Discrete Hirota’s equation in quantum integrable models

We show that the set of transfer matrices of an arbitrary fusion type for an integrable quantum model obeys these bilinear functional relations, which are identified with an integrable dynamical system on a Grassmann manifold (higher Hirota equation). The bilinear relations were previously known for a particular class of transfer matrices corresponding to rectangular Young diagrams. We extend this result for general Young diagrams. A general solution of the bilinear equations is presented.

arXiv:solv-int/9704015Modern Physics Letters AJun 21, 199716 citations

Functional methods

Gleb Eduardovich Arutyunov

Cambridge University Press eBooksApr 27, 20260 citations

Discrete Hirota’s equation in quantum integrable models

Anton Vladimirovich Zabrodin

Published Jan 1, 1996• 55 citations

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Abstract

Related papers

Discrete Hirota's Equation in Quantum Integrable Models

A. Zabrodin

The recent progress in revealing classical integrable structures in quantum models solved by Bethe ansatz is reviewed. Fusion relations for eigenvalues of quantum transfer matrices can be written in the form of classical Hirota's bilinear difference equation. This equation is also known as the completely discretized version of the 2D Toda lattice. We explain how one obtains the specific quantum results by solving the classical equation. The auxiliary linear problem for the Hirota equation is shown to generalize Baxter's T-Q relation.The recent progress in revealing classical integrable structures in quantum models solved by Bethe ansatz is reviewed. Fusion relations for eigenvalues of quantum transfer matrices can be written in the form of classical Hirota's bilinear difference equation. This equation is also known as the completely discretized version of the 2D Toda lattice. We explain how one obtains the specific quantum results by solving the classical equation. The auxiliary linear problem for the Hirota equation is shown to generalize Baxter's T-Q relation.

arXiv:hep-th/9610039International Journal of Modern Physics BOct 30, 199736 citations

Bethe Ansatz and Classical Hirota Equations

Anton Vladimirovich Zabrodin

A brief non-technical review of the recent study of classical integrable structures in quantum integrable systems is given. It is explained how to identify the standard objects of quantum integrable systems (transfer matrices, Baxter's $Q$ -operators, etc) with elements of classical non-linear integrable difference equations ( $τ$ -functions, Baker-Akhiezer functions, etc). The nested Bethe ansatz equations for $A_{k-1}$ -type models emerge as discrete time equations of motion for zeros of classical $τ$ -functions and Baker-Akhiezer functions. The connection with discrete time Ruijsenaars-Schneider system of particles is discussed.A brief non-technical review of the recent study of classical integrable structures in quantum integrable systems is given. It is explained how to identify the standard objects of quantum integrable systems (transfer matrices, Baxter's $Q$-operators, etc) with elements of classical non-linear integrable difference equations ($τ$-functions, Baker-Akhiezer functions, etc). The nested Bethe ansatz equations for $A_{k-1}$-type models emerge as discrete time equations of motion for zeros of classical $τ$-functions and Baker-Akhiezer functions. The connection with discrete time Ruijsenaars-Schneider system of particles is discussed.

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Bethe Ansatz and Classical Hirota Equation

Paul B. Wiegmann

We discuss an interrelation between quantum integrable models and classical soliton equations with discretized time. It appeared that spectral characteristics of quantum integrable systems may be obtained from entirely classical set up. namely, the eigenvalues of the quantum transfer matrix and the scattering S-matrix itself are identified with a certain τ-functions of the discrete Liouville equation. The Bethe ansatz equations are obtained as dynamics of zeros. For comparison we also present the Bethe ansatz equations for elliptic solutions of the classical discrete Sine-Gordon equation. The paper is based on the recent study of classical integrable structures in quantum integrable systems.1We discuss an interrelation between quantum integrable models and classical soliton equations with discretized time. It appeared that spectral characteristics of quantum integrable systems may be obtained from entirely classical set up. namely, the eigenvalues of the quantum transfer matrix and the scattering S-matrix itself are identified with a certain τ-functions of the discrete Liouville equation. The Bethe ansatz equations are obtained as dynamics of zeros. For comparison we also present the Bethe ansatz equations for elliptic solutions of the classical discrete Sine-Gordon equation. The paper is based on the recent study of classical integrable structures in quantum integrable systems.1

arXiv:cond-mat/9610132International Journal of Modern Physics BJan 20, 199719 citations

