Bloch bundles, Marzari-Vanderbilt functional and maximally localized Wannier functions

TL;DR

This paper investigates the properties of maximally localized Wannier functions derived from Bloch bands of periodic Schrödinger operators, proving existence and exponential localization of minimizers in dimensions less than four.

Contribution

It establishes the existence and exponential localization of minimizers of the Marzari-Vanderbilt functional for Wannier functions in low dimensions, using harmonic map techniques.

Findings

Existence of minimizers for the localization functional in dimensions d<4.

Exponential decay of Wannier functions at the minimizers.

Application of harmonic map theory to quantum localization problems.

Abstract

We consider a periodic Schroedinger operator and the composite Wannier functions corresponding to a relevant family of its Bloch bands, separated by a gap from the rest of the spectrum. We study the associated localization functional introduced by Marzari and Vanderbilt, and we prove some results about the existence and exponential localization of its minimizers, in dimension d < 4. The proof exploits ideas and methods from the theory of harmonic maps between Riemannian manifolds.