Integrable spin chains and cellular automata with medium range interaction

TL;DR

This paper develops an algebraic framework for integrable spin chains and cellular automata with medium-range interactions, classifies specific models, and identifies new integrable models including certain cellular automata.

Contribution

It introduces a new integrability condition for medium-range Hamiltonians and provides a partial classification, including the discovery of new models and integrable deformations.

Findings

Rule150 and Rule105 automata are Yang-Baxter integrable with three-site interactions.

New integrable models with medium-range interactions are identified.

The framework extends known methods to classify and analyze such models.

Abstract

We study integrable spin chains and quantum and classical cellular automata with interaction range $\ell\ge 3$. This is a family of integrable models for which there was no general theory so far. We develop an algebraic framework for such models, generalizing known methods from nearest neighbor interacting chains. This leads to a new integrability condition for medium range Hamiltonians, which can be used to classify such models. A partial classification is performed in specific cases, including $U(1)$-symmetric three site interacting models, and Hamiltonians that are relevant for interaction-round-a-face models. We find a number of models which appear to be new. As an application we consider quantum brickwork circuits of various types, including those that can accommodate the classical elementary cellular automata on light cone lattices. In this family we find that the so-called Rule150 and Rule105 models are Yang-Baxter integrable with three site interactions. We present integrable quantum deformations of these models, and derive a set of local conserved charges for them. For the famous Rule54 model we find that it does not belong to the family of integrable three site models, but we can not exclude Yang-Baxter integrability with longer interaction ranges.