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 feature of the low-temperature phase is a large self-formed inhomogeneity of the order-parameter distribution near the critical point K >= K-c(g), where the critical temperature T-c of the ordering transition vanishes. We find that the local probability distribution P(B) of the order parameter B has a long power-law tail in the region where B is much larger than its typical value B-0. Near the quantum-critical point, at K -> K-c(g), the typical value of the order parameter vanishes exponentially, B-0 proportional to e(-C/[K-Kc(g)]) while the spatial scale N-inh of the order parameter inhomogeneities diverges as [K-K-c(g)](-2). In the disordered regime, realized at K