Topological insulators and topological nonlinear σ models

In this paper we link the physics of topological nonlinear σ models with that of Chern-Simons insulators. We show that corresponding to every 2n-dimensional Chern-Simons insulator there is a (n-1)-dimensional topological nonlinear σ model with the Wess-Zumino-Witten term. Breaking internal symmetry in these nonlinear σ models leads to nonlinear σ models with the θ term. [This is analogous to the dimension reduction leading from 2n-dimensional Chern-Simons insulators to (2n-1) and (2n-2)-dimensional topological insulators protected by discrete symmetries.] The correspondence described in this paper allows one to derive the topological term in a theory involving fermions and order parameters (we shall referred to them as ''fermion-σ models'') when the conventional gradient-expansion method fails. We also discuss the quantum number of solitons in topological nonlinear σ model and the electromagnetic action of the (2n-1)-dimensional topological insulators. Throughout the paper we use a simple model to illustrate how things work.