Superconductor-Insulator transition and energy localization

We develop an analytical theory for generic disorder-driven quantum phase transitions. We apply this formalism to the superconductor-insulator transition and we briefly discuss the applications to the order-disorder transition in quantum magnets. The effective spin-1/2 models for these transitions are solved in the cavity approximation which becomes exact on a Bethe lattice with large branching number K >> 1 and weak dimensionless coupling g << 1. The characteristic features of the low temperature phase is a large self-formed inhomogeneity of the order-parameter distribution near the critical point K_{c}(g) where the critical temperature T_{c} of the ordering transition vanishes. Near the quantum critical point, the typical value of the order parameter vanishes exponentially, B_{0}\propto e^{-C/(K-K_{c}(g))}. In the disordered regime, realized at K<K_{c}(g) we find actually two distinct phases characterized by different behavior of relaxation rates. The first phase exists in an intermediate range of K^{*}(g)<K<K_{c}(g). It has two regimes of energies: at low excitation energies, \omega<\omega_{d}(K,g), the many-body spectrum of the model is discrete, with zero level widths, while at \omega>\omega_{d} the level acquire a non-zero width which is self-generated by the many-body interactions. In this phase the spin model provides by itself an intrinsic thermal bath. Another phase is obtained at smaller K<K^{*}(g), where all the eigenstates are discrete, corresponding to full many-body localization. These results provide an explanation for the activated behavior of the resistivity in amorphous materials on the insulating side near the SI transition and a semi-quantitative description of the scanning tunneling data on its superconductive side.