Spin generalizations of the classical elliptic Calogero--Inozemtsev model and integrable models of interacting tops associated with the $\rm{BC}_n$ root system

Abstract

We construct Lax pairs with a spectral parameter for a family of models of interacting tops associated with the $\rm{BC}_n$ root system. The construction uses the parametric $\rm{BC}_n$ associative Yang--Baxter equations, which involve both quantum $R$-matrices and boundary $K$-matrices satisfying the reflection equation. Special cases yield spin generalizations of the elliptic Calogero--Inozemtsev model. We also show how quantum Hamiltonians of interacting tops, combined with the Polychronakos freezing prescription, reproduce recently introduced anisotropic quantum long-range spin chains of Haldane--Shastry--Inozemtsev type.