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Journal of the Physical Society of Japan

The Quantum Volterra Model and the Lattice Sine–Gordon System. – Construction of the BaxterQOperator and the Integrals of Motion

Kazuhiro Hikami, Rei Inoue

Published Feb 15, 1999• 0 citations
Publisher

Abstract

We investigate the integrable structure of the quantum Volterra model, which is a discretization of the Korteweg–de Vries equation. Based on Sklyanin's method we derive the Baxter equation, and then clarify a role of the “fundamental transfer matrix”. We show that the fundamental transfer matrix also generates the integrals of motion for a lattice analogue of the sine–Gordon system.

DOI
10.1143/jpsj.68.380

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We investigate the integrable structure of the quantum Volterra model, which is a discretization of the Korteweg–de Vries equation. Based on Sklyanin's method we derive the Baxter equation, and then clarify a role of the “fundamental transfer matrix”. We show that the fundamental transfer matrix also generates the integrals of motion for a lattice analogue of the sine–Gordon system.
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We develop a method for computing form factors of local operators in the framework of Sklyanin's separation of variables (SOV) approach to quantum integrable systems.For that purpose, we consider the sine-Gordon model on a finite lattice and in finite dimensional cyclic representations as our main example.We first build our two central tools for computing matrix elements of local operators, namely, a generic determinant formula for the scalar products of states in the SOV framework and the reconstruction of local fields in terms of the separate variables.The general form factors are then obtained as sums of determinants of finite dimensional matrices, their matrix elements being given as weighted sums running over the separate variables and involving the Baxter Q-operator eigenvalues.
Using the Baxter's T-Q relation derived from the transfer matrix technique, we consider the diagonalization problem of discrete quantum pendulum and discrete quantum sine-Gordon Hamiltonian from the algebraic geometry aspect. For a finite chain system of the size L, when the spectral curve degenerates into rational curves, we have reduced Baxter's T-Q relation into a polynomial equation; the connection of T-Q polynomial equation with the algebraic Bethe Ansatz is clearly established . In particular, for L=4 it is the case of rational spectral curves for the discrete quantum pendulum and discrete sine-Gordon model. To these Baxter's T-Q polynomial equations, we have obtained the complete and explicit solutions with a detailed understanding of their quantitative and qualitative structure. In general the model possesses a spectral curve with a generic parameter. We have conducted certain qualitative study on the algebraic geometry of this high-genus Riemann surface incorporating Baxter's T-Q relation.
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Various aspects of the theory of quantum integrable systems are reviewed. Basic ideas behind the construction of integrable ultralocal and nonultralocal quantum models are explored by exploiting the underlying algebraic structures related to the Yang-Baxter equations. Physical meaning of abstract mathematical notions like universal R-matrix, quantized algebras, Sklyanin algebra, braided algebra, Hopf algebra etc. and the role played by them in integrable systems are highlighted. Systematic construction of quantum integrable lattice as well as field models and their exact excitation spectra are presented through examples. The coordinate and algebraic formulations of the Bethe ansatz are illustrated with comparison, along with the description of nested and functional Bethe ansatzes. The techniques for deriving quantum Hamiltonians from the Lax operators are demonstrated on concrete models. The exposition of this review is kept in a fairly elementary level with emphasis on the physical contents.