The Fermi-Pasta-Ulam paradox, Anderson Localization problem and the generalized diffusion approach

TL;DR

This paper explores the analogy between Anderson localization and the Fermi-Pasta-Ulam effect using a classical Hamiltonian map, providing new insights into localization phenomena and deriving exact equations for the localization operator.

Contribution

It introduces a novel interpretation of Anderson localization as a FPU-like effect in a modified dynamical system and derives exact equations for the localization operator using classical Hamilton maps.

Findings

Localized states correspond to thermalization in the dynamical system

Delocalized states are analogous to stable quasi-periodic motion

Counter-intuitive results in Anderson localization mirror FPU counter-intuitivity

Abstract

The goal of this paper is two-fold. First, based on the interpretation of a quantum tight-binding model in terms of a classical Hamiltonian map, we consider the Anderson localization (AL) problem as the Fermi-Pasta-Ulam (FPU) effect in a modified dynamical system containing both stable and unstable (inverted) modes. Delocalized states in the AL are analogous to the stable quasi-periodic motion in FPU; whereas localized states are analogous to thermalization, respectively. The second aim is to use the classical Hamilton map for a simplified derivation of \textit{exact} equations for the localization operator $H(z)$. The letter was presented earlier [J.Phys.: Condens. Matter {\bf 14} (2002) 13777] treating the AL as a generalized diffusion in a dynamical system. We demonstrate that counter-intuitive results of our studies of the AL are similar to the FPU counter-intuitivity.