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Cambridge University Press eBooks

Separation of variables

Gleb Eduardovich Arutyunov

Published Apr 27, 2026• 0 citations
Publisher

Abstract

This chapter introduces an alternative framework for solving spectral problems in classical and quantum integrable systems via separation of variables, grounded in the Hamilton–Jacobi theory. The method is first outlined in the classical context and then extended to quantum integrable spin chains with gl ell symmetry and associated Yangians. A key objective is to construct separated representations for the spectrum, reducing multidimensional eigenvalue problems to a set of one-dimensional ones solvable through algebraic and representation-theoretic techniques. This approach proves especially effective in cases where the completeness of the Bethe ansatz is unclear, offering complementary insights. Focusing on the classical gl ell magnets, the chapter details the construction of separated variables using the Baker–Akhiezer function and its role in identifying canonical coordinates on the spectral curve. In the quantum case, complete diagonalization of the gl 2 spin chain is achieved through separation of variables. For higher-rank models and generic local representations of gl ell , the full construction remains an open and active area of research. The chapter closes with an overview of current developments and challenges in advancing this method.

DOI
10.1017/9781009664837.010

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This chapter introduces an alternative framework for solving spectral problems in classical and quantum integrable systems via separation of variables, grounded in the Hamilton–Jacobi theory. The method is first outlined in the classical context and then extended to quantum integrable spin chains with gl ell symmetry and associated Yangians. A key objective is to construct separated representations for the spectrum, reducing multidimensional eigenvalue problems to a set of one-dimensional ones solvable through algebraic and representation-theoretic techniques. This approach proves especially effective in cases where the completeness of the Bethe ansatz is unclear, offering complementary insights. Focusing on the classical gl ell magnets, the chapter details the construction of separated variables using the Baker–Akhiezer function and its role in identifying canonical coordinates on the spectral curve. In the quantum case, complete diagonalization of the gl 2 spin chain is achieved through separation of variables. For higher-rank models and generic local representations of gl ell , the full construction remains an open and active area of research. The chapter closes with an overview of current developments and challenges in advancing this method.
. 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...
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.
This article, based on the author's PhD thesis, reviews recent advancements in the field of quantum integrability, in particular the separation of variables (SoV) program for high-rank integrable spin chains and the boost mechanism for solving the Yang-Baxter equation. We begin with a general overview of quantum integrable systems with special emphasis on their description in terms of quantum algebras. We then provide a detailed account of the Yangian of gl(n)\mathfrak{gl}(n)gl(n) in particular the Bethe algebra, fusion, and T- and Q-systems. We then introduce the notion of separation of variables in integrable systems and build on Sklyanin's work in rank 1 models and extend to higher rank. By exploiting a novel link between SoV and quantum algebra representation theory we construct the separated variables for gl(n)\mathfrak{gl}(n)gl(n) spin chains for arbitrary compact representations of the symmetry algebra and develop various new tools along the way. Next, we build on the previous part and develop a new technique for the computation of scalar products in the SoV framework which we call Functional SoV or FSoV. Unlike the work in the previous, operatorial, part this approach is based on the Baxter TQ equations. After developing this technique we supplement it with a new operator construction providing a unified view of functional and operatorial SoV. Then, we generalise the results of the previous part from compact spin chains to non-compact spin chains. The final part of this work is based on the development of tools for solving the Yang- Baxter equation. We develop a bottom-up approach for this based on the so-called Boost automorphism and uses the spin chain Hamiltonian as a starting point. Our approach allows us to classify numerous families of solutions in particular a complete classification of 4 x 4 solutions which preserve fermion number which have applications in the AdS/CFT correspondence.
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 describe the extension, beyond fundamental representations of the Yang-Baxter algebra, of our new construction of separation of variables bases for quantum integrable lattice models. The key idea underlying our approach is to use the commuting conserved charges of the quantum integrable models to generate bases in which their spectral problem is separated, i.e. in which the wave functions are factorized in terms of specific solutions of a functional equation. For the so-called “non-fundamental” models we construct two different types of SoV bases. The first is given from the fundamental quantum Lax operator having isomorphic auxiliary and quantum spaces and that can be obtained by fusion of the original quantum Lax operator. The construction essentially follows the one we used previously for fundamental models and allows us to derive the simplicity and diagonalizability of the transfer matrix spectrum. Then, starting from the original quantum Lax operator and using the full tower of the fused transfer matrices, we introduce a second type of SoV bases for which the proof of the separation of the transfer matrix spectrum is naturally derived. We show that, under some special choice, this second type of SoV bases coincides with the one associated to Sklyanin’s approach. Moreover, we derive the finite difference type (quantum spectral curve) functional equation and the set of its solutions defining the complete transfer matrix spectrum. This is explicitly implemented for the integrable quantum models associated to the higher spin representations of the general quasi-periodic Y(gl_{2}) Y ( g l 2 ) Yang-Baxter algebra. Our SoV approach also leads to the construction of a Q Q -operator in terms of the fused transfer matrices. Finally, we show that the Q Q -operator family can be equivalently used as the family of commuting conserved charges enabling to construct our SoV bases.
The analysis of the transfer matrices associated to the most general representations of the 8-vertex reflection algebra on spin-1/2 chains is here implemented by introducing a quantum separation of variables (SOV) method which generalizes to these integrable quantum models the method first introduced by Sklyanin. More in detail, for the representations reproducing in their homogeneous limits the open XYZ spin-1/2 quantum chains with the most general integrable boundary conditions, we explicitly construct representations of the 8-vertex reflection algebras for which the transfer matrix spectral problem is separated. Then, in these SOV representations we get the complete characterization of the transfer matrix spectrum (eigenvalues and eigenstates) and its non-degeneracy. Moreover, we present the first fundamental step toward the characterization of the dynamics of these models by deriving determinant formulae for the matrix elements of the identity on separated states, which apply in particular to transfer matrix eigenstates. The comparison of our analysis for the 8-vertex reflection algebra with that of [1, 2] for the 6-vertex one leads to the interesting remark that a profound similarity in both the characterization of the spectral problems and of the scalar products exists for these two different realizations of the reflection algebra once they are described by SOV method. As it will be shown in a future publication, this remarkable similarity will be at the basis of the simultaneous determination of form factors of local operators of integrable quantum models associated to general reflection algebra representations of both 8-vertex and 6-vertex type.