A Spatial Localizer for Electrons in Insulators

TL;DR

This paper introduces a general framework called Spatial Localizers for determining electron locations in higher-dimensional insulators, extending Wannier functions to complex systems with boundaries and disorder.

Contribution

The authors develop a spectral operator-based method to find electron positions in 2D and 3D insulators, generalizing Wannier centers to complex scenarios.

Findings

Framework extends Wannier functions to insulators with boundaries and disorder.

Maximally localized states are derived, matching Wannier functions in atomic insulators.

In Chern insulators, states resemble Landau level coherent states.

Abstract

The location of electrons governs phenomena ranging from chemical bonding and electric polarization to the topological classification of band insulators and the emergence of correlated states in quantum matter. While a prescription exists for finding local state representations of electrons in one-dimensional insulators, no comparably general theory exists in higher dimensions. Here, we introduce a general framework for finding the location of electrons in insulators in two and three dimensions based on the spectral properties of quantum-mechanical operators that we term Spatial Localizers. This framework naturally extends the notion of Wannier centers to insulators with boundaries, defects, and disorder, which we use to establish a position-space formulation of the bulk-defect correspondence for electronic charge. This framework also yields maximally localized electronic states. As two representative examples, we show that these states reduce to maximally localized Wannier functions in atomic insulators, whereas in Chern insulators they form coherent states that mirror the coherent-state structure of Landau levels in the quantum Hall effect.