Instantons in the Hofstadter butterfly: difference equation, resurgence and quantum mirror curves

TL;DR

This paper investigates the non-perturbative spectrum of the Harper-Hofstadter Hamiltonian, revealing instanton effects, their explicit calculations, and their role in resolving ambiguities in the quantum butterfly spectrum.

Contribution

It provides an explicit instanton analysis of the Hofstadter butterfly spectrum, including fluctuation determinants and the connection to topological string theory.

Findings

Instanton corrections match numerical butterfly spectrum data.

Explicit calculation of one-loop fluctuation determinants.

Ambiguities in instanton-anti-instanton contributions cancel perturbative series ambiguities.

Abstract

We study the Harper-Hofstadter Hamiltonian and its corresponding non-perturbative butterfly spectrum. The problem is algebraically solvable whenever the magnetic flux is a rational multiple of $2\pi$. For such values of the magnetic flux, the theory allows a formulation with two Bloch or $\theta$-angles. We treat the problem by the path integral formulation, and show that the spectrum receives instanton corrections. Instantons as well as their one loop fluctuation determinants are found explicitly and the finding is matched with the numerical band width of the butterfly spectrum. We extend the analysis to all 2-instanton sectors with different $\theta$-angle dependence to leading order and show consistency with numerics. We further argue that the instanton--anti-instanton contributions are ambiguous and cancel the ambiguity of the perturbation series, as they should. We hint at the possibility of exact 2-instanton solutions responsible for such contributions via Picard-Lefschetz theory. We also present a powerful way to compute the perturbative fluctuations around the 1-instanton saddle as well as the instanton--anti-instanton ambiguity by using the topological string formulation.