Interference in disordered systems: A particle in a complex random landscape

TL;DR

This paper analyzes a one-dimensional particle in a disordered landscape, revealing three distinct phases and universal behaviors, with implications for understanding interference effects in complex disordered systems.

Contribution

It provides an analytical study of a disordered particle model, identifying phase transitions and universal correlator forms, linking interference phenomena to complex Burgers dynamics.

Findings

Identification of three distinct phases: high-temperature, pinned, and diffusive.

Derivation of a universal form for the Burgers velocity correlator in phase III.

Discovery of a logarithmic singularity related to interference effects.

Abstract

We consider a particle in one dimension submitted to amplitude and phase disorder. It can be mapped onto the complex Burgers equation, and provides a toy model for problems with interplay of interferences and disorder, such as the NSS model of hopping conductivity in disordered insulators and the Chalker-Coddington model for the (spin) quantum Hall effect. The model has three distinct phases: (I) a {\em high-temperature} or weak disorder phase, (II) a {\em pinned} phase for strong amplitude disorder, and (III) a {\em diffusive} phase for strong phase disorder, but weak amplitude disorder. We compute analytically the renormalized disorder correlator, equivalent to the Burgers velocity-velocity correlator at long times. In phase III, it assumes a universal form. For strong phase disorder, interference leads to a logarithmic singularity, related to zeroes of the partition sum, or poles of the complex Burgers velocity field. These results are valuable in the search for the adequate field theory for higher-dimensional systems.