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Universal Lax pairs for Spin Calogero-Moser Models and Spin Exchange Models | | V. I. Inozemtsev
; R. Sasaki
; | | Date: |
17 May 2001 | | Journal: | J.Phys. A34 (2001) 7621-7632 | | Subject: | High Energy Physics - Theory; Mathematical Physics; Exactly Solvable and Integrable Systems; Dynamical Systems | hep-th cond-mat math-ph math.DS math.MP nlin.SI | | Abstract: | 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 ${f V}_{mu}$ whose element is called a "spin". Its dynamical variables are the canonical coordinates ${q_j,p_j}$ of a particle in ${f R}^r$, ($r=$ rank of $Delta$), and spin exchange operators ${hat{cal P}_
ho}$ ($
hoinDelta$) which exchange the spins at the sites $mu$ and $s_{
ho}(mu)$. Here $s_
ho$ is the reflection generated by $
ho$. 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. | | Source: | arXiv, hep-th/0105164 | | Services: | Forum | Review | PDF | Favorites | |
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