Superconductor-insulator transition and energy localization

Abstract

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-$\frac{1}{2}$ 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 $K\ensuremath{\ge}{K}_{c}(g)$, 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 $K\ensuremath{\rightarrow}{K}_{c}(g)$, the typical value of the order parameter vanishes exponentially, ${B}_{0}\ensuremath{\propto}{e}^{\ensuremath{-}C/[K\ensuremath{-}{K}_{c}(g)]}$ while the spatial scale ${N}_{inh}$ of the order parameter inhomogeneities diverges as ${[K\ensuremath{-}{K}_{c}(g)]}^{\ensuremath{-}2}$. 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 ${K}^{\ensuremath{\ast}}(g)<K<{K}_{c}(g)$. It has two regimes of energies: at low excitation energies, $\ensuremath{\omega}<{\ensuremath{\omega}}_{d}(K,g)$, the many-body spectrum of the model is discrete, with zero-level widths, while at $\ensuremath{\omega}>{\ensuremath{\omega}}_{d}$ 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 $K<{K}^{\ensuremath{\ast}}(g)$, 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.