Classical tau-function for quantum spin chains

A. Alexandrov, Vladimir Kazakov, Sébastien Leurent +2 more

Abstract For an arbitrary generalized quantum integrable spin chain we introduce a “masterT-operator” which represents a generating function for commuting quantum transfer matrices constructed by means of the fusion procedure in the auxiliary space. We show that the functional relations for the transfer matrices are equivalent to an infinite set of model-independent bilinear equations of the Hirota form for the masterT-operator, which allows one to identify it withτ-function of an integrable hierarchy of classical soliton equations. In this paper we consider spin chains with rationalGL(N)-invariantR-matrices but the result is independent of a particular functional form of the transfer matrices and directly applies to quantum integrable models with more general (trigonometric and elliptic)R-matrices and to supersymmetric spin chains.Abstract For an arbitrary generalized quantum integrable spin chain we introduce a “masterT-operator” which represents a generating function for commuting quantum transfer matrices constructed by means of the fusion procedure in the auxiliary space. We show that the functional relations for the transfer matrices are equivalent to an infinite set of model-independent bilinear equations of the Hirota form for the masterT-operator, which allows one to identify it withτ-function of an integrable hierarchy of classical soliton equations. In this paper we consider spin chains with rationalGL(N)-invariantR-matrices but the result is independent of a particular functional form of the transfer matrices and directly applies to quantum integrable models with more general (trigonometric and elliptic)R-matrices and to supersymmetric spin chains.

arXiv:1112.3310Journal of High Energy PhysicsSep 1, 201360 citations

Fusion Rules for Quantum Transfer Matrices as a Dynamical System on Grassmann Manifolds

O. Lipan, Paul B. Wiegmann, A. Zabrodin

arXiv:solv-int/9704015Modern Physics Letters AJun 21, 199716 citations

Functional methods

Gleb Eduardovich Arutyunov

This chapter focuses on the derivation and interpretation of functional relations among commuting transfer matrices, which underpin a wide range of powerful techniques in quantum integrable systems. These include spectral equations, Baxter’s TQ-relations, and the analytic Bethe ansatz, as well as methods based on wronskian Bethe equations. Rather than isolated tools, these structures are intricately connected and reveal deep algebraic insights with significant implications, including quantum–classical duality and the completeness problem. Further emphasis is placed on the group-theoretic structure underlying these functional relations. Connections are drawn with the classical theory of characters, where objects such as Schur polynomials and Jacobi–Trudi formulae lead to bilinear relations naturally interpreted as Hirota equations in the context of transfer matrices. This perspective offers a conceptual bridge between quantum integrable models and classical representation theory, highlighting the unifying role of functional relations in both structure and application. Special attention is given to Q-functions governed by Baxter’s TQ-relations and, more broadly, by determinant identities from the wronskian formalism. These functions encode the transfer matrix eigenvalues and serve as fundamental algebraic objects. Functional methods thus offer powerful tools for exploring integrable models, especially when conventional Bethe ansatz techniques become impractical.This chapter focuses on the derivation and interpretation of functional relations among commuting transfer matrices, which underpin a wide range of powerful techniques in quantum integrable systems. These include spectral equations, Baxter’s TQ-relations, and the analytic Bethe ansatz, as well as methods based on wronskian Bethe equations. Rather than isolated tools, these structures are intricately connected and reveal deep algebraic insights with significant implications, including quantum–classical duality and the completeness problem. Further emphasis is placed on the group-theoretic structure underlying these functional relations. Connections are drawn with the classical theory of characters, where objects such as Schur polynomials and Jacobi–Trudi formulae lead to bilinear relations naturally interpreted as Hirota equations in the context of transfer matrices. This perspective offers a conceptual bridge between quantum integrable models and classical representation theory, highlighting the unifying role of functional relations in both structure and application. Special attention is given to Q-functions governed by Baxter’s TQ-relations and, more broadly, by determinant identities from the wronskian formalism. These functions encode the transfer matrix eigenvalues and serve as fundamental algebraic objects. Functional methods thus offer powerful tools for exploring integrable models, especially when conventional Bethe ansatz techniques become impractical.

Cambridge University Press eBooksApr 27, 20260 citations

Q

-operators, etc) with elements of classical non-linear integrable difference equations (

τ

-functions, Baker-Akhiezer functions, etc). The nested Bethe ansatz equations for

A_{k-1}

-type models emerge as discrete time equations of motion for zeros of classical

τ

-functions and Baker-Akhiezer functions. The connection with discrete time Ruijsenaars-Schneider system of particles is discussed.