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 $\mu \propto U^{ -2}$ 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 $E_{ \mu}$. We find that $E_{ \mu} \propto 1/\sqrt{ \mu}$ when $\Gamma$ (the inverse lifetime of the states of the preferential basis) is smaller than $\Delta_2$ (the level spacing), and $E_{ \mu} \propto 1/\mu $ when $\Gamma > \Delta_2$. This implies that the two-particle localization length $L_2$ first increases as $|U|$ before eventually behaving as $U^2$.
Categories: cond-mat