Matrix product operator representations for the local conserved quantities of the spin-$1/2$ XYZ chain

TL;DR

This paper develops explicit matrix product operator representations for the conserved quantities of the spin-1/2 XYZ chain, revealing their combinatorial structure and providing a new simple Lax operator without elliptic functions.

Contribution

It introduces explicit MPO representations for XYZ chain conserved quantities, simplifies their coefficients, and presents a novel Lax operator free of elliptic functions.

Findings

Coefficients are polynomial generalizations of Catalan numbers.

MPO representations simplify the structure of conserved quantities.

A new 3x3 Lax operator for the XYZ chain is derived.

Abstract

We present explicit matrix product operator (MPO) representations for the local conserved quantities of the spin-$1/2$ XYZ chain. Through these MPO representations, we simplify the coefficients appearing in the local conserved quantities originally derived by one of the authors, and reveal their combinatorial meaning: the coefficients prove to be a polynomial generalization of the Catalan numbers, defined via weighted monotonic lattice paths. Furthermore, we obtain a new simple $3 \times 3$ Lax operator for the XYZ chain that, unlike Baxter's R-matrix, does not involve elliptic functions.