Level Statistics and Localization for Two Interacting Particles in a Random Potential

Abstract

We consider two particles with a local interaction $U$ in a random potential at a scale $L_1$ (the one particle localization length). A simplified description is provided by a Gaussian matrix ensemble with a preferential basis. We define the symmetry breaking parameter $μ\propto U^{-2}$ associated to the statistical invariance under change of basis. We show that the Wigner-Dyson rigidity of the energy levels is maintained up to an energy $E_μ$. We find that $E_μ \propto 1/\sqrtμ$ when $Γ$ (the inverse lifetime of the states of the preferential basis) is smaller than $Δ_2$ (the level spacing), and $E_μ \propto 1/μ$ when $Γ> Δ_2$. This implies that the two-particle localization length $L_2$ first increases as $|U|$ before eventually behaving as $U^2$.