QCD phase transition at finite isospin density and magnetic field

The QCD phase transition is explored at finite isospin density and magnetic field within the extended two-flavor Nambu--Jona-Lasinio model. By adopting the Ginzburg-Landau approximation, we study the transitions from normal chiral symmetry breaking phase to pion superfluidity or rho superconductivity. To avoid the artificial divergence for a large isospin chemical potential, we adopt the Landau representation rather than the proper-time one for the fermion propagators in a constant magnetic field. For the Landau representation, the same cutoff to the Landau energies, rather than to Landau levels, should be adopted to regularize the divergences from the summations over Landau levels. Then, the Ginzburg-Landau coefficients for pion and rho mesons are worked out both analytically and numerically in random phase approximation. The results show that pion superfluidity is favored for a small magnetic field while rho superconductivity is favored for a large magnetic field when increasing isospin chemical potential, in line with the magnetic enhancement (deduction) of the lowest energy of $\pi^+ ({\rho}^{+})$ meson. The novel rho superconductivity phase at large magnetic field and finite isospin density implies an interesting and nontrivial interplay between QCD and QED.