QCD at finite isospin chemical potential µ I possesses a positively definite fermion determinant and the lattice simulation can be successfully performed. While the two-flavor effective models may be sufficient to describe the phenomenon of pion condensation, it is interesting to study the roles of the strangeness degree of freedom and the U A (1) anomaly. In this paper, we present a systematic study of the three-flavor Nambu-Jona-Lasinio model with a Kobayashi-Maskawa-'t Hooft (KMT) term that mimics the U A (1) anomaly at finite isospin chemical potential. In the mean-field approximation, the model predicts a phase transition from the vacuum to the pion superfluid phase, which takes place at µ I equal to the pion mass m π . Due to the U A (1) anomaly, the strangeness degree of freedom couples to the light quark degrees of freedom and the strange quark effective mass depends on the pion condensate. However, the strange quark condensate and the strange quark effective mass change slightly in the pion superfluid phase, which verifies the validity of the two-flavor models. The effective four-fermion interaction of the Kobayashi-Maskawa-'t Hooft term in the presence of the pion condensation is constructed. Due to the U A (1) anomaly, the pion condensation generally induces scalar-pseudoscalar interaction. The Bethe-Salpeter equation for the mesonic excitations is established and the meson mass spectra are obtained at finite isospin chemical potential and temperature. Finally, the general expression for the topological susceptibility χ at finite isospin chemical potential µ I is derived. In contrast to the finite temperature effect which suppresses χ, the isospin density effect leads to an enhancement of χ.

We study UA(1) symmetry and its relation to chiral symmetry at finite temperature through the application of functional renormalization group to the SU (3) quark meson model. Very different from the mass gap and mixing angel between η and η ′ mesons which are defined at mean field level and behavior like the chiral condensates, the topological susceptibility includes a fluctuations induced part which becomes dominant at high temperature. As a result, the UA(1) symmetry is still considerably broken in the chiral symmetry restoration phase.

We calculate the meson screening mass in a pion superfluid in the framework of Nambu-Jona-Lasinio model. The minimum of the attractive quark potential is always located at the phase boundary of pion superfluid. Different from the temperature and baryon density effect, the potential at finite isospin density can not be efficiently suppressed and the matter is always in a strongly coupled phase due to the Goldstone mode in the pion superfluid.