Fock-space correlations and the origins of many-body localization

We consider the problem of many-body localization on Fock space, focusing on the essential features of the Hamiltonian which stabilize a localized phase. Any many-body Hamiltonian has a canonical representation as a disordered tight-binding model on the Fock-space graph. The underlying physics is, however, fundamentally different from that of conventional Anderson localization on high-dimensional graphs because the Fock-space graph possesses nontrivial correlations. These correlations are shown to lie at the heart of whether or not a stable many-body localized phase can be sustained in the thermodynamic limit, and a theory is presented for the conditions the correlations must satisfy for a localized phase to be stable. Our analysis is rooted in a probabilistic, self-consistent mean-field theory for the local Fock-space propagator and its associated self-energy, in which the Fock-space correlations, together with the extensive nature of the connectivity of Fock-space nodes, are key ingredients. The origins of many-body localization in typical local Hamiltonians where the correlations are strong, as well as its absence in uncorrelated random energy models, emerge as predictions from the same overarching theory. To test these, we consider three specific microscopic models, first establishing in each case the nature of the associated Fock-space correlations. Numerical exact diagonalization is then used to corroborate the theoretical predictions for the occurrence or otherwise of a stable many-body localized phase, with mutual agreement found in each case.