We investigate Anderson transitions for a system of two particles moving in a three-dimensional disordered lattice and subject to on-site (Hubbard) interactions of strength U . The two-body problem is exactly mapped into an effective single-particle equation for the center of mass motion, whose localization properties are studied numerically. We show that, for zero total energy of the pair, the transition occurs in a regime where all single-particle states are localized. In particular the critical disorder strength exhibits a non-monotonic behavior as a function of |U |, increasing sharply for weak interactions and converging to a finite value in the strong coupling limit. Within our numerical accuracy, short-range interactions do not affect the universality class of the transition.

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Most of our quantitative understanding of disorder-induced metal-insulator transitions comes from numerical studies of simple noninteracting tight-binding models, like the Anderson model in three dimensions. An important outstanding problem is the fate of the Anderson transition in the presence of additional Hubbard interactions of strength U between particles. Based on large-scale numerics, we compute the position of the mobility edge for a system of two identical bosons or two fermions with opposite spin components. The resulting phase diagram in the interaction-energy-disorder space possesses a remarkably rich and counterintuitive structure, with multiple metallic and insulating phases. We show that this phenomenon originates from the molecular or scatteringlike nature of the pair states available at given energy E and disorder strength W. The disorder-averaged density of states of the effective model for the pair is also investigated. Finally, we discuss the implications of our results for ongoing research on many-body localization.