Two interacting particles in a disordered chain I: Multifractality of the interaction matrix elements

TL;DR

This paper investigates the multifractal structure of interaction matrix elements in a disordered one-dimensional chain with two particles, revealing how interactions influence localization and decay properties.

Contribution

It introduces a numerical analysis of the multifractality of interaction matrix elements and links fractal dimensions to decay rates and localization length enhancement.

Findings

Interaction matrix elements are multifractal when states are localized.

Fractal dimension relates to Golden rule decay of non-interacting states.

Interaction enhances localization length in the disordered chain.

Abstract

For $N$ interacting particles in a one dimensional random potential, we study the structure of the corresponding network in Hilbert space. The states without interaction play the role of the ``sites''. The hopping terms are induced by the interaction. When the one body states are localized, we numerically find that the set of directly connected ``sites'' is multifractal. For the case of two interacting particles, the fractal dimension associated to the second moment of the hopping term is shown to characterize the Golden rule decay of the non interacting states and the enhancement factor of the localization length.