Wannier representation of Z_2 topological insulators

TL;DR

This paper investigates the construction of Wannier functions in two-dimensional Z_2 topological insulators, revealing a topological obstruction under time-reversal symmetry and providing a modified approach to obtain Wannier representations.

Contribution

It demonstrates that Wannier functions can be constructed for Z_2 topological insulators if the time-reversal symmetry gauge condition is relaxed, and provides an explicit construction method.

Findings

Wannier functions exist for Z_2-even phases using standard methods.

A topological obstruction exists for Z_2-odd phases under time-reversal symmetric gauges.

Modified projection scheme successfully constructs Wannier functions for Z_2-odd phases.

Abstract

We consider the problem of constructing Wannier functions for Z_2 topological insulators in two dimensions. It is well known that there is a topological obstruction to the construction of Wannier functions for Chern insulators, but it has been unclear whether this is also true for the Z_2 case. We consider the Kane-Mele tight-binding model, which exhibits both normal (Z_2-even) and topological (Z_2-odd) phases as a function of the model parameters. In the Z_2-even phase, the usual projection-based scheme can be used to build the Wannier representation. In the Z_2-odd phase, we do find a topological obstruction, but only if one insists on choosing a gauge that respects the time-reversal symmetry, corresponding to Wannier functions that come in time-reversal pairs. If instead we are willing to violate this gauge condition, a Wannier representation becomes possible. We present an explicit construction of Wannier functions for the Z_2-odd phase of the Kane-Mele model via a modified projection scheme followed by maximal localization, and confirm that these Wannier functions correctly represent the electric polarization and other electronic properties of the insulator.