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BCS-BEC crossover in relativistic superfluid
Yusuke Nishida (University of Tokyo) with Hiroaki Abuki (Yukawa Institute)
Seminar @ KEK 3 October, 2005
hep-ph/0504083
2/22
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…
3/22Idea 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
4/22
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
5/22Contents
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
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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
7/22Why Tc in BEC is constant ? Bound boson’s mass : density :
Tc of ideal BEC : independent of coupling
mB decreases as increasing the coupling
• 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)
Non-relativistic superfluids :
In relativistic system, binding effect on the Boson mass appears :
8/22Ideal 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
9/22Ideal 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
10/22BCS-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
11/22Our 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_
12/22Stable (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
13/22
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)
14/22Critical temperature vs. coupling
weak intermediate strong
BCS BEC RBEC
Superfluid phase
Tc
Chemical potential and densities on Tc line
Normal phase
15/22Weak coupling region
• Exponentially increasing Tc
• Mean field result is valid• ~ EF
• Fermion density is dominant
BCS phase
Mean field result
16/22Intermediate 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
17/22Strong 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
18/22Entropy vs. coupling
: total
: fermion
: boson
BCS > BEC << RBEC
FF_
Stotal = SF + SB
At T=Tc
19/22Dissociation 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
_
20/22Phase 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
21/22Summary2 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
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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
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Backup Slides
24/22Hints from lattice simulationsNakamura and Saito, PTP111 (2004) 733; PTP 112 (2004) 183
Q-Q potentialsQ-Q potentials_
25/22Size of pairingEngelbrecht et al., PRB 55 (1997) 15153
26/22Idea 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.
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