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

Using a numerical decimation method, we compute the localisation length $λ_{2}$ for two onsite interacting particles (TIP) in a one-dimensional random potential. We show that an interaction $U>0$ does lead to $λ_2(U) > λ_2(0)$ for not too large $U$ and test the validity of various proposed fit functions for $λ_2(U)$. Finite-size scaling allows us to obtain infinite sample size estimates $ξ_{2}(U)$ and we find that $ξ_{2}(U) \sim ξ_2(0)^{α(U)}$ with $α(U)$ varying between $α(0)\approx 1$ and $α(1) \approx 1.5$. We observe that all $ξ_2(U)$ 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.

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