q-Q potentialsQ-Q potentials_

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