Elliptic Ruijsenaars-Toda and elliptic Toda chains: classical r-matrix structure and relation to XYZ chain

TL;DR

This paper analyzes elliptic Toda and Ruijsenaars-Toda chains, deriving their classical r-matrix structures and revealing their gauge equivalence to XYZ spin chains with specific parameters.

Contribution

It demonstrates how these chains are special cases of the elliptic Ruijsenaars chain and establishes their gauge equivalence to XYZ models.

Findings

Elliptic Toda and Ruijsenaars-Toda chains are derived from the elliptic Ruijsenaars chain.

Classical r-matrix structures are explicitly obtained for these chains.

Elliptic Ruijsenaars-Toda chain is gauge equivalent to the XYZ Landau-Lifshitz model.

Abstract

We discuss the classical elliptic Toda chain introduced by Krichever and the elliptic Ruijsenaars-Toda chain introduced by Adler, Shabat and Suris. It is shown that these models can be obtained as particular cases of the elliptic Ruijsenaars chain. We explain how the classical $r$-matrix structures are derived for these chains. Also, as a by-product, we prove that the elliptic Ruijsenaars-Toda chain is gauge equivalent to discrete Landau-Lifshitz model of XYZ type. The elliptic Toda chain is also gauge equivalent to XYZ chain with special values of the Casimir functions at each site.