Optimally Localized Wannier Functions for 2D Chern Insulators

TL;DR

This paper introduces a method to construct optimally localized Wannier functions for 2D Chern insulators by regularizing divergence issues, providing analytical solutions and applications to electric polarization evaluation.

Contribution

It presents a novel regularization approach for Wannier functions in Chern insulators, enabling their optimal localization and analytical treatment.

Findings

Regularization of diverging variance in Wannier functions.

Analytical solutions for localized Wannier functions.

Application to electric polarization calculation.

Abstract

The construction of optimally localized Wannier functions (and Wannier functions in general) for a Chern insulator has been considered to be impossible owing to the fact that the second moment of such functions is generally infinite. In this manuscript, we propose a solution to this problem in the case of a single band. We accomplish this by drawing an analogy between the minimization of the variance and the minimization of the electrostatic energy of a periodic array of point charges in a smooth neutralizing background. In doing so, we obtain a natural regularization of the diverging variance and this leads to an analytical solution to the minimization problem. We demonstrate our results numerically for a particular model system. Furthermore, we show how the optimally localized Wannier functions provide a natural way of evaluating the electric polarization for a Chern insulator.