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

TL;DR

This paper develops $R$-matrix-valued Lax pairs for the Calogero-Moser model and links them to long-range quantum spin chains, including Inozemtsev and Haldane-Shastry types, revealing new integrable structures and extensions.

Contribution

It constructs $R$-matrix-valued Lax pairs for the Calogero-Moser model and connects them to long-range spin chains, introducing anisotropic extensions and verifying integrability.

Findings

Reproduces Inozemtsev chain Hamiltonians from $R$-matrix choices.

Introduces anisotropic extensions of the Inozemtsev chain.

Verifies commutativity of Hamiltonians numerically.

Abstract

In this paper we discuss $R$-matrix-valued Lax pairs for ${\rm sl}_N$ 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.