Quantum chains with a Catalan tree pattern of conserved charges: the $\Delta = -1$ XXZ model and the isotropic octonionic chain

TL;DR

This paper explores quantum chains with a unique Catalan tree pattern of conserved charges, identifying new integrable models including the XXZ model at    and a novel octonionic chain, expanding understanding of algebraic structures in quantum integrability.

Contribution

It establishes conditions for the existence of Catalan tree-structured conserved charges and introduces two new models with this pattern, broadening the class of integrable quantum chains.

Findings

Identified Catalan tree pattern in conserved charges of certain quantum chains.

Discovered the    XXZ model with   .

Introduced a new octonionic isotropic chain with noncommuting conserved quantities.

Abstract

A class of quantum chains possessing a family of local conserved charges with a Catalan tree pattern is studied. Recently, we have identified such a structure in the integrable $SU(N)$-invariant chains. In the present work we find sufficient conditions for the existence of a family of charges with this structure in terms of the underlying algebra. Two additional systems with a Catalan tree structure of conserved charges are found. One is the spin 1/2 XXZ model with $\Delta=-1$. The other is a new octonionic isotropic chain, generalizing the Heisenberg model. This system provides an interesting example of an infinite family of noncommuting local conserved quantities.