R-matrix-valued Lax pairs and long-range spin chains

Abstract

In this paper we discuss $R$-matrix-valued Lax pairs for ${\mathrm{sl}}_N$ Calogero-Moser model and their relation to integrable quantum long-range spin chains of the Haldane-Shastry-Inozemtsev type. First, we construct the $R$-matrix-valued Lax pairs for the third flow of the classical Calogero-Moser model. Then we notice that the scalar parts (in the auxiliary space) of the $M$-matrices corresponding to the second and third flows have form of special spin exchange operators. The freezing trick restricts them to quantum Hamiltonians of long-range spin chains. We show that for a special choice of the $R$-matrix these Hamiltonians reproduce those for the Inozemtsev chain. In the general case related to the Baxter's elliptic $R$-matrix we obtain a natural anisotropic extension of the Inozemtsev chain. Commutativity of the Hamiltonians is verified numerically. Trigonometric limits lead to the Haldane-Shastry chains and their anisotropic generalizations.

12 pages, Introduction added, minor corrections

References in corpus (5)

• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• On AGT relation in the case of U(3)
• Physics and Mathematics of Calogero particles
• Resurrecting the partially isotropic Haldane-Shastry model
• On R-matrix valued Lax pairs for Calogero-Moser models

Cited by in corpus (15)

• A slow review of the AGT correspondence
• Quantum chaos and ensemble inequivalence of quantum long-range Ising chains
• Trigonometric integrable tops from solutions of associative Yang-Baxter equation
• Generalized model of interacting integrable tops
• Resurrecting the partially isotropic Haldane-Shastry model
• On R-matrix valued Lax pairs for Calogero-Moser models
• Anisotropic spin generalization of elliptic Macdonald-Ruijsenaars operators and R-matrix identities
• Elliptic generalization of integrable q-deformed anisotropic Haldane-Shastry long-range spin chain
• Integrable System of Generalized Relativistic Interacting Tops
• Quadratic algebras based on SL(NM) elliptic quantum R-matrices
• Quasiballistic transport in long-range anisotropic Heisenberg model
• Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation
• Odd supersymmetrization of elliptic R-matrices
• R-matrix valued Lax pair for elliptic Calogero-Inozemtsev system and associative Yang-Baxter equations of ${\mathrm{BC}}_n$ type
• Multi-pole extension for elliptic models of interacting integrable tops