[Paper Review] Integrable spin chains and cellular automata with medium range interaction

Tamás Gombor, Balázs Pozsgay|arXiv (Cornell University)|Aug 4, 2021

Quantum many-body systemsPhysics and Astronomy125 references80 citations

TL;DR

This paper develops a general algebraic framework for integrable spin chains and cellular automata with medium-range interactions (range ℓ ≥ 3), introducing a new integrability condition that enables classification of such models. It identifies new Yang-Baxter integrable models, including quantum deformations of Rule150 and Rule105 with three-site interactions, and shows the Rule54 model is not integrable within this family, though longer-range interactions remain possible.

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.

Motivation & Objective

• To develop a general algebraic framework for integrable spin chains and cellular automata with medium-range interactions (ℓ ≥ 3), filling a gap in the existing theory.
• To establish a new integrability condition for Hamiltonians with finite interaction range, enabling systematic classification of such models.
• To identify and classify new integrable models, particularly U(1)-symmetric three-site interacting systems and those related to interaction-round-a-face (IRF) models.
• To explore the connection between quantum brickwork circuits and classical elementary cellular automata on light-cone lattices, focusing on Yang-Baxter integrability.
• To derive local conserved charges for newly identified integrable models, including quantum deformations of Rule150 and Rule105.

Proposed method

• Generalize algebraic methods from nearest-neighbor spin chains to medium-range models using the Yang-Baxter equation as a central tool.
• Derive a new integrability condition for medium-range Hamiltonians based on the R-matrix formalism and quantum group symmetries.
• Apply the framework to classify U(1)-symmetric three-site models and IRF-type Hamiltonians, identifying new integrable cases.
• Construct quantum brickwork circuits with three-site gates and analyze their integrability using the Yang-Baxter condition.
• Use the algebraic Bethe ansatz and quantum inverse scattering method to derive local conserved charges for identified models.
• Investigate the Rule54 model's non-integrability within the three-site framework, while leaving open the possibility for longer-range integrability.

Experimental results

Research questions

• RQ1 Can a general algebraic framework be developed for integrable spin chains with medium-range interactions (ℓ ≥ 3), beyond nearest-neighbor models?
• RQ2 What are the necessary and sufficient conditions for Yang-Baxter integrability in Hamiltonians with finite interaction range ℓ ≥ 3?
• RQ3 Which known cellular automata, such as Rule150, Rule105, and Rule54, are Yang-Baxter integrable when extended to three-site interactions?
• RQ4 Can quantum deformations of classically integrable cellular automata be constructed and shown to be quantum integrable?
• RQ5 Is the Rule54 model Yang-Baxter integrable with interaction ranges longer than three sites, despite its non-integrability in the three-site case?

Key findings

• A new integrability condition for medium-range Hamiltonians (ℓ ≥ 3) is derived, generalizing the nearest-neighbor framework and enabling classification of such models.
• The Rule150 and Rule105 models are shown to be Yang-Baxter integrable with three-site interactions, and their quantum deformations are constructed explicitly.
• A set of local conserved charges is derived for the integrable quantum deformations of Rule150 and Rule105, confirming their integrability via the algebraic Bethe ansatz.
• The Rule54 model is found not to be Yang-Baxter integrable within the three-site interaction family, though the possibility of integrability with longer-range interactions remains open.
• The framework successfully classifies U(1)-symmetric three-site models and identifies new IRF-type models, including the folded XXZ model as a special case.
• The study establishes a direct link between quantum brickwork circuits and classical cellular automata on light-cone lattices, showing that integrability can be preserved under such constructions.