Optimally localized Wannier functions for quasi one-dimensional nonperiodic insulators

TL;DR

This paper proves that for general quasi one-dimensional insulators, the band position operator has discrete spectrum and yields an eigenbasis of optimally localized Wannier functions, confirming a longstanding conjecture.

Contribution

It establishes the existence of an eigenbasis of the band position operator with optimal localization for nonperiodic quasi one-dimensional insulators, confirming the strong conjecture of Marzari and Vanderbilt.

Findings

Band position operator has discrete spectrum in quasi 1D systems.

Eigenfunctions of the operator are as localized as the spectral projection.

Wannier basis inherits translation symmetries when present.

Abstract

It is proved that for general, not necessarily periodic quasi one dimensional systems, the band position operator corresponding to an isolated part of the energy spectrum has discrete spectrum and its eigenfunctions have the same spatial localization as the corresponding spectral projection. As a consequence, an eigenbasis of the band position operator provides a basis of optimally localized (generalized) Wannier functions for quasi one dimensional systems, thus proving the "strong conjecture" of Marzari and Vanderbilt. If the system has some translation symmetries (e.g. usual translations, screw transformations), they are "inherited" by the Wannier basis.