A universal graph description for one-dimensional exchange models

TL;DR

This paper introduces a unified graph-based framework using Cayley graphs of permutation groups to analyze one-dimensional exchange models, enabling efficient spectral analysis and insights into quantum and classical processes.

Contribution

It establishes a universal graph description for a broad class of exchange models, linking spectral properties to physical and algebraic characteristics, and extends to non-Hermitian systems.

Findings

Spectral gap can be computed in polynomial time.

Framework applies to classical and quantum exchange models.

Provides new mathematical insights into unweighted Cayley graphs.

Abstract

We demonstrate that a large class of one-dimensional quantum and classical exchange models can be described by the same type of graphs, namely Cayley graphs of the permutation group. Their well-studied spectral properties allow us to derive crucial information about those models of fundamental importance in both classical and quantum physics, and to completely characterize their algebraic structure. Notably, we prove that the spectral gap can be obtained in polynomial computational time, which has strong implications in the context of adiabatic quantum computing with quantum spin-chains. This quantity also characterizes the rate to stationarity of some important classical random processes such as interchange and exclusion processes. Reciprocally, we use results derived from the celebrated Bethe ansatz to obtain original mathematical results about these graphs in the unweighted case. We also discuss extensions of this unifying framework to other systems, such as asymmetric exclusion processes -- a paradigmatic model in non-equilibrium physics, or the more exotic non-Hermitian quantum systems.