Localised Wannier functions in metallic systems

TL;DR

This paper proves that for metallic systems, a set of energy bands can be represented by a small number of Wannier functions with super-polynomial decay, highlighting differences from insulators.

Contribution

It establishes the existence of localized Wannier functions for metals with super-polynomial decay, extending the understanding beyond insulators.

Findings

N energy bands in metals can be represented by N+1 Wannier functions

Wannier functions decay faster than any polynomial

Spectral gap absence prevents exponential decay

Abstract

The existence and construction of exponentially localised Wannier functions for insulators is a well-studied problem. In comparison, the case of metallic systems has been much less explored, even though localised Wannier functions constitute an important and widely used tool for the numerical band interpolation of metallic condensed matter systems. In this paper we prove that, under generic conditions, $N$ energy bands of a metal can be exactly represented by $N+1$ Wannier functions decaying faster than any polynomial. We also show that, in general, the lack of a spectral gap does not allow for exponential decay.