Quadrupole moments, edge polarizations, and corner charges in the Wannier representation

TL;DR

This paper develops a Wannier-based formalism to predict corner charges in 2D insulators with inversion symmetry, revealing gauge dependencies and proposing methods for consistent calculations, validated on tight-binding models.

Contribution

It introduces a gauge-consistent Wannier approach to determine corner charges from bulk and edge properties in 2D insulators with inversion symmetry.

Findings

Edge polarization is gauge-dependent and not a physical observable.

Sum of quadrupole and edge dipole contributions is gauge-independent.

The formalism accurately predicts corner charges in several tight-binding models.

Abstract

The modern theory of polarization allows for the determination of the macroscopic end charge of a truncated one-dimensional insulator, modulo the charge quantum $e$, from a knowledge of bulk properties alone. A more subtle problem is the determination of the corner charge of a two-dimensional insulator, modulo $e$, from a knowledge of bulk and edge properties alone. While previous works have tended to focus on the quantization of corner charge in the presence of symmetries, here we focus on the case that the only bulk symmetry is inversion, so that the corner charge can take arbitrary values. We develop a Wannier-based formalism that allows the corner charge to be predicted, modulo $e$, only from calculations on ribbon geometries of two different orientations. We elucidate the dependence of the interior quadrupole and edge dipole contributions upon the gauge used to construct the Wannier functions, finding that while these are individually gauge-dependent, their sum is gauge-independent. From this we conclude that the edge polarization is not by itself a physical observable, and that any Wannier-based method for computing the corner charge requires the use of a common gauge throughout the calculation. We satisfy this constraint using two Wannier construction procedures, one based on projection and another based on a gauge-consistent nested Wannier construction. We validate our theory by demonstrating the correct prediction of corner charge for several tight-binding models. We comment on the relations between our approach and previous ones that have appeared in the literature.