For any root system $\Delta$ and an irreducible representation ${ \cal R}$ of the reflection (Weyl) group $G_\Delta$ generated by $\Delta$, a { \em spin Calogero-Moser model} can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member $\mu$ of ${ \cal R}$, to be called a "site", we associate a vector space ${ \bf V}_{ \mu}$ whose element is called a "spin". Its dynamical variables are the canonical coordinates $\{ q_j,p_j\}$ of a particle in ${ \bf R}^r$, ($r=$ rank of $\Delta$), and spin exchange operators $\{ \hat{ \cal P}_\rho\}$ ($\rho\in\Delta$) which exchange the spins at the sites $\mu$ and $s_{ \rho}(\mu)$. Here $s_\rho$ is the reflection generated by $\rho$. For each $\Delta$ and ${ \cal R}$ a { \em 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 $\Delta=A_r$ and ${ \cal R}=$ 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 { \em degenerate} potentials.
Categories: hep-th cond-mat math-ph math.DS math.MP nlin.SI