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 B₀ . Near the quantum-critical point, at K→K_c(g) , the typical value of the order parameter vanishes exponentially, B₀∝e^{-C/[K-K_c(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<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, ω<ω_d(K,g) , the many-body spectrum of the model is discrete, with zero-level widths, while at ω>ω_d the level acquire a nonzero 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 superconductor-insulator transition and a semiquantitative description of the scanning tunneling data on its superconductive side.