Two interacting particles in a disordered chain II: Critical statistics and maximum mixing of the one body states

TL;DR

This paper investigates how two interacting particles in a disordered chain exhibit critical spectral statistics and maximum mixing of states near a fixed point of a duality transformation, revealing limitations in achieving full chaos in 1D.

Contribution

It identifies the conditions under which weak chaos and multifractal wave functions occur due to interactions, and clarifies the limitations of local interactions in 1D systems.

Findings

Maximum mixing occurs at the duality fixed point.

Weak chaos and multifractality are observed near the localization length.

Full Wigner-Dyson chaos cannot be achieved in 1D with local interactions.

Abstract

For two particles in a disordered chain of length $L$ with on-site interaction $U$, a duality transformation maps the behavior at weak interaction onto the behavior at strong interaction. Around the fixed point of this transformation, the interaction yields a maximum mixing of the one body states. When $L \approx L_1$ (the one particle localization length), this mixing results in weak chaos accompanied by multifractal wave functions and critical spectral statistics, as in the one particle problem at the mobility edge or in certain pseudo-integrable billiards. In one dimension, a local interaction can only yield this weak chaos but can never drive the two particle system to full chaos with Wigner-Dyson statistics.