Dynamics of a quantum particle in low-dimensional disordered systems with extended states

TL;DR

This paper studies quantum particle dynamics in low-dimensional disordered systems with extended states, revealing anomalous bi-fractal diffusion and critical density profiles at special energies, challenging typical Anderson localization expectations.

Contribution

It provides a unified analysis of anomalous diffusion and critical behavior in specific disordered models with diverging localization lengths, supported by analytical and numerical results.

Findings

Quantum motion shows bi-fractal anomalous diffusion.

Density profile becomes critical at long times.

Divergence of moments occurs for q > q0, with q0=2 for off-diagonal disorder and q0=1/2 for the random-dimer model.

Abstract

We investigate the dynamics of a quantum particle in disordered tight-binding models in one and two dimensions which are exceptions to the common wisdom on Anderson localization, in the sense that the localization length diverges at some special energies. We provide a consistent picture for two well-known one-dimensional examples: the chain with off-diagonal disorder and the random-dimer model. In both cases the quantum motion exhibits a peculiar kind of anomalous diffusion which can be referred to as bi-fractality. The disorder-averaged density profile of the particle becomes critical in the long-time regime. The $q$-th moment of the position of the particle diverges with time whenever $q$ exceeds some $q_0$. We obtain $q_0=2$ for off-diagonal disorder on the chain (and conjecturally on two-dimensional bipartite lattices as well). For the random-dimer model, our result $q_0=1/2$ corroborates known rigorous results.