# Taming two interacting particles with disorder

## TL;DR

This paper investigates how two interacting particles in a disordered one-dimensional chain exhibit localization properties, revealing how interaction strength and disorder influence localization length and the transition to asymptotic scaling regimes.

## Contribution

It provides large-scale numerical analysis of two-particle localization, identifying the conditions for maximum enhancement and the crossover to asymptotic scaling behavior.

## Key findings

- Maximum localization length enhancement occurs at specific interaction strengths and energies.
- A crossover to asymptotic scaling occurs for large single-particle localization lengths.
- A nonlinear scaling function describes the relationship between two-particle and single-particle localization lengths.

## Abstract

We compute the scaling properties of the localization length $\xi_2$ of two interacting particles in a one-dimensional chain with diagonal disorder, and the connectivity properties of the Fock states. We analyze record large system sizes (up to $N=20000$) and disorder strengths (down to $W=0.5$). We vary the energy $E$ and the on-site interaction strength $u$. At a given disorder strength the largest enhancement of $\xi_2$ occurs for $u$ of the order of the single particle band width, and for two-particle states with energies at the center of the spectrum, $E=0$. We observe a crossover in the scaling of $\xi_2$ with the single particle localization length $\xi_1$ into the asymptotic regime for $\xi_1 > 100$ ($W < 1.0$). This happens due to the recovery of translational invariance and momentum conservation rules in the matrix elements of interconnected Fock eigenstates for $u=0$. The entrance into the asymptotic scaling is manifested through a nonlinear scaling function $\xi_2/\xi_1=F(u\xi_1)$.