Level Statistics and Localization for Two Interacting Particles in a Random Potential

TL;DR

This paper investigates how two interacting particles in a random potential exhibit changes in energy level statistics and localization length, revealing a transition governed by interaction strength and lifetime effects.

Contribution

It introduces a simplified Gaussian matrix model with a symmetry-breaking parameter to analyze the spectral and localization properties of two interacting particles in disorder.

Findings

Wigner-Dyson level rigidity persists up to a certain energy scale.

The two-particle localization length initially increases with interaction strength.

Different regimes of energy scale behavior depending on lifetime and level spacing.

Abstract

We consider two particles with a local interaction $U$ in a random potential at a scale $L_1$ (the one particle localization length). A simplified description is provided by a Gaussian matrix ensemble with a preferential basis. We define the symmetry breaking parameter $\mu \propto U^{-2}$ associated to the statistical invariance under change of basis. We show that the Wigner-Dyson rigidity of the energy levels is maintained up to an energy $E_{\mu}$. We find that $E_{\mu} \propto 1/\sqrt{\mu}$ when $\Gamma$ (the inverse lifetime of the states of the preferential basis) is smaller than $\Delta_2$ (the level spacing), and $E_{\mu} \propto 1/\mu $ when $\Gamma > \Delta_2$. This implies that the two-particle localization length $L_2$ first increases as $|U|$ before eventually behaving as $U^2$.