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arXiv:2010.08077 · SciPost Physics

Characteristic determinant and Manakov triple for the double elliptic integrable system

A. Grekov, Andrei Zotov

Published Mar 4, 2021• 9 citations
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Abstract

Using the intertwining matrix of the IRF-Vertex correspondence we propose a determinant representation for the generating function of the commuting Hamiltonians of the double elliptic integrable system. More precisely, it is a ratio of the normally ordered determinants, which turns into a single determinant in the classical case. With its help we reproduce the recently suggested expression for the eigenvalues of the Hamiltonians for the dual to elliptic Ruijsenaars model. Next, we study the classical counterpart of our construction, which gives expression for the spectral curve and the corresponding L-matrix. This matrix is obtained explicitly as a weighted average of the Ruijsenaars and/or Sklyanin type Lax matrices with the weights as in the theta function series definition. By construction the L-matrix satisfies the Manakov triple representation instead of the Lax equation. Finally, we discuss the factorized structure of the L-matrix.

DOI
10.21468/scipostphys.10.3.055

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Using the intertwining matrix of the IRF-Vertex correspondence we propose a determinant representation for the generating function of the commuting Hamiltonians of the double elliptic integrable system. More precisely, it is a ratio of the normally ordered determinants, which turns into a single determinant in the classical case. With its help we reproduce the recently suggested expression for the eigenvalues of the Hamiltonians for the dual to elliptic Ruijsenaars model. Next, we study the classical counterpart of our construction, which gives expression for the spectral curve and the corresponding L-matrix. This matrix is obtained explicitly as a weighted average of the Ruijsenaars and/or Sklyanin type Lax matrices with the weights as in the theta function series definition. By construction the L-matrix satisfies the Manakov triple representation instead of the Lax equation. Finally, we discuss the factorized structure of the L-matrix.
Abstract We describe the most general GL NM classical elliptic finite-dimensional integrable system, which Lax matrix has n simple poles on elliptic curve. For M = 1 it reproduces the classical inhomogeneous spin chain, for N = 1 it is the Gaudin type (multispin) extension of the spin Ruijsenaars–Schneider model, and for n = 1 the model of M interacting relativistic GL N tops emerges in some particular case. In this way we present a classification for relativistic Gaudin models on GL-bundles over elliptic curve. As a by-product we describe the inhomogeneous Ruijsenaars chain. We show that this model can be considered as a particular case of multispin Ruijsenaars–Schneider model when residues of the Lax matrix are of rank one. An explicit parametrization of the classical spin variables through the canonical variables is obtained for this model. Finally, the most general GL NM model is also described through R -matrices satisfying associative Yang–Baxter equation. This description provides the trigonometric and rational analogues of GL NM models.
In the present paper we construct separation of variables (SoV) for all Lax-integrable systems, two by two Lax matrix of which enjoys classical reflection equation algebra with the elliptic r−s matrices. We show that, similar to the cases of SoV for the classical XYZ and XXZ models [1], [2], the constructed SoV admits two types of momenta, which are important in the quantum case [1], [2]. We consider two examples of such the integrable hamiltonian systems governed by the simplest Poisson brackets connected with classical reflection equation algebra with the elliptic r−s matrices. They are the classical Sklyanin algebra and its four-dimensional extension which provide a quadratic structure for the classical Steklov top. We explicitly construct the variables of separation, Abel equations and reconstruction formulae for the corresponding integrable models.
Algebraic Bethe ansatz for the elliptic quantum group Eτ,η(sl2)E_{\tau,\eta}(sl_2)Eτ,η​(sl2​)
To each representation of the elliptic quantum group Eτ,η(sl2)E_{\tau,\eta}(sl_2)Eτ,η​(sl2​) is associated a family of commuting transfer matrices. We give common eigenvectors by a version of the algebraic Bethe ansatz method. Special cases of this construction give eigenvectors for IRF models, for the eight-vertex model and for the two-body Ruijsenaars operator. The latter is a qqq-deformation of Hermite's solution of the Lam\'e equation.
Fukai and Yamada introduced a Lax operator with a three-dimensional auxiliary space for the spin-1/2 XYZ chain whose entries are algebraic rather than elliptic. We give an explicit algebraic 9 × 9 intertwiner for this operator. The matrix has the 21-vertex pattern and is written directly in terms of the two spectral parameters and the three XYZ couplings. Direct reduction in the radical function field proves the local RLL relation. We also prove an exchange-unitarity identity, generic uniqueness up to scalar, and regularity after a scalar renormalization. Its invariant blocks give a factorization of the characteristic polynomial and show that, on the common finite chart, the rank-loss locus is exactly 1 − R² = 0. We make the constant gauge to the fused eight-vertex matrix explicit and obtain regular XXZ and scaled rational XXX limits. The result supplies the local intertwiner left implicit by the fusion argument; it does not constitute a new integrable model.
We describe classical top-like integrable systems arising from the quantum exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed in terms of the quantum non-dynamical R-matrix even at the classical level, where the Planck constant plays the role of the relativistic deformation parameter in the sense of Ruijsenaars and Schneider (RS). The integrable systems (relativistic tops) are described as multidimensional Euler tops, and the inertia tensors are written in terms of the quantum and classical R-matrices. A particular case of gl N system is gauge equivalent to the N-particle RS model while a generic top is related to the spin generalization of the RS model. The simple relation between quantum R-matrices and classical Lax operators is exploited in two ways. In the elliptic case we use the Belavin’s quantum R-matrix to describe the relativistic classical tops. Also by the passage to the noncommutative torus we study the large N limit corresponding to the relativistic version of the nonlocal 2d elliptic hydrodynamics. Conversely, in the rational case we obtain a new gl N quantum rational non-dynamical R-matrix via the relativistic top, which we get in a different way — using the factorized form of the RS Lax operator and the classical Symplectic Hecke (gauge) transformation. In particular case of gl2 the quantum rational R-matrix is 11-vertex. It was previously found by Cherednik. At last, we describe the integrable spin chains and Gaudin models related to the obtained R-matrix.