Towards full instanton trans-series in Hofstadter's butterfly

TL;DR

This paper advances the understanding of the Harper-Hofstadter model by constructing its full energy trans-series, linking it to supersymmetric Wilson loops, and exploring its self-similarity and non-perturbative corrections.

Contribution

It introduces a complete energy trans-series for the Hofstadter butterfly, connecting quantum spectra to supersymmetric gauge theory and analyzing its self-similar structure.

Findings

Constructed the full energy trans-series for the Harper-Hofstadter model.

Linked the perturbative series to 5d SYM Wilson loops.

Provided high-precision numerical evidence for the trans-series.

Abstract

The trans-series completion of perturbative series of a wide class of quantum mechanical systems can be determined by combining the resurgence program and extra input coming from exact WKB analysis. In this paper, we reexamine the Harper-Hofstadter model and its spectrum, Hofstadter's butterfly, in light of recent developments. We demonstrate the connection between the perturbative energy series of the Harper-Hofstadter model and the vev of $1/2$-BPS Wilson loop of 5d SYM and clarify the differences between their non-perturbative corrections. Taking insights from the cosine potential model, we construct the full energy trans-series for flux $\phi=2\pi/Q$ and provide numerical evidence with remarkably high precision. Finally, we revisit the problem of self-similarity of the butterfly and discuss the possibility of a completed version of the Rammal-Wilkinson formula.