QCD phase diagram in a magnetic field with baryon and isospin chemical potentials

Based on the chiral perturbation theory at the leading order, we present the phase diagram of low-energy QCD in a magnetic field at finite baryon and isospin chemical potentials. The phase diagram consists of the QCD vacuum, the chiral soliton lattice, the uniform charged pion condensation, an Abrikosov vortex lattice of the charged pions, a baryonic vortex lattice composed of neutral and charged pion vortices with their topological linking number being the baryon number, and a hybrid phase of chiral soliton and vortex lattices, with their intersections carrying the baryon number. While the chiral soliton lattice demands ultra-strong magnetic field $\sim 10^{19}$ G,the intersection phase appears at $\sim10^{17}$ G, which is more realistic in neutron stars.