The localization spread and polarizability of rings and periodic chains

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.