Origin of magnetic field in compact stars and magnetic properties of quark matter

Microscopic origin of the magnetic field observed in compact stars is studied in quark matter. Spontaneous spin polarization appears in high-density region due to the Fock exchange term. On the other hand, quark matter becomes unstable to form spin density wave in the moderate density region, where restoration of chiral symmetry plays an important role. Coexistence of the magnetism and color superconductivity is briefly discussed.