Quantum group deformation of the Kittel--Shore model

Abstract

The Kittel--Shore (KS) Hamiltonian describes $N$ spins with long-range interactions that are identically coupled; therefore, this (mean-field) model is also known as the Heisenberg XXX model on the complete graph. In this paper, the underlying $U(\mathfrak{su}(2))$ coalgebra symmetry of the KS model is demonstrated for arbitrary spins, and the quantum deformation of the KS Hamiltonian ($q$-KS model) is obtained using the corresponding $U_q(\mathfrak{su}(2))$ quantum group. By construction, the existence of such a symmetry guarantees that all integrability properties of the KS model are preserved under $q$-deformation. In particular, the $q$-KS model for spin-$1/2$ particles is analysed, the cases with $N=2$ and $3$ spins are studied in detail, and higher-spin $q$-KS models are sketched. As a first excursion into the thermodynamic properties of the spin-$1/2$ $q$-KS model, the dependence of the Curie temperature on the deformation parameter is studied through numerical analysis.