Coulomb repulsion versus Hubbard repulsion in a disordered chain

We study the difference between on site Hubbard and long range Coulomb repu lsions for two interacting particles in a disordered chain. The system size L tin units of the lattice spacing) is of the order of the one particle lo calization length and the energies are taken near the band center. In the t wo cases, the limits of weak and strong interactions are characterized by u ncorrelated energy levels and are separated by a crossover regime where the states are more extended and the spectra more rigid. U denoting the intera ction strength and t the kinetic energy scale, the crossovers take place fo r interaction energy to kinetic energy ratios U/t and U/(2tL) of order one, for Hubbard and Coulomb repulsions respectively. While Hubbard repulsion c an only yield weak critical chaos with intermediate spectral statistics, Co ulomb repulsion can drive the two particle system to quantum chaos with Wig ner-Dyson spectral statistics. The interaction matrix elements are studied to explain this difference.