Anatomy of localisation protected quantum order on Hilbert space

TL;DR

This paper explores how many-body localised phases exhibit quantum order by analyzing eigenstate correlations and their relation to Hilbert-space structure, revealing insights into phase transitions and entanglement.

Contribution

It introduces a novel Hilbert-space correlation framework to characterize localisation protected quantum order and phase transitions in many-body systems.

Findings

Eigenstate spread correlates with order parameters.

Higher-point correlations distinguish phases.

Scaling of correlation lengths signals phase transitions.

Abstract

Many-body localised phases of disordered, interacting quantum systems allow for exotic localisation protected quantum order in eigenstates at arbitrarily high energy densities. In this work, we analyse the manifestation of such order on the Hilbert-space anatomy of eigenstates. Quantified in terms of non-local Hilbert-spatial correlations of eigenstate amplitudes, we find that the spread of the eigenstates on the Hilbert-space graph is directly related to the order parameters which characterise the localisation protected order, and hence these correlations, in turn, characterise the order or lack thereof. Higher-point eigenstate correlations also characterise the different entanglement structures in the many-body localised phases, with and without order, as well as in the ergodic phase. The results pave the way for characterising the transitions between many-body localised phases and the ergodic phase in terms of scaling of emergent correlation lengthscales on the Hilbert-space graph.