Structure of the conservation laws in integrable spin chains with short range interactions

TL;DR

This paper analyzes the structure of conservation laws in quantum integrable spin chains, deriving explicit forms of conserved charges and their recursive relations, with special focus on XYZ, XXX, XY, and Hubbard models, revealing new algebraic structures.

Contribution

It introduces a general polynomial form for conserved charges in XYZ-type chains and derives recursion relations, providing explicit solutions for specific submodels and extending to su(M) chains.

Findings

Conserved charges in XYZ chains can be expressed as simple polynomials in spin variables.

Explicit closed-form expressions for charges in XXX and XY models are obtained.

The ladder operator's role is clarified, showing its absence in quantum continuous limits and the Hubbard model.

Abstract

We present a detailed analysis of the structure of the conservation laws in quantum integrable chains of the XYZ-type and in the Hubbard model. With the use of the boost operator, we establish the general form of the XYZ conserved charges in terms of simple polynomials in spin variables and derive recursion relations for the relative coefficients of these polynomials. For two submodels of the XYZ chain - namely the XXX and XY cases, all the charges can be calculated in closed form. For the XXX case, a simple description of conserved charges is found in terms of a Catalan tree. This construction is generalized for the su(M) invariant integrable chain. We also indicate that a quantum recursive (ladder) operator can be traced back to the presence of a hamiltonian mastersymmetry of degree one in the classical continuous version of the model. We show that in the quantum continuous limits of the XYZ model, the ladder property of the boost operator disappears. For the Hubbard model we demonstrate the non-existence of a ladder operator. Nevertheless, the general structure of the conserved charges is indicated, and the expression for the terms linear in the model's free parameter for all charges is derived in closed form.