Superconductor-metal transition in a Hubbard model with random interactions.

We present a BCS theory of superconductivity for a Hubbard model with attractive on-site interaction U along with random and all-to-all hopping, exchange as well as Cooper-pair hopping. The analysis is carried out in the large-M limit, where M corresponds to generalizing the SU(2) spin symmetry to USp(M). Within the mean-field treatment we find that the effect of Cooper-pair hopping is on the same footing as that of the exchange interaction. In our model, we prove the existence of a BCS-type log divergence in the gap equation, which implies superconductivity at zero temperature in the presence of an infinitesimal attractive Hubbard interaction. The superconductivity in our model arises out of a Fermi-liquid metal for small exchange interaction, while it arises from a SYK-like non-Fermi-liquid metal at very large interactions. We calculate the value of the critical temperature (T_c) and the superconducting gap as a function of the exchange interaction and find that T_c decreases for intermediate values of exchange interactions. Our model may be relevant to two-dimensional disordered superconductors.