Classical functional Bethe ansatz for SL(N): Separation of variables for the magnetic chain

Abstract

It is desirable for one to be able to construct separation variables for systems with SL(N) R matrices or Lax pairs. This article relates and extends the two known methods of construction. The Sklyanin’s functional Bethe ansatz (FBA) [Commun. Math. Phys. 150, 181–191 (1992)] was previously only known for SL(2) and SL(3) (and associated) R matrices. The algebraic geometric method of Adams, Harnad, and Hurtubise [Commun. Math. Phys. 155, 385–413 (1993)] has been shown to work for SL(N) but up until now only for linear Poisson brackets. In this article Sklyanin’s program is advanced 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 xi of ℬ(u) are the separation coordinates and the Pi=𝒜(xi) provide their conjugate momenta. It is shown that the separation variables thus defined are the same as those given (as the simultaneous solutions of several equations in two variables) by the other method. The crucial calculation of Adams et al. of the commutation relations is also adapted to the case of quadratic Poisson brackets. The method is illustrated with the magnetic chain and the Gaudin model, and its wider applicability is discussed. No knowledge of algebraic geometry is required.