Group word dynamics from local random matrix Hamiltonians and beyond

TL;DR

This paper explores quantum spin chains with random matrix interactions, mapping their dynamics to a single-particle hopping model on a Cayley graph, revealing insights into thermalization and integrability.

Contribution

It introduces a novel mapping from many-body quantum dynamics to single-particle hopping on Cayley graphs using free probability theory, and investigates the effects of group modifications on integrability.

Findings

Density of states is approximately Gaussian.

Energy density thermalizes as expected.

Adding braid relations may induce integrability.

Abstract

We study one dimensional quantum spin chains whose nearest neighbor interactions are random matrices that square to one. By employing free probability theory, we establish a mapping from the many-body quantum dynamics of energy density in the original chain to a single-particle hopping dynamics when the local Hilbert space dimension is large. The hopping occurs on the Cayley graph of an infinite Coxeter reflection group. Adjacency matrices on large finite clusters of this Cayley graph can be constructed numerically by leveraging the automatic structure of the group. The density of states and two-point functions of the local energy density are approximately computed and consistent with the physics of a generic local Hamiltonian: Gaussian density of states and thermalization of energy density. We then ask what happens to the physics if we modify the group on which the hopping dynamics occurs, and conjecture that adding braid relations into the group leads to integrability. Our results put into contact ideas in free probability theory, quantum mechanics of hyperbolic lattices, and the physics of both generic and integrable Hamiltonian dynamics.