Integrability of planar-algebraic models

TL;DR

This paper extends the Quantum Inverse Scattering Method to planar algebra-based models with non-separable R-operators, characterizing polynomial integrability and demonstrating it in specific models like the eight-vertex and Temperley--Lieb loop models.

Contribution

It introduces a framework for integrable models using planar algebras, relaxing tensorial R-matrix properties, and characterizes polynomial integrability in these models.

Findings

Transfer operator is polynomial in the Hamiltonian for the eight-vertex model.

In the Temperley--Lieb loop model, transfer operators are polynomial in the Hamiltonian for most loop fugacities.

The model admits a second Hamiltonian that also acts as a polynomial integrability generator.

Abstract

The Quantum Inverse Scattering Method is a scheme for solving integrable models in $1+1$ dimensions, building on an $R$-matrix that satisfies the Yang--Baxter equation and in terms of which one constructs a commuting family of transfer matrices. In the standard formulation, this $R$-matrix acts on a tensor product of vector spaces. Here, we relax this tensorial property and develop a framework for describing and analysing integrable models based on planar algebras, allowing non-separable \textit{$R$-operators} satisfying \textit{generalised} Yang--Baxter equations. We also re-evaluate the notion of integrals of motion and characterise when an (algebraic) \textit{transfer operator} is polynomial in a single integral of motion. We refer to such models as {\em polynomially integrable}. In an eight-vertex model, we demonstrate that the corresponding transfer operator is polynomial in the natural hamiltonian. In the Temperley--Lieb loop model with loop fugacity $\beta\in\mathbb{C}$, we likewise find that, for all but finitely many $\beta$-values, the transfer operator is polynomial in the usual hamiltonian element of the Temperley--Lieb algebra $\mathrm{TL}_n(\beta)$, at least for $n\leq17$. Moreover, we find that this model admits a second canonical hamiltonian, and that this hamiltonian also acts as a polynomial integrability generator for small $n$ and all but finitely many $\beta$-values.