Closed hierarchy of Heisenberg equations in integrable models with Onsager algebra

TL;DR

This paper shows that in certain integrable quantum models with Onsager algebra, the Heisenberg equations form a closed, manageable system, allowing explicit solutions for operators in these models, exemplified by the Ising and Potts models.

Contribution

The paper demonstrates the closure of Heisenberg equations in integrable models with Onsager algebra, enabling explicit analytical solutions for these systems.

Findings

Heisenberg equations form a closed system in these models.

Explicit solutions are obtained for operators in the Onsager algebra.

Applications shown for the transverse field Ising and Potts models.

Abstract

Dynamics of a quantum system can be described by coupled Heisenberg equations. In a generic many-body system these equations form an exponentially large hierarchy that is intractable without approximations. In contrast, in an integrable system a small subset of operators can be closed with respect to commutation with the Hamiltonian. As a result, the Heisenberg equations for these operators can form a smaller closed system amenable to an analytical treatment. We demonstrate that this indeed happens in a class of integrable models where the Hamiltonian is an element of the Onsager algebra. We explicitly solve the system of Heisenberg equations for operators from this algebra. Two specific models are considered as examples: the transverse field Ising model and the superintegrable chiral 3-state Potts model.