Defining relations on the Hamiltonians of XXX and XXZ R-matrices and new integrable spin-orbital chains

Abstract

We present several complete systems of integrability conditions on the density of the Hamiltonian of a spin chain matrix. The corresponding formulas for R-matrices are also given. The latter are expressed via the local Hamiltonian density in a form similar to spin one half XXX and XXZ models. The result is applied to the problem of integrability of SU(2) × SU(2)-and SU(2) × U(1)-invariant spin-orbital chains (the Kugel-Homskii-Inagaki model). Eight new integrable cases are found. One of these cases corresponds to the Temperley-Lieb algebra, three cases correspond to the algebra associated with the XXX model, one case corresponds to the algebra associated with the XXZ model, and one case corresponds to the algebra associated with the graded XXZ model. The remaining two R-matrices are also presented. Bibliography: 19 titles.