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

Abstract

We describe the level statistics of two interacting particles in a disordered conductor by a Gaussian matrix ensemble with preferential basis. This simplified model allows us to identify the energy scale $E_U$, below which the spectrum exhibits the universal Wigner-Dyson rigidity. $E_U\propto |U|$, when a small interaction $|U|$ yields “Rabi oscillations” between only two eigenstates at $U=0\mathrm{}$, while $E_U\propto U^2$ for larger $|U|$, as numerically confirmed for conductors. Scaling to insulators, this yields a localization length increasing as $|U|$ before eventually behaving as $U^2$.