Title: Taming two interacting particles with disorder
Abstract: We compute the scaling properties of the localization length ${\ensuremath{\xi}}_{2}$ 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 $N=20\phantom{\rule{0.16em}{0ex}}000$) 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 ${\ensuremath{\xi}}_{2}$ 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 ${\ensuremath{\xi}}_{2}$ with the single-particle localization length ${\ensuremath{\xi}}_{1}$ into the asymptotic regime for ${\ensuremath{\xi}}_{1}>100$ ($W<1.0$). 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 ${\ensuremath{\xi}}_{2}/{\ensuremath{\xi}}_{1}=F(u{\ensuremath{\xi}}_{1})$.