Smooth gauge and Wannier functions for topological band structures in arbitrary dimensions

TL;DR

This paper introduces a general method for constructing smooth Wannier functions in topological insulators with zero net Chern number, aiding in the analysis of topological properties and responses.

Contribution

It presents a new technique for obtaining smooth Bloch functions and Wannier functions in topological bands with zero net Chern number, even when symmetries are broken.

Findings

Verified against Kane-Mele model results

Applied to Bi2Se3 topological insulator

Facilitates calculation of magnetoelectric response

Abstract

The construction of exponentially localized Wannier functions for a set of bands requires a choice of Bloch-like functions that span the space of the bands in question, and are smooth and periodic functions of k in the entire Brillouin zone. For bands with nontrivial topology, such smooth Bloch functions can only be chosen such that they do not respect the symmetries that protect the topology. This symmetry breaking is a necessary, but not sufficient condition for smoothness, and, in general, finding smooth Bloch functions for topological bands is a complicated task. We present a generic technique for finding smooth Bloch functions and constructing exponentially localized Wannier functions in the presence of nontrivial topology, given that the net Chern number of the bands in question vanishes. The technique is verified against known results in the Kane-Mele model. It is then applied to the topological insulator Bi2Se3, where the topological state is protected by two symmetries: time reversal and inversion. The resultant exponentially localized Wannier functions break both these symmetries. Finally, we illustrate how the calculation of the Chern-Simons orbital magnetoelectric response is facilitated by the proposed smooth gauge construction.