The localization spread and polarizability of rings and periodic chains

TL;DR

This paper explores how the localization spread criterion can be applied to finite ring systems and periodic chains, providing formulas for metallic versus insulating behavior and extending to polarizability.

Contribution

It demonstrates that the localization spread on finite rings converges to known formulas for infinite systems and introduces a new sum-over-states formula for polarizability.

Findings

Localization spread on finite rings matches infinite system formulas in the large R limit.

A new sum-over-states formula for polarizability is proposed.

The approach bridges finite and periodic boundary condition analyses.

Abstract

The localization spread gives a criterion to decide between metallic versus insulating behaviour of a material. It is defined as the second moment cumulant of the many-body position operator, divided by the number of electrons. Different operators are used for systems treated with Open or Periodic Boundary Conditions. In particular, in the case of periodic systems, we use the complex-position definition, that was already used in similar contexts for the treatment of both classical and quantum situations. In this study, we show that the localization spread evaluated on a finite ring system of radius $R$ with Open Boundary Conditions leads, in the large $R$ limit, to the same formula derived by Resta et al. for 1D systems with periodic Born-von K\'arm\'an boundary conditions. A second formula, alternative to the Resta's one, is also given, based on the sum-over-state formalism, allowing for an interesting generalization to polarizability and other similar quantities.