Localization-landscape generalized Mott-Berezinski\u{i} formula

TL;DR

This paper reformulates the Mott-Berezinski theory of AC conductivity in disordered systems using localization landscape theory, introducing a geometric approach that accounts for inhomogeneity and extends applicability near the mobility edge.

Contribution

It presents a novel geometric framework based on localization landscapes that generalizes the Mott-Berezinski formula for disordered quantum materials.

Findings

Replaces classical hopping length with a generalized Mott scale

Incorporates spatial inhomogeneity into AC conductivity modeling

Recovers standard MB result as a special case

Abstract

We introduce a conceptual reformulation of the Mott-Berezinski\u{i} (MB) theory of low-frequency AC conductivity in disordered systems based on localization landscape theory. Instead of assuming uniform localization and fixed hopping distances, transport is described through an effective potential whose geometry encodes the spatial organization and energy-dependent localization of quantum states. Using the associated Agmon metric, we define a generalized Mott scale that replaces the classical hopping length with a geometric criterion set by the disorder landscape. This framework naturally incorporates strong spatial inhomogeneity and yields the AC conductivity directly from the effective potential. The standard MB result is recovered as a limiting case. Our approach extends the conceptual foundation of MB theory to arbitrary disordered media and energies approaching the mobility edge, providing a unified description of AC transport in complex quantum materials.