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

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.

arXiv comment41 pages, 9 figures; v2: references added

References in corpus (25)

• Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes
• Topological Strings and Integrable Hierarchies
• Analytic Continuation of Liouville Theory
• Quantization of Integrable Systems and Four Dimensional Gauge Theories
• Quantum Geometry of Refined Topological Strings
• Multi-Instantons and Exact Results I: Conjectures, WKB Expansions, and Instanton Interactions
• Multi-Instantons and Exact Results II: Specific Cases, Higher-Order Effects, and Numerical Calculations
• Extended Holomorphic Anomaly in Gauge Theory
• Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local CP2
• Matrix models from operators and topological strings, 2
• New Exact Quantization Condition for Toric Calabi-Yau Geometries
• Complexified path integrals, exact saddles and supersymmetry
• Operators from mirror curves and the quantum dilogarithm
• Hidden topological angles and Lefschetz thimbles
• Matrix models from operators and topological strings
• Cheshire Cat resurgence, Self-resurgence and Quasi-Exact Solvable Systems
• Hofstadter's Butterfly in Quantum Geometry
• The curious incident of multi-instantons and the necessity of Lefschetz thimbles
• Quantization condition from exact WKB for difference equations
• Exact Quantization Conditions, Toric Calabi-Yau and Nonperturbative Topological String
• Critical Points at Infinity, Non-Gaussian Saddles, and Bions
• Landau levels, response functions and magnetic oscillations from a generalized Onsager relation
• Bethe/Gauge correspondence in odd dimension: modular double, non-perturbative corrections and open topological strings
• Perturbative/nonperturbative aspects of Bloch electrons in a honeycomb lattice
• High order perturbation theory for difference equations and Borel summability of quantum mirror curves

Cited by