The algebraic geometry of Harper operators

TL;DR

This paper explores the algebraic geometry of Harper operators in one and two dimensions, analyzing their spectral properties, Bloch varieties, and density of states, revealing phenomena analogous to the Hofstadter butterfly.

Contribution

It introduces an algebraic geometric framework for Harper operators, describing Bloch varieties and spectral functions, and distinguishes rational and irrational parameter cases.

Findings

Bloch varieties are algebraic or ind-pro-varieties depending on parameters.

Density of states can be expressed via elliptic integrals in 2D.

Spectral density depends on parameters in 1D, illustrating Hofstadter butterfly.

Abstract

Following an approach developed by Gieseker, Kn\"orrer and Trubowitz for discretized Schr\"odinger operators, we study the spectral theory of Harper operators in dimension two and one, as a discretized model of magnetic Laplacians, from the point of view of algebraic geometry. We describe the geometry of an associated family of Bloch varieties and compute their density of states. Finally, we also compute some spectral functions based on the density of states. We discuss the difference between the cases with rational or irrational parameters: for the two dimensional Harper operator, the compactification of the Bloch variety is an ordinary variety in the rational case and an ind-pro-variety in the irrational case. This gives rise, at the algebro-geometric level of Bloch varieties, to a phenomenon similar to the Hofstadter butterfly in the spectral theory. In dimension two, the density of states can be expressed in terms of period integrals over Fermi curves, where the resulting elliptic integrals are independent of the parameters. In dimension one, for the almost Mathieu operator, with a similar argument we find the usual dependence of the spectral density on the parameter, which gives rise to the well known Hofstadter butterfly picture.