Towards full instanton trans-series in Hofstadter's butterfly

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 $φ=2π/Q$ 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.

Figures (9)

• Figure 2.1: Hofstadter. We plot the band structure for $P/Q$ with $(P,Q)=1$ and $Q$ up to 60.
• Figure 3.1: Classically allowed and forbidden regions
• Figure 3.2: Borel singularities for Wilson loop vev in the 5d SYM theory with (a) $z<1/16$ and (b) $z>1/16$ respectively. In the left figure with $z<1/16$, the Borel singularities marked by black dots have charge vectors $\gamma = (2,-1,0)$ (on the real axis), $(2,0,0)$ (slightly away), $(2,-1,1)$ (far off in the first quadrant). In the right figure with $z>1/16$, the Borel singularities marked by black dots have charge vectors $\gamma = (2,0,0)$ (on the positive real axis), $(2,1,1)$ (far off in the first quadrant).
• Figure 4.1: Borel singularities of perturbative energy series at Landau levels $N=0,1,2$. The singularities marked by black dots on the positive real axis and off in the first quadrant in all three plots are $16C$ and $16C+4\pi^2{\mathsf{i}}$, where $C$ is the Catalan number. The arcs of singular points on the right periphery of each plot are due to numerical instability and thus are spurious.
• Figure 4.2: The order of magnitude ($-\log_{10}(|*|)$, vertical axis) of the difference between the exact spectrum and the Borel resummation of full energy trans-series in the form of \ref{['eq:tensor']} at Landau level 0 with varying $\Theta$ (horizontal axis). We include progressively contributions of increasing instanton orders $n=0,1,2,\ldots$ from lower data points to higher data points.