Inhomogeneous Chiral Phase in Quark Matter

Recent development of inhomogeneous chiral phase in Kyoto group is briefly reviewed. First, the nesting effect of the Fermi surface is emphasized as a key mechanism leading to inhomogeneous chiral phase. After introduction of inhomogeneous chiral phase, some topological aspect is discussed in 1+1 dimensions and in the presence of the magnetic field in 1+3 dimensions. Spectral asymmetry gives rise to anomalous quark number and is closely related to chiral anomaly. It may induce spontaneous magnetization and a novel Lifshitz point may appear on the line $\mu=0$. Some astronomical implications are briefly discussed. Finally, the effects of the fluctuations around the order parameter are discussed; the propagator of the fluctuations exhibits a singularity at finite momentum and changes the properties of the phase transition through loop diagrams.