Topological insulators and topological non-linear sigma models

Abstract

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.

18 pages, 11 figures (figure problem fixed)

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• Skyrmions with quadratic band touching fermions: A way to achieve charge 4e superconductivity
• Geometric phases and competing orders in two dimensions
• Stripe melting, a transition between weak and strong symmetry protected topological phases
• Zero modes and charged Skyrmions in graphene bilayer
• The landscape of the Hubbard model
• Unified Topological Response Theory for Gapped and Gapless Free Fermions
• Competing orders in the Dirac-like electronic structure and the non-linear sigma model with the topological terms
• Deconfined criticality in bilayer graphene
• Topological defects of Néel order and Kondo singlet formation for Kondo-Heisenberg model on a honeycomb lattice
• Origin of Topological Order in a Cooper Pair Insulator
• Unidirectional magneto-transport of linearly dispersing topological edge states
• Mechanism of skyrmion condensation and pairing for twisted bi-layer graphene