Chengfu Mu, Peking University Evaluating the Phase Diagram at finite Isospin and Baryon Chemical Potentials in NJL model Chengfu Mu, Peking University Collaborated with Lianyi He , J.W.Goethe University Prof. Yu-xin Liu, Peking University Ref: Phys. Rev. D 82, 056006 (2010)

Outline Introduction: exotic pair states Theoretical work: NJL Model at finite chemical potential Phase diagram at finite chemical potential Summary

Introduction QCD at finite density possesses a rich phase structure: 1. Cold quark matter forms color superconductivity at high baryon density, for example, CFL, 2SC et al. 2. Pion superfluid occurs when the isospin chemical potential is larger than pion mass in vacuum. At low , the ground state is a superfluid pion condensate, but at high , it is a Fermi liquid with Cooper pairing. They are connected by BCS-BEC crossover.

BCS-BEC crossover crossover Qijin Chen et. al.: Physics Reports 412, 1-88 (2005) crossover In the BCS and BEC regimes: only one excitation spectrum with a single component. For the crossover case the excitations have two distinct components.

In the standard BCS theory, the two fermionic species form a Cooper pair with same magnitude but opposite direction in momenta. But in real world, the two paired species have mismatched Fermi surface because of different chemical potential, unequal number densities or an external magnetic field et.al.. Clogston limit: upper limit of mismatch. What is the ground state in the asymmetric matter? This is a hot topic in both quark matter and condensed matter. Some candidates are proposed. We mainly discuss two of them: the Sarma phase and Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) phase.

The Breached Paired phase (also called Sarma phase) Superfluid component is breached by normal one in the region where gapless excitations happen. W.Yi and L.M. Duan: Phys. Rev. Lett. 97, 120401 (2006) SF BP k1 k2

LOFF state is a spatially anisotropic ground state where the rotational symmetry and/or the translational symmetry are spontaneously broken. Each Cooper pair carries a total momentum . M.Alford et. al. PRD63, 074016(2001)

Two Flavor Nambu–Jona-Lasinio Model The lagrangian density : Chiral condensate : Pion condensates : u and d quark potential : Partition function :

The gap equations for m and : The thermodynamic potential: where and The gap equations for m and :

The explicit form of the gap equations: BEC BCS

Single-particle excitation gap:

Sarma phase at finite isospin and quark potential From the conditions: We obtain the gapless points: where

Only in the case , there is the possibility to realize the Sarma phase Only in the case , there is the possibility to realize the Sarma phase. There are 3 types: Only the branch has two gapless nodes at . Only the branch has one gapless node at . This is in our case. The branch has one gapless node at , the branch has one gapless node at

The LOFF phase

where

The unphysical term from q does not affect the gap equations Fukushima and Iida, PRD76, 054004 (2007); J. O. Andersen and T. Brauner, PRD81, 096004 (2010). The unphysical term from q does not affect the gap equations for m and : The optimal value of q obtained via minimizing

Phase Diagram: Phase transition in the low isospin chemical potential

Chiral phase transition

Superuid-Normal Phase Transition and Tricritical Point

Gapless pion condensate and topological quantum phase transition

LOFF phase The phase diagram at large isospin chemical potential 1st 2nd Normal SF The LOFF window in the weak coupling case: Our calculation indicates the LOFF window in the strong coupling region.

Quasiparticle dispersions in LOFF phase: in the weak coupling case:

Quasiparticle dispersions in LOFF phase: Three gapless nodes Two blocking regions

Contour plot of the thermodynamic potential

1st VI VI VI

Summary In this work, we focus on the case with arbitrary isospin chemical potential and small baryon chemical potential . The phase diagram shows a rich phase structure since the system undergoes a crossover from a Bose-Einstein condensate of charged pions to a BCS superfluid with condensed quark–antiquark Cooper pairs when increases at , and a nonzero baryon chemical potential serves as a mismatch between the pairing species. We observe a gapless pion condensation phase near the quadruple point . The first order chiral phase transition becomes a smooth crossover when .

At very large isospin chemical potential , an inhomogeneous LOFF superfluid phase appears in a window of , which should in principle exist for arbitrary large . Between the gapless and the LOFF phases, the pion superfluid phase and the normal quark matter phase are connected by a first order phase transition. In the normal phase above the superfluid domain, we find that charged pions are still bound states even though becomes very large, which is quite different from that at finite temperature. Our phase diagram is in good agreement with that found in imbalanced cold atomic systems. Thank you !