Scaling the localisation lengths for two interacting particles in one-dimensional random potentials

Using a numerical decimation method, we compute the localisation length $\lambda_{2}$ for two onsite interacting particles (TIP) in a one-dimensional random potential. We show that an interaction $U>0$ does lead to $\lambda_2(U) > \lambda_2(0)$ for not too large $U$ and test the validity of various proposed fit functions for $\lambda_2(U)$. Finite-size scaling allows us to obtain infinite sample size estimates $\xi_{2}(U)$ and we find that $\xi_{2}(U) \sim \xi_2(0)^{\alpha(U)}$ with $\alpha(U)$ varying between $\alpha(0)\approx 1$ and $\alpha(1) \approx 1.5$. We observe that all $\xi_2(U)$ data can be made to coalesce onto a single scaling curve. We also present results for the problem of TIP in two different random potentials corresponding to interacting electron-hole pairs.