Taming two interacting particles with disorder

We compute the scaling properties of the localization length ξ2 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 = 20 000) 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 ξ2 occurs for u of the order of the single-particle bandwidth and for two-particle states with energies at the center of the spectrum, E = 0. We observe a crossover in the scaling of ξ2 with the single-particle localization length ξ1 into the asymptotic regime for ξ1 > 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 ξ2 /ξ1 = F (uξ1 ). ©2019 American Physical Society11sciescopu