Topological Aspects of the Imhomogeneous Chiral Phases and Implication on Compact Stars

Recent studies of our group are reviewed about the inhomogeneous chiral phase. Appearance of the inhomogeneous chiral phase is a new paradigm in the field of the QCD phase diagram. We consider the dual-chiral-density-wave in the presence of the magnetic field. Some topological effect manifests in the form of spectral asymmetry of the quark single-particle energy.