1. Lianyi He and Pengfei Zhuang Physics Department, Tsinghua U.
BCS-BEC Crossover in Relativistic System and Its Possible Realization in QCD
Lianyi He and Pengfei Zhuang
Physics Department, Tsinghua U.
References: Phys.Rev.D75:096003,2007
Phys.Rev.D75:096004,2007
Phys.Rev.D76:056003,2007
E-print Arxiv:
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2. Impact of Superconductivity
Phenomena
Order parameter
Pair Breaking effect
Metallic superconductivity
Zeeman Splitting
Superfluidity in Ultracold Fermi atoms
Population imbalance
Chiral symmetry
breaking in QCD
Baryon chemical potential
Color superconductivity
in dense quark matter
Isospin chemical potential
Strange quark mass
flavor superfluidity
in isospin matter
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3. Theoretical History of Crossover:
Introduction to BCS-BEC Crossover
Theoretical History of Crossover:
Leggett (1980) noted that BCS T=0 wavefunction could be
generalized to arbitrary attraction: a smooth BCS-BEC crossover !
Let us consider the history of theoretical studies of BCS-BEC crossover. Of course, the first theory is a mean-field theory. Here is the phase diagram at zero temperature. As the attractive interaction U is changed from a small negative value to a large negative value, we go from the BCS regime to the BEC regime. For the BCS regime, the BCS wave-function is valid. While for the BEC regime, we have the BEC variational wave-function. The important discovery by Eagles and Leggett is that these two wave-functions actually are connected. For example, in the BEC limit, the vk in the BCS wave-function becomes very small, we can rewrite it into this form. Therefore, the BCS wave-function can be transformed into the BEC wave-function. This observation suggests that from BCS to BEC it is not an abrupt phase transition, rather, we have a smooth crossover. The mean-field theory has been generalized by Holland and Levin to the cold atomic gases.
strong coupling
weak coupling
BEC
BCS
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4. Eagles (1969): Pairing without superconductivity
So, what is the BCS-BEC crossover? Consider a two-component Fermi gas with attractive interactions. When the interaction is weak, we have the BCS fermionic superfluidity of Cooper-pairs, and we know that these Cooper-pairs are long-range objects. As the interaction increases, the size of Cooper-pairs becomes smaller and smaller, and finally can be regarded as a point-like structure, or molecules. We then can have a condensate of molecules. These two limits, BCS and BEC limits, are actually well-understood, however, the regime between these limits, is not so clear, and is to explored.
To access the crossover, of course, the interaction strength should be enhanced enough large.
In HTSC, the root of PG is not
quite clear. The pre-formed boson
mechanism is only one explanation.
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5. Experimental evidence for pseudogap
in ultracold fermions
Science 305, 1128 (2004)
Science 307, 1296 (2005)
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6. m T BEC? Relativistic BCS-BEC Crossover Motivation:
1) QCD phase diagram at moderate temperature and
density, strong coupling nature!
2) BCS-BEC crossover in CSC? Pseudogap of CSC
3) NJL model: BCS theory of chiral symmetry breaking.
Chiral pesudogap above Tc? T m
Quark Spectrum and pseudogap, M. Kitazawa, et.al. 2005
SQGP?
Structure change of Cooper pairs,
H.Abuki, et.al.2002
pseudogap? SB
CSC
BEC?
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7. Progress in this field of research
H.Abuki and Y.Nishida: NSR approach,
new crossover: BCS-BEC-RBEC
L.He and P.Zhuang: BCS-Leggett theory at zero T
and pesudogap theory at finite T
BCS-BEC vs. Chiral Restoration
J.Deng and Q.Wang: Boson-fermion model
in mean field and beyond mean field
M.Kitazawa, D.Rischke and I.Shovkovy:
Possible Diquark BEC in NJL
T.Brauner: Collective Excitation
1/N expansion including quantum fluctuation
(with H.Abuki)

8. Zero Temperature: BCS-Leggett mean Field Theory
Model Lagrangian
Mean Field Equations
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9. Fermionic Excitation Spectrum
anti-fermion
Non-relativistic limit
Criterion for BCS and BEC: Minimum of the dispersion
BCS: at , existence of Fermi surface
BEC: at , molecule condensation
The role of fermion mass (new aspect in QCD)
BCS-BEC Chiral symmetry breaking& restoration
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10. The Non-Relativistic Limit
For small value of the parameter , the NR version of
BCS-Leggett result is recovered
The condition or is needed
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11. Relativistic Effect I: BCS-NBEC-RBEC Crossover
Physical picture:
RBEC: , anti-particles available!
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12. Relativistic Effect II: Density Induced Crossover
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13. Collective Mode Evolution
Effective action
Dispersion of the collective mode
Goldstone mode
A-B mode
BCS
NBEC
Bogoliubov theory
RBEC
Ultra-relativistic BEC
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14. K.Levin and Qijin Chen, et.al,
Nonzero Temperature:
Pseudogap theory
K.Levin and Qijin Chen, et.al,
Phys.Rept. 412, 1(2005)
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15. Tc Calculation NBEC regime RBEC regime Bound states above Tc
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16. BCS-BEC Crossover in QCD?
Two color QCD
QCD at finite isospin density
In the BEC region, quark energy gap is different from
the order parameter! Can be confirmed in Lattice!
Phys.Rev.D75:096004,2007
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17. Chiral restoration vs. BCS-BEC crossover
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18. Condensation and Chiral Phase Transition
Pseudogap in QCD?
Condensation and Chiral Phase Transition
NJL Model
Nc dependence
Quark mass is still large above
chiral phase transition!
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19. Diquark condensation and color superconductivity
NJL model, 2SC phase
Quark pesudogap remarkable!
A high Tc superconductor!
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20. Non-Fermi liquid behavior
at low T!
(High T, see M.Kitazawa, 2005)

21. 1. BCS-BEC crossover theory has been generalized
Summary
1. BCS-BEC crossover theory has been generalized
to relativistic system, both at zero and at finite
temperature.
2. Proper NR limit is recovered and new relativistic
effects are found.
3. Possible BCS-BEC crossover in diquark and
isospin matter is discussed.
4. The size of quark pseudogap in chiral and CSC
phase transition is calculated.
Outlook
Determine the pseudogap phase in QCD phase diagram
Signals of diquark fluctuation and pesudogap in HIC
Effect of the pseudogap on compact star cooling
More precise theory, including quantum fluctuation
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22. Thank You!
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