Chern-Simons term at finite density

Abstract The Chern-Simons topological term coefficient is derived at arbitrary finite density. As it happens μ 2 = m 2 is the crucial point for Chern-Simons. So when μ 2 m 2 the μ-influence disappears and we get the usual Chern-Simons term. On the other hand when μ 2 > m 2 the Chern-Simons term vanishes because of the non-zero density of background fermions. In particular for the massless case the parity anomaly is absent at any finite density. This result holds in any odd dimension both in the abelian and in the nonabelian case.