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arXiv · 2406.18098

Towards full instanton trans-series in Hofstadter's butterfly

Jie Gu·Zhaojie Xu
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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.

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Jie Gu, Zhaojie Xu. 2024-09-19. Towards full instanton trans-series in Hofstadter's butterfly. https://arxiv.org/abs/2406.18098

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