Fluctuation effect in relativistic Fluctuation effect in relativistic BCSBCS--BECBEC Crossover Crossover

Jian Deng, Department of Modern Physics, USTCJian Deng, Department of Modern Physics, USTC

2008, 7, 12 @ QCD workshop, Hefei 2008, 7, 12 @ QCD workshop, Hefei

Introduction Boson-fermion model for BCS-BEC crossover beyond MFA, fluctuation effect Discussions and outlooks

• J. Deng, A. Schmitt, Q. Wang, Phys.Rev.D76:034013,2007• J. Deng, J.-C. Wang, Q. Wang, arXiv:0803.4360 • J. Deng et al., in preparation

BCSBCS--BECBEC CrossoverCrossover

Relativistic Relativistic case Icase I

CCoolloorr superconductivity in superconductivity in neutron starsneutron stars

Possible Possible sstrong coupling trong coupling QQuark uark GGluon luon PPlasmalasma

In In RRelativistic elativistic HHeavy eavy IIon on CCollisionsollisions

Relativistic Relativistic case IIcase II

See e.g.

• Braun-Munzinger, Wambach, 2008 (review)

• Ruester,Werth,Buballa, Shovkovy,Rischke,2005

•Fukushima, Kouvaris, Rajagopal, 2005

•Blaschke, Fredriksson, Grigorian, Oztas, Sandin, 2005

Recent works by other group:

• Nishida & Abuki, PRD 2007 -- NJL approach • Abuki, NPA 2007 – Static and Dynamic

properties • Sun, He & Zhuang, PRD 2007 – NJL

approach • He & Zhuang, PRD 2007 – Beyond mean

field• Kitazawa, Rischke & Shovkovy,

arXiv:0709.2235v1 – NJL+phase diagram• Brauner, arXiv:0803.2422 – Collective

excitations

Relativistic BCS-BEC Relativistic BCS-BEC crossover for quark-crossover for quark-

quark pairingquark pairing

BosonBoson--fermionfermion model: model: setting upsetting up

Global Global U(1)U(1) symmetry: conserved symmetry: conserved currentcurrent

Phase diagram Phase diagram

Non-relativistic Non-relativistic Relativistic Relativistic

Shadowed region stand for unstable Shadowed region stand for unstable solutions, which will collapse to solutions, which will collapse to LOFF LOFF

statestate or or separating phaseseparating phaseWilfgang Ketterle (MIT) arXiv:0805.0623 Wilfgang Ketterle (MIT) arXiv:0805.0623 Realization of a strongly interacting Realization of a strongly interacting Bose-Bose-

Fermi mixtureFermi mixture from a two-component from a two-component Fermi gasFermi gas

BeyondBeyond MFAMFA

The fluctuation of The fluctuation of condensate sets in condensate sets in

Higgs Higgs andand Nambu-Nambu-

GoldstoneGoldstone fields:fields:

The The CJTCJT formalism formalism (J. M. (J. M. CCornwall, R. ornwall, R. JJackiw and E. ackiw and E. TTomboulis, 1974 )omboulis, 1974 )

Full Full propagator:propagator:Tree-level Tree-level

propagator:propagator:

2PI 2PI diagrams and diagrams and DSDS equations equations

Pseudo-Pseudo-gapgap

First order phase First order phase transition with transition with

fixed chemical potentialfixed chemical potential

Introduction of term in : Introduction of term in :

B.I.Halperin, T.C.Lubensky and S. Ma 1974B.I.Halperin, T.C.Lubensky and S. Ma 1974 (magnetic field fluctuations)(magnetic field fluctuations)I. Giannakis, D. f. Hou, H. c. Ren and D. H. Rischke, 2004I. Giannakis, D. f. Hou, H. c. Ren and D. H. Rischke, 2004 ((Gauge Field Fluctuations)Gauge Field Fluctuations)Sasaki, Friman, Redlich, 2007Sasaki, Friman, Redlich, 2007 (baryon number fluctuation in 1st chiral phase transition)(baryon number fluctuation in 1st chiral phase transition)

gap and density gap and density equationsequations

At small TAt small T

The results are similar to the The results are similar to the MFAMFA results results

At T=TcAt T=Tc

Fluctuations become important in Fluctuations become important in BECBEC regime.regime.In In BECBEC regime T*>Tc. regime T*>Tc.

T-dependenceT-dependence

The fluctuation effects become larger.The fluctuation effects become larger.BEC criterion is related to the minimization BEC criterion is related to the minimization of the thermodynamics potential.of the thermodynamics potential.

Summary Summary

1. Relativistic boson-fermion model can well 1. Relativistic boson-fermion model can well describe the describe the BCSBCS- - BECBEC crossover within or crossover within or beyond beyond MFAMFA..

2.As an fluctuation effect, the pseudo-gap become more important for larger temperature.

3.3.FluctuationFluctuation changes the phase transition to be changes the phase transition to be first-orderfirst-order..

Outlook Outlook Full self-consistencyFull self-consistency is needed. is needed.

BEC criterionBEC criterion for interacting bosons need more for interacting bosons need more close look.close look.

Anti-particlesAnti-particles and and finite size of bosonsfinite size of bosons should should be be considered carefullyconsidered carefully

Our model can be extended to discuss Our model can be extended to discuss quarkoynic quarkoynic continuity with finite chemical continuity with finite chemical potential where the confinement and chiral potential where the confinement and chiral symmetry breaking are not coincide (L. symmetry breaking are not coincide (L. Mclerran and R. D. Pisarski ).Mclerran and R. D. Pisarski ).

Thanks a Thanks a lotlot

BEC: BEC: condensate, number density condensate, number density conservation,conservation,

critical temperaturecritical temperature

Distribution Distribution function:function:

Density Density conservation:conservation:

Thermal bosons at Thermal bosons at most:most:

Temperature Temperature dependence:dependence:

BosonBoson--fermionfermion model model ((MFAMFA))With With bosonicbosonic and and fermionicfermionic degrees of degrees of

freedom and their coupling, but neglect freedom and their coupling, but neglect the coupling of thermal bosons and the coupling of thermal bosons and

fermions as fermions as MMean ean FField ield AApproximation pproximation

Pairing with imbalance Pairing with imbalance population population

Fermi surface Fermi surface topologies topologies

ApproximatioApproximationn

Ensure the reliability of gap Ensure the reliability of gap equationequation

Continuous changing of Continuous changing of gap with fixed number gap with fixed number

densitydensity

But still But still first-orderfirst-order phase phase transitiontransition