Topological insulators and topological non-linear sigma models

TL;DR

This paper establishes a theoretical link between topological insulators and nonlinear sigma models, revealing how topological terms and symmetries relate across different dimensions, with implications for fermion-{ ext} models and soliton quantum numbers.

Contribution

It introduces a novel correspondence between Chern-Simons insulators and topological nonlinear sigma models, enabling derivation of topological terms where traditional methods fail.

Findings

Correspondence between 2n-dimensional Chern-Simons insulators and (n-1)-dimensional sigma models.

Breaking symmetry in sigma models yields models with theta terms.

Demonstrates how to derive topological terms in fermion-{ ext} models.

Abstract

In this paper we link the physics of topological nonlinear {\sigma} 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 {\sigma} model with the Wess-Zumino-Witten term. Breaking internal symmetry in these nonlinear {\sigma} models leads to nonlinear {\sigma} models with the {\theta} 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-{\sigma} models") when the conventional gradient-expansion method fails. We also discuss the quantum number of solitons in topological nonlinear {\sigma} 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.