Spectral theory of extended Harper's model and a question by Erd\H{o}s and Szekeres

TL;DR

This paper completes the spectral analysis of the extended Harper's model across all parameter regions, demonstrating a spectral collapse phenomenon and addressing a problem posed by Erdős and Szekeres.

Contribution

It provides a full description of spectral measures for the extended Harper's model in the self-dual region, including a proof of spectral collapse and resolving a number theoretic problem.

Findings

Spectral collapse from absolutely continuous to singular continuous spectrum occurs when NNN interaction is symmetric.

Spectral measures are singular continuous in the remaining part of the self-dual region.

The analysis relies on number theoretic estimates solving a problem posed by Erdős and Szekeres.

Abstract

The extended Harper's model, proposed by D.J. Thouless in 1983, generalizes the famous almost Mathieu operator, allowing for a wider range of lattice geometries (parametrized by three coupling parameters) by permitting 2D electrons to hop to both nearest and next nearest neighboring (NNN) lattice sites, while still exhibiting its characteristic symmetry (Aubry duality). Previous understanding of the spectral theory of this model was restricted to two dual regions of the parameter space, one of which is characterized by the positivity of the Lyapunov exponent. In this paper, we complete the picture with a description of the spectral measures over the entire remaining (self-dual) region, for all irrational values of the frequency parameter (the magnetic flux in the model). Most notably, we prove that in the entire interior of this regime, the model exhibits a collapse from purely ac spectrum to purely sc spectrum when the NNN interaction becomes symmetric. In physics literature, extensive numerical analysis had indicated such "spectral collapse," however so far not even a heuristic argument for this phenomenon could be provided. On the other hand, in the remaining part of the self-dual region, the spectral measures are singular continuous irrespective of such symmetry. The analysis requires some rather delicate number theoretic estimates, which ultimately depend on the solution of a problem posed by Erd\H{o}s and Szekeres.