Properties of quark matter and the origin of the magnetic field in compact stars

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Using an effective theory of QCD, possible ferromagnetic transition is discussed in quark matter within the Fermi-liquid theory. We draw the magnetic phase diagram in the temperature-density plane. As a theoretical consequence, it is shown that the screening effect for the interaction produces a novel non-Fermi-liquid effect in the magnetic susceptibility. Hadron-quark transition is studied in compact stars to reveal the properties. Signatures of the phase transition are suggested in the spectra of the gravitational waves. Emergence of an inhomogeneous phase is discussed following the chiral transition and a new theoretical framework is proposed to properly take into account the symmetry breaking effect.