Chern-Simons term at finite density and temperature

The Chern-Simons topological term dynamical generation in the effective action is obtained at arbitrary finite density and temperature. By using a few different approaches it is shown that at zero temperature μ2=m2 is the crucial point for Chern-Simons. So when μ2 < m2, μ-influence disappears and we get the usual Chern-Simons term. On the other hand, when μ2 > m2, the Chern-Simons term vanishes because of non-zero density of back-ground fermions. In particular, for massless case parity anomaly is absent at any finite density or temperature. This results holds in any odd dimension as in abelian so as in nonabelian case