Two interacting particles in a random potential: The random matrix model revisited

TL;DR

This paper critically examines the effectiveness of using an effective random matrix model to describe two interacting particles in a random potential, revealing significant discrepancies and potential inaccuracies in the model's predictions.

Contribution

The study provides a detailed numerical analysis showing that the coupling matrix elements decrease faster than assumed and that the model's predictions for localization length can be qualitatively incorrect.

Findings

Coupling matrix elements decrease faster with localization length than in the model.

The model's predictions for localization length can be qualitatively wrong.

Mapping to an effective random matrix model can be potentially misleading.

Abstract

We reinvestigate the validity of mapping the problem of two onsite interacting particles in a random potential onto an effective random matrix model. To this end we first study numerically how the non-interacting basis is coupled by the interaction. Our results indicate that the typical coupling matrix element decreases significantly faster with increasing single-particle localization length than is assumed in the random matrix model. We further show that even for models where the dependency of the coupling matrix element on the single-particle localization length is correctly described by the corresponding random matrix model its predictions for the localization length can be qualitatively incorrect. These results indicate that the mapping of an interacting random system onto an effective random matrix model is potentially dangerous. We also discuss how Imry's block-scaling picture for two interacting particles is influenced by the above arguments.