Structure of the Conservation Laws in Quantum Integrable Spin Chains with Short Range Interactions

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. The essential tool for the former class of models is the boost operator, which provides a recursive way of calculating the integrals of motion. With its help, 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. Although these relations are difficult to solve in general, a subset of the coefficients can be determined. Moreover, for two submodels of the XYZ chain, namely the XXX and XY cases, all the charges can be calculated in closed form. Using this approach, we rederive the known expressions for the XY charges in a novel way. 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 drain. We also investigate the circumstances permitting the existence of a recursive (ladder) operator in general quantum integrable systems. We indicate that a quantum ladder operator can be traced back to the presence of a hamiltonian mastersymmetry of degree one in the classical continuous version of the model. In this way, quantum chains endowed with a recursive structure can be identified from the the properties of their classical relatives. We also 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.