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

  • Zhihao DuanEmail author
  • Jie Gu
  • Yasuyuki Hatsuda
  • Tin Sulejmanpasic
Open Access
Regular Article - Theoretical Physics


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 2π. For such values of the magnetic flux, the theory allows a formulation with two Bloch or θ-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 θ-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.


Nonperturbative Effects Solitons Monopoles and Instantons Topological Strings 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique & Institut de Physique Théorique Philippe Meyer, École Normale Supérieure, CNRS, PSL Research University, Sorbonne Universités, UPMCParis Cedex 05France
  2. 2.Department of PhysicsRikkyo UniversityTokyoJapan
  3. 3.Institute for Nuclear PhysicsUniversity of MainzMainzGermany

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