Universal Lax pairs for Spin Calogero-Moser Models and Spin Exchange Models

Abstract

For any root system ∆ and an irreducible representation R of the reflection (Weyl) group G ∆ generated by ∆, a spin Calogero-Moser model can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member µ of R, to be called a “site”, we associate a vector space Vµ whose element is called a “spin”. Its dynamical variables are the canonical coordinates {qj,pj} of a particle in R r, (r = rank of ∆), and spin exchange operators { ˆ Pρ} (ρ ∈ ∆) which exchange the spins at the sites µ and sρ(µ). Here sρ is the reflection generated by ρ. For each ∆ and R a spin exchange model can be defined. The Hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only. It is obtained by “freezing ” the canonical variables at the equilibrium point of the corresponding classical Calogero-Moser model. For ∆ = Ar and R = vector representation it reduces to the well-known Haldane-Shastry model. Universal Lax pair operators for both spin Calogero-Moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for degenerate potentials.