Smooth gauge for topological insulators

TL;DR

This paper introduces a method to construct smooth Bloch functions for 2D topological insulators, enabling better Wannier function construction by decomposing the occupied space into topologically trivial parts.

Contribution

The authors develop a robust technique to decompose the occupied subspace of 2D Z_2-insulators into trivial parts, facilitating the construction of smooth gauge functions.

Findings

Method successfully applied to the Kane-Mele model.

Decomposition remains robust regardless of symmetries or model specifics.

Enables construction of smooth Wannier functions for topological insulators.

Abstract

We develop a technique for constructing Bloch-like functions for 2D Z_2-insulators (i.e., quantum spin-Hall insulators) that are smooth functions of k on the entire Brillouin-zone torus. As the initial step, the occupied subspace of the insulator is decomposed into a direct sum of two "Chern bands," i.e., topologically nontrivial subspaces with opposite Chern numbers. This decomposition remains robust independent of underlying symmetries or specific model features. Starting with the Chern bands obtained in this way, we construct a topologically nontrivial unitary transformation that rotates the occupied subspace into a direct sum of topologically trivial subspaces, thus facilitating a Wannier construction. The procedure is validated and illustrated by applying it to the Kane-Mele model.