Topology of delocalization in the nonlinear Anderson model and anomalous diffusion on finite clusters

TL;DR

This paper investigates how nonlinearity affects Anderson localization, revealing a unique transition at quadratic nonlinearity and proposing a topological analytical method to predict transport behaviors, including anomalous diffusion on finite clusters.

Contribution

It introduces a topological approximation method for the nonlinear Anderson model that predicts transport exponents for integer and half-integer nonlinearities.

Findings

Quadratic nonlinearity permits an abrupt localization-delocalization transition.

Super-quadratic nonlinearity leads to diffusion on finite clusters.

The proposed method accurately predicts transport exponents using a triangulation procedure.

Abstract

This study is concerned with destruction of Anderson localization by a nonlinearity of the power-law type. We suggest using a nonlinear Schr\"odinger model with random potential on a lattice that quadratic nonlinearity plays a dynamically very distinguished role in that it is the only type of power nonlinearity permitting an abrupt localization-delocalization transition with unlimited spreading already at the delocalization border. For super-quadratic nonlinearity the borderline spreading corresponds to diffusion processes on finite clusters. We have proposed an analytical method to predict and explain such transport processes. Our method uses a topological approximation of the nonlinear Anderson model and, if the exponent of the power nonlinearity is either integer or half-integer, will yield the wanted value of the transport exponent via a triangulation procedure in an Euclidean mapping space. A kinetic picture of the transport arising from these investigations uses a fractional extension of the diffusion equation to fractional derivatives over the time, signifying non-Markovian dynamics with algebraically decaying time correlations.