Beyond the universal Dyson singularity for 1-D chains with hopping disorder

TL;DR

This paper investigates how the universal Dyson singularity in the density of states and localization length for 1D disordered chains can be violated when the hopping terms follow divergent probability distributions, linking quantum and classical anomalous behaviors.

Contribution

It demonstrates that the universal Dyson singularity can be broken in a tunable way by using divergent hopping distributions, connecting quantum localization phenomena with classical anomalous diffusion.

Findings

Universal Dyson singularity can be violated with divergent hopping distributions

Connection established between quantum localization breakdown and classical anomalous diffusion

Tunable control over singularity behavior in 1D disordered systems

Abstract

We study a simple non-interacting nearest neighbor tight-binding model in one dimension with disorder, where the hopping terms are chosen randomly. This model exhibits a well-known singularity at the band center both in the density of states and localization length. If the probability distribution of the hopping terms is well-behaved, then the singularities exhibit universal behavior, the functional form of which was first discovered by Freeman Dyson in the context of a chain of classical harmonic oscillators. We show here that this universal form can be violated in a tunable manner if the hopping elements are chosen from a divergent probability distribution. We also demonstrate a connection between a breakdown of universality in this quantum problem and an analogous scenario in the classical domain - that of random walks and diffusion with anomalous exponents.