Algebraic Geometry and Hofstadter Type Model

TL;DR

This paper explores the algebraic geometry underlying Hofstadter models by solving Bethe equations on high genus spectral curves, providing explicit solutions in special cases and discussing geometric properties relevant to the physical model.

Contribution

It introduces a novel algebraic geometric approach to solving Bethe equations for Hofstadter models, especially on high genus curves, linking physical models with complex algebraic structures.

Findings

Explicit solutions for Bethe equations on rational curves

Analysis of algebraic geometry properties on high genus curves

Connection between spectral curve geometry and physical Hamiltonians

Abstract

In this report, we study the algebraic geometry aspect of Hofstadter type models through the algebraic Bethe equation. In the diagonalization problem of certain Hofstadter type Hamiltonians, the Bethe equation is constructed by using the Baxter vectors on a high genus spectral curve. When the spectral variables lie on rational curves, we obtain the complete and explicit solutions of the polynomial Bethe equation; the relation with the Bethe ansatz of polynomial roots is discussed. Certain algebraic geometry properties of Bethe equation on the high genus algebraic curves are discussed in cooperation with the consideration of the physical model.