Bloch and Bethe ansatze for the Harper model: A butterfly with a boundary

TL;DR

This paper develops an analytical framework for the Harper model with arbitrary magnetic flux, revealing boundary-dependent edge states, a universal bulk spectrum, and introducing a Bethe ansatz applicable to different geometries.

Contribution

It presents a generalized Bloch ansatz for the Harper model, analytically solves the equations, and introduces a Bethe ansatz applicable to various geometries, advancing understanding of edge and bulk spectra.

Findings

Edge state energies are independent of cylinder length.

Bulk spectrum depends on a single spectral parameter and is identical for cylinder and torus.

The bulk spectrum forms a Cantor set in irrational magnetic fields.

Abstract

Based on a recent generalization of Bloch's theorem, we present a Bloch ansatz for the Harper model with an arbitrary rational magnetic flux in various geometries, and solve the associated ansatz equations analytically. In the case of a cylinder and a particular boundary condition, we find that the energy spectrum of edge states has no dependence on the length of the cylinder, which allows us to construct a quasi-one-dimensional edge theory that is exact and describes two edges simultaneously. We prove that energies of bulk states, generating the so-called Hofstadter's butterfly, depend on a single geometry-dependent spectral parameter and have exactly the same functional form for the cylinder and the torus with general twisted boundary conditions, and argue that the (edge) bulk spectrum of a semi-infinite cylinder in an irrational magnetic field is (the complement of) a Cantor set. Finally, realizing that the bulk projection of the Harper Hamiltonian is a linear form over a deformed Weyl algebra, we introduce a Bethe ansatz valid for both cylinder and torus geometries.