Taming two interacting particles with disorder

Abstract

We compute the scaling properties of the localization length $ξ_2$ 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 $ξ_2$ 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 $ξ_2$ with the single particle localization length $ξ_1$ into the asymptotic regime for $ξ_1 > 100$ ($W < 1.0$). 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 $ξ_2/ξ_1=F(uξ_1)$.