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

TL;DR

This paper explores the algebraic geometric approach to solving the spectral problem of discrete quantum integrable models using Baxter's T-Q relation, revealing explicit solutions and their relation to algebraic Bethe Ansatz.

Contribution

It provides a detailed algebraic geometric analysis of Baxter's T-Q relation for discrete quantum models, including explicit solutions in special cases and insights into spectral curve properties.

Findings

Explicit solutions for T-Q polynomial equations in rational spectral curve cases

Clear connection established between Baxter's T-Q relation and algebraic Bethe Ansatz

Discussion of spectral curve geometry and solution properties for general models

Abstract

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.