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arXiv:hep-th/0107047 · Czechoslovak Journal of Physics

Algebraic Geometry in Discrete Quantum Integrable Model and Baxter's T–Q Relation

Shao-Shiung Lin, Shi-Shyr Roan

Published Jun 1, 2002• 0 citations
arXivHTMLPDF
DOI
10.1023/a:1016233911940

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We study the diagonalization problem of certain discrete quantum integrable models by the method of Baxter’s T-Q relation from the algebraic geometry aspect. Among those the Hofstadter type model (with the rational magnetic flux), discrete quantum pendulum and discrete sine-Gordon model are our main concern in this report. By the quantum inverse scattering method, the Baxter’s T-Q relation is formulated on the associated spectral curve, a high genus Riemann surface in general, arisen from the study of the spectrum problem of the system. In the case of degenerated spectral curve where the spectral variables lie on rational curves, we obtain the complete and explicit solution of the T-Q polynomial equation associated to the model, and the intimate relation between the Baxter’s T-Q relation and algebraic Bethe Ansatz is clearly revealed. The algebraic geometry of a general spectral curve attached to the model and certain qualitative properties of solutions of the Baxter’s T-Q relation are discussed incorporating the physical consideration.
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.
In this paper we investigate the mathematical properties of the integrability of the symmetric six-vertex model towards the view of algebraic geometry. We show that the algebraic variety originated from Baxter’s commuting transfer method is birationally isomorphic to a ubiquitous threefold known as Segre cubic primal. This relation makes it possible to present the most generic solution for the Yang–Baxter triple associated to this lattice model. The respective -matrix and Lax operators are parameterized by three independent affine spectral variables.