Evaluating the phase diagram at finite isospin and baryon chemical potentials in the Nambu–Jona-Lasinio model

Abstract

We study the phase diagram of two-flavor dense QCD at finite isospin and baryon chemical potentials in the framework of the Nambu--Jona-Lasinio model. We focus on the case with arbitrary isospin chemical potential ${\ensuremath{\mu}}_{\mathrm{I}}$ and small baryon chemical potential ${\ensuremath{\mu}}_{\mathrm{B}}\ensuremath{\le}{\ensuremath{\mu}}_{\mathrm{B}}^{\ensuremath{\chi}}$ where ${\ensuremath{\mu}}_{\mathrm{B}}^{\ensuremath{\chi}}$ is the critical chemical potential for the first-order chiral phase transition to happen at ${\ensuremath{\mu}}_{\mathrm{I}}=0$. The ${\ensuremath{\mu}}_{\mathrm{I}}\ensuremath{-}{\ensuremath{\mu}}_{\mathrm{B}}$ 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 ${\ensuremath{\mu}}_{\mathrm{I}}$ increases at ${\ensuremath{\mu}}_{\mathrm{B}}=0$, 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 $({\ensuremath{\mu}}_{\mathrm{I}},{\ensuremath{\mu}}_{\mathrm{B}})=({m}_{\ensuremath{\pi}},{M}_{\mathrm{N}}\ensuremath{-}1.5{m}_{\ensuremath{\pi}})$ where ${m}_{\ensuremath{\pi}}$, ${M}_{\mathrm{N}}$ are the vacuum masses of pions and nucleons, respectively. The first-order chiral phase transition becomes a smooth crossover when ${\ensuremath{\mu}}_{\mathrm{I}}>0.82{m}_{\ensuremath{\pi}}$. At very large isospin chemical potential, ${\ensuremath{\mu}}_{\mathrm{I}}>6.36{m}_{\ensuremath{\pi}}$, an inhomogeneous Larkin-Ovchinnikov-Fulde-Ferrell superfluid phase, appears in a window of ${\ensuremath{\mu}}_{\mathrm{B}}$, which should in principle exist for arbitrary large ${\ensuremath{\mu}}_{\mathrm{I}}$. Between the gapless and the Larkin-Ovchinnikov-Fulde-Ferrell 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 ${\ensuremath{\mu}}_{\mathrm{I}}$ 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 atom systems.