Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states

TL;DR

This paper demonstrates a finite-temperature metal-insulator transition in a disordered, weakly interacting electron system, showing that electron-electron interactions alone cannot induce relaxation or thermalization in the localized phase.

Contribution

It provides a theoretical proof of a finite-temperature metal-insulator transition driven by many-body localization in Fock space, with a detailed analysis of the system's low-energy effective Hamiltonian.

Findings

Electrical conductivity vanishes below a critical temperature $T_c$.

Electron-electron interactions alone do not cause relaxation in the localized phase.

Weak coupling to a bath induces finite conductivity in the insulating phase.

Abstract

We consider low-temperature behavior of weakly interacting electrons in disordered conductors in the regime when all single-particle eigenstates are localized by the quenched disorder. We prove that in the absence of coupling of the electrons to any external bath dc electrical conductivity exactly vanishes as long as the temperatute $T$ does not exceed some finite value $T_c$. At the same time, it can be also proven that at high enough $T$ the conductivity is finite. These two statements imply that the system undergoes a finite temperature Metal-to-Insulator transition, which can be viewed as Anderson-like localization of many-body wave functions in the Fock space. Metallic and insulating states are not different from each other by any spatial or discrete symmetries. We formulate the effective Hamiltonian description of the system at low energies (of the order of the level spacing in the single-particle localization volume). In the metallic phase quantum Boltzmann equation is valid, allowing to find the kinetic coefficients. In the insulating phase, $T<T_c$, we use Feynmann diagram technique to determine the probability distribution function for quantum-mechanical transition rates. The probability of an escape rate from a given quantum state to be finite turns out to vanish in every order of the perturbation theory in electron-electron interaction. Thus, electron-electron interaction alone is unable to cause the relaxation and establish the thermal equilibrium. As soon as some weak coupling to a bath is turned on, conductivity becomes finite even in the insulating phase.