Many-body systems with random spatially local interactions

TL;DR

This paper extends random matrix theory to analyze quantum many-body systems with local interactions, introducing new diagrammatic and algebraic methods to evaluate correlators in large local Hilbert space limits.

Contribution

It develops three novel methods—stacked planar diagrams, heap freeness, and dependency partitions—for systematically evaluating correlators in spatially local quantum systems.

Findings

Correlators are expressed via sums over stacked planar diagrams.

Heap freeness generalizes free independence for local systems.

Large-N quantum satisfiability relates to the independence polynomial of graphs.

Abstract

We extend random matrix theory to consider randomly interacting spin systems with spatial locality. We develop several methods by which arbitrary correlators may be systematically evaluated in a limit where the local Hilbert space dimension $N$ is large. First, the correlators are given by sums over 'stacked' planar diagrams which are completely determined by the spectra of the individual interactions and a dependency graph encoding the locality in the system. We then introduce 'heap freeness' as a generalization of free independence, leading to a second practical method to evaluate the correlators. Finally, we generalize the cumulant expansion to a sum over 'dependency partitions', providing the third and most succinct of our methods. Our results provide tools to study dynamics and correlations within extended quantum many-body systems which conserve energy. We further apply the formalism to show that quantum satisfiability at large-$N$ is determined by the evaluation of the independence polynomial on a wide class of graphs.