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

TL;DR

This paper develops a probabilistic mean-field theory to understand how correlations in Fock space influence the stability of many-body localisation, contrasting it with Anderson localisation and validating predictions with numerical models.

Contribution

It introduces a theory linking Fock-space correlations to many-body localisation stability, highlighting their essential role and providing a framework to predict localisation in various models.

Findings

Fock-space correlations are crucial for many-body localisation stability.

Uncorrelated models generally do not support stable localisation.

Numerical results confirm the theory's predictions across different models.

Abstract

We consider the problem of many-body localisation on Fock space, focussing on the essential features of the Hamiltonian which stabilise a localised 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 localisation on high-dimensional graphs because the Fock-space graph possesses non-trivial correlations. These correlations are shown to lie at the heart of whether or not a stable many-body localised phase can be sustained in the thermodynamic limit, and a theory is presented for the conditions the correlations must satisfy for a localised 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 localisation in typical local Hamiltonians where the correlations are strong, as well as its absence in uncorrelated random energy-like 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 diagonalisation is then used to corroborate the theoretical predictions for the occurrence or otherwise of a stable many-body localised phase; with mutual agreement found in each case.