Pion Condensation in a two-flavor NJL model: the role of charge neutrality

We study pion condensation and the phase structure in a two-flavor Nambu-Jona-Lasinio model in the presence of baryon chemical potential $\mu$ and isospin chemical potential $\mu_I$at zero and finite temperature. There is a competition between the chiral condensate and a Bose-Einstein condensate of charged pions. In the chiral limit, the chiral condensate vanishes for any finite value of the isospin chemical potential, while there is a charged pion condensate that depends on the chemical potentials and the temperature. At the physical point, the chiral condensate is always nonzero, while the charged pion condensate depends on $\mu_I$ and $T$. For $T=\mu=0$, the critical isospin chemical potential $\mu_I^c$ for the onset of Bose-Einstein condensation is always equal to the pion mass. For $\mu=0$, we compare our results with chiral perturbation theory, sigma-model calculations, and lattice simulations. We examine the effects of imposing electric charge neutrality and weak equilibrium on the phase structure of the model. In the chiral limit, there is a window of baryon chemical potential and temperature where the charged pions condense. At the physical point, the charged pions do not condense.