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≥Kc(g), where the critical temperature Tc 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→Kc(g), the typical value of the order parameter vanishes exponentially, B₀∝e^-C/[K-Kc(g)] while the spatial scale Nᵢₙₕ of the order parameter inhomogeneities diverges as [K-Kc(g)]^-2. In the disordered regime, realized at K<Kc(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<Kc(g). It has two regimes of energies: at low excitation energies, ω<ωd(K,g), the many-body spectrum of the model is discrete, with zero-level widths, while at ω>ωd 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.