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)
1
Outline
Introduction: exotic pair states
 Theoretical work: NJL Model at finite
chemical potential
 Phase diagram at finite chemical potential
 Summary

2
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 I , the ground state is a superfluid pion
condensate, but at high I , it is a Fermi liquid with
Cooper pairing. They are connected by BCS-BEC
crossover.
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BCS-BEC 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 LarkinOvchinnikov-Fulde-Ferrell (LOFF) phase.
5

The Breached Paired phase (also called Sarma phase)
Superfluid component is breached by normal one in the
region k1  k  k2 where gapless excitations happen.
W.Yi and L.M. Duan: Phys. Rev. Lett. 97, 120401 (2006)
SF
BP
SF
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 2q .
M.Alford et. al. PRD63, 074016(2001)
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Two Flavor Nambu–Jona-Lasinio Model

The lagrangian density :

Chiral condensate :

Pion condensates :

u and d quark potential :

Partition function :
mf
8
The thermodynamic potential:
where

and
g ( x)  x / 2  T ln(1  e x /T )
The gap equations for m and  :
9

The explicit form of the gap equations:
BCS
BEC
10
Single-particle excitation gap:
 0

Sarma phase at finite isospin and quark potential
From the conditions:
We obtain the gapless points:
where
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Only in the case
, there is the possibility to
realize the Sarma phase. There are 3 types:
Only the branch
2
has two gapless nodes at
.
2
Only the branch 2 has one gapless node at
The branch 2 has one gapless node at
branch has one gapless node at
. This is in our case.
, the
4
13

The LOFF phase
14
where
15
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
 s
 2 s
 0,
0
2
q
q
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Phase Diagram:

Phase transition in the low isospin chemical potential
nB  0
17
Chiral phase transition
18
Superuid-Normal Phase Transition and Tricritical Point
  0
19

Gapless pion condensate and topological quantum phase transition
20
LOFF phase
The phase diagram at large isospin chemical potential
Normal
2nd
1st
SF
The LOFF window in the weak coupling case:
Our calculation indicates the LOFF window in the strong coupling region.
21
Quasiparticle dispersions in LOFF phase:
in the weak coupling case:
22
Quasiparticle dispersions in LOFF phase:
Three gapless nodes
Two blocking regions
23
Contour plot of the thermodynamic potential
24
 
25
VI
VI
VI
1st
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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
.
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
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 !
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