Classical Functional Bethe Ansatz for $SL(N)$: separation of variables for the magnetic chain

TL;DR

This paper extends Sklyanin's Functional Bethe Ansatz to $SL(N)$ systems, providing a new method for separation of variables applicable to magnetic chains and Gaudin models, with explicit construction of separation variables.

Contribution

The paper develops a generalized FBA for $SL(N)$ $R$-matrices, advancing the separation of variables method beyond $SL(2)$ and $SL(3)$ cases.

Findings

Constructed rational functions $ extA(u)$ and $ extB(u)$ for $SL(N)$ systems.

Identified zeros of $ extB(u)$ as separation coordinates.

Demonstrated the method on magnetic chain and Gaudin model.

Abstract

The Functional Bethe Ansatz (FBA) proposed by Sklyanin is a method which gives separation variables for systems for which an $R$-matrix is known. Previously the FBA was only known for $SL(2)$ and $SL(3)$ (and associated) $R$-matrices. In this paper I advance Sklyanin's program by giving the FBA for certain systems with $SL(N)$ $R$-matrices. This is achieved by constructing rational functions $\A(u)$ and $\B(u)$ of the matrix elements of $T(u)$, so that, in the generic case, the zeros $x_i$ of $\B(u)$ are the separation coordinates and the $P_i=\A(x_i)$ provide their conjugate momenta. The method is illustrated with the magnetic chain and the Gaudin model, and its wider applicability is discussed.