Magnetic field effect on the pion superfluid

Magnetic field effect on pion superfluid phase transition is investigated in frame of a Pauli-Villars regularized NJL model. Instead of directly dealing with charged pion condensate, we apply the Goldstone's theorem (massless Goldstone boson $\pi^+$) to determine the onset of pion superfluid phase, and obtain the phase diagram in magnetic field, temperature, isospin and baryon chemical potential space. At weak magnetic field, it is analytically proved that the critical isospin chemical potential of pion superfluid phase transition is equal to the mass of $\pi^+$ meson in magnetic field. The pion superfluid phase is retarded to higher isospin chemical potential, and can survive at higher temperature and higher baryon chemical potential under external magnetic field.