Calculation of Anderson localization criterium for a one dimensional chain with diagonal disorder

TL;DR

This paper derives analytical expressions for the Anderson localization criterion in a one-dimensional disordered chain, focusing on the edge excitation density and its dependence on disorder type and energy, validated by simulations.

Contribution

The paper provides new analytical formulas for the edge excitation density in a disordered chain, including binary and small disorder cases, extending understanding of Anderson localization criteria.

Findings

Derived exact expression for edge excitation density in binary disordered chain.

Obtained formula for small disorder case.

Confirmed results with computer simulations.

Abstract

For a one dimensional half-infinite chain with diagonal disorder we calculated the ultimate at $t\to\infty$ value of the average excitation density at the edge site $D$ if at $t=0$ the excitation was localised at the edge site (Anderson' s creterium). We obtained the following results: i) for the binary disordered chain we derived the close expression for $D$ which is exact in the limit of low concentration of defects and is valid for an arbitrary energy of defects $\varepsilon$. In this case $D$ demonstrated the non analytical dependence on $\varepsilon$. ii) The close expression for $D$ is obtained for the case of an arbitrary small disorder. iii) The relative contribution of states with specified energy to $D$ is calculated. All the results obtained are in complete agreement with computer simulation.