We extend Sklyanin's method of separation of variables to quantum integrable models associated to elliptic curves. After reviewing the differential case, the elliptic Gaudin model studied by Enriquez, Feigin and Rubtsov, we consider the difference case and find a class of transfer matrices whose eigenvalue problem can be solved by separation of variables. These transfer matrices are associated to representations of the elliptic quantum group E , ( sl 2 ) by difference operators. One model of statistical mechanics to which this method applies is the interaction-round-a-face model with antiperiodic boundary conditions. The eigenvalues of the transfer matrix are given as solutions of a system of quadratic equations in a space of higher-order theta functions.
We extend Sklyanin's method of separation of variables to quantum integrable models associated to elliptic curves. After reviewing the differential case, the elliptic Gaudin model studied by Enriquez, Feigin and Rubtsov, we consider the difference case and find a class of transfer matrices whose eigenvalue problem can be solved by separation of variables. These transfer matrices are associated to representations of the elliptic quantum group E , ( sl 2 ) by difference operators. One model of statistical mechanics to which this method applies is the interaction-round-a-face model with antiperiodic boundary conditions. The eigenvalues of the transfer matrix are given as solutions of a system of quadratic equations in a space of higher-order theta functions.
This work deals on the one hand with how to use the representation theory of a quantum group ( [14]) to investigate a statistical mechanical model and on the other hand with how to solve the statistical mechanical model by Sklyanin's method of separation of variables.[47].More concretely, we use the elliptic quantum group E y,'I)(sl2) established by Felder in [29, 25] to investigate the SOS eight-vertex model established by Date, Jimbo, Miwa and Okado [10] with antiperiodic boundary conditions which are the reason that Bethe ansatz fails and we have to use Sklyanin's method of separation of variables [47].The SOS eight-vertex model is a face-model version of Baxter's original eight-vertex model [3].It is related to the elliptic quantum group E y,'I)(sl2), since we rediscover its Boltzmann weights We(c, b, a, dlz) by a suitably discretized version of the R-matrix Re(z,.\)defining E Y,'I)(sl2)' This relation reads
Abstract We implement our new separation of variables (SoV) approach for open quantum integrable models associated to higher rank representations of the reflection algebras. We construct the (SoV) basis for the fundamental representations of the Y ( gl n ) reflection algebra associated to general integrable boundary conditions. Moreover, we give the conditions on the boundary K -matrices allowing for the transfer matrix to be diagonalizable with simple spectrum. These SoV basis are then used to completely characterize the transfer matrix spectrum for the rank one and two reflection algebras. The rank one case is developed for both the rational and trigonometric fundamental representations of the 6-vertex reflection algebra. On the one hand, we extend the complete spectrum characterization to representations previously lying outside the SoV approach, e.g. those for which the standard algebraic Bethe Ansatz applies. On the other hand, we show that our new SoV construction can be reduced to the generalized Sklyanin’s one whenever it is applicable. The rank two case is developed explicitly for the fundamental representations of the Y ( gl 3 ) reflection algebra associated to general integrable boundary conditions. For both rank one and two our SoV approach leads to a complete characterization of the transfer matrix spectrum in terms of a set of polynomial solutions to the corresponding quantum spectral curve equation. Those are finite difference functional equations of order equal to the rank plus one, i.e. here two and three respectively for the Y ( gl 2 ) and Y ( gl 3 ) reflection algebras.
We present a new approach to construct the separate variables basis leading to the full characterization of the transfer matrix spectrum of quantum integrable lattice models. The basis is generated by the repeated action of the transfer matrix itself on a generically chosen state of the Hilbert space. The fusion relations for the transfer matrix, stemming from the Yang-Baxter algebra properties, provide the necessary closure relations to define the action of the transfer matrix on such a basis in terms of elementary local shifts, leading to a separate transfer matrix spectral problem. Hence our scheme extends to the quantum case a key feature of the Liouville-Arnold classical integrability framework where the complete set of conserved charges defines both the level manifold and the flows on it leading to the construction of action-angle variables. We work in the framework of the quantum inverse scattering method. As a first example of our approach, we give the construction of such a basis for models associated with Y(gln) and argue how it extends to their trigonometric and elliptic versions. Then we show how our general scheme applies concretely to fundamental models associated with the Y(gl2) and Y(gl3) R-matrices leading to the full characterization of their spectrum. For Y(gl2) and its trigonometric deformation, a particular case of our method reproduces Sklyanin’s construction of separate variables. For Y(gl3), it gives new results, in particular, through the proper identification of the shifts acting on the separate basis. We stress that our method also leads to the full characterization of the spectrum of other known quantum integrable lattice models, including, in particular, trigonometric and elliptic spin chains, open chains with general integrable boundaries, and further higher rank cases that we will describe in forthcoming publications.
We apply our new approach of quantum Separation of Variables (SoV) to the complete characterization of the transfer matrix spectrum of quantum integrable lattice models associated to \bm{gl_n} 𝐠 𝐥 𝐧 -invariant \bm{R} 𝐑 -matrices in the fundamental representations. We consider lattices with \bm{N} 𝐍 -sites and general quasi-periodic boundary conditions associated to an arbitrary twist matrix \bm{K} 𝐊 having simple spectrum (but not necessarily diagonalizable). In our approach the SoV basis is constructed in an universal manner starting from the direct use of the conserved charges of the models, e.g. from the commuting family of transfer matrices. Using the integrable structure of the models, incarnated in the hierarchy of transfer matrices fusion relations, we prove that our SoV basis indeed separates the spectrum of the corresponding transfer matrices. Moreover, the combined use of the fusion rules, of the known analytic properties of the transfer matrices and of the SoV basis allows us to obtain the complete characterization of the transfer matrix spectrum and to prove its simplicity. Any transfer matrix eigenvalue is completely characterized as a solution of a so-called quantum spectral curve equation that we obtain as a difference functional equation of order \bm{n} 𝐧 . Namely, any eigenvalue satisfies this equation and any solution of this equation having prescribed properties that we give leads to an eigenvalue. We construct the associated eigenvector, unique up to normalization, of the transfer matrices by computing its decomposition on the SoV basis that is of a factorized form written in terms of the powers of the corresponding eigenvalues. Finally, if the twist matrix \bm{K} 𝐊 is diagonalizable with simple spectrum we prove that the transfer matrix is also diagonalizable with simple spectrum. In that case, we give a construction of the Baxter \bm{Q} 𝐐 -operator and show that it satisfies a \bm{T} 𝐓 - \bm{Q} 𝐐 equation of order \bm{n} 𝐧 , the quantum spectral curve equation, involving the hierarchy of the fused transfer matrices.
Abstract In this paper we apply our new separation of variables approach to completely characterize the transfer matrix spectrum for quantum integrable lattice models associated to fundamental evaluation representations of with quasi-periodic boundary conditions. We consider here the case of generic deformations associated to a parameter q which is not a root of unity. The separation of variables (SoV) basis for the transfer matrix spectral problem is generated by using the action of the transfer matrix itself on a generic co-vector of the Hilbert space, following the general procedure described in our paper (Maillet and Niccoli 2018 J. Math. Phys . 59 091417). Such a SoV construction allows to prove that for general values of the parameters defining the model the transfer matrix is diagonalizable and with simple spectrum for any twist matrix which is also diagonalizable with simple spectrum. Then, using together the knowledge of such a SoV basis and of the fusion relations satisfied by the hierarchy of transfer matrices, we derive a complete characterization of the transfer matrix eigenvalues and eigenvectors as solutions of a system of polynomial equations of order n + 1. Moreover, we show that such a SoV discrete spectrum characterization is equivalently reformulated in terms of a finite difference functional equation, the quantum spectral curve equation, under a proper choice of the set of its solutions. A construction of the associated Q -operator induced by our SoV approach is also presented.
. For the elliptic Gaudin model (a degenerate case of XYZ integrable spin chain) a separation of variables is constructed in the classical case. The corresponding separated coordinates are obtained as the poles of a suitably normalized Baker-Akhiezer function. The classical results are generalized to the quantum case where the kernel of separating integral operator is constructed. The simplest onedegree -of-freedom case is studied in detail. 1 Introduction The quantum elliptic (or XYZ) Gaudin model was introduced in [1], see also [2], as a limiting case of the integrable XYZ spin chain [3]. The commuting Hamiltonians H n of the model are expressed as quadratic combinations of sl 2 spin operators. Determining the spectrum of H n turned out to be a difficult problem like the original XYZ spin chain. Let us list the known facts related to this problem. ffl A solution by means of Algebraic Bethe Ansatz has been obtained only recently [4]. See also [5] ffl As shown in [6], in the SU(2...