Localization Tensor Revisited: Geometric-Probabilistic Foundations and a Structure-Factor Criterion under Periodic Boundaries

TL;DR

This paper revisits the localization tensor from geometric and probabilistic perspectives, extending it to periodic boundary conditions, and introduces a structure-factor-based criterion to distinguish localization from dimerization.

Contribution

It develops new PBC-compatible extensions of the localization tensor and relates it to the structure factor, providing a sharp criterion to differentiate localization from dimerization.

Findings

Localization tensor can be expressed as a covariance of a probability distribution.

Two PBC-compatible extensions of the localization tensor are proposed: geometric and mutual information based.

A structure-factor-based criterion effectively distinguishes localized phases from dimerized phases.

Abstract

We revisit the localization tensor (LT) from geometric and probabilistic perspectives and construct extensions that are naturally compatible with periodic boundary conditions (PBC), without redefining the position operator. In open boundary conditions, we show that the LT can be written exactly as the covariance of a bivariate probability distribution built from density-density correlations. This leads to two conceptually distinct extensions to PBC: (i) a geometric one based on the Riemannian center (Frechet mean) on the circle, and (ii) a metric-free one based on the mutual information I, which treats the configuration space purely as a probability space. We then relate the LT to the static structure factor by identifying the diagonal part, Cpp, as a "localization function" C(p), whose small-momentum behavior determines the LT in the thermodynamic limit. This clarifies why the LT is sensitive to transitions out of the extended phase but by itself cannot distinguish Anderson-type localization from dimerization: both share the same low-momentum asymptotics. We show that the finite-momentum behavior of C(p), together with an inverse participation ratio (IPR)-based upper bound valid in localized phases, provides a sharp criterion that discriminates localization from dimerization. These results are illustrated on the Su-Schrieffer-Heeger and Aubry-Andre models, with and without interactions, and suggest that structure factor-based probes offer robust and experimentally accessible diagnostics of localized and dimerized phases under PBC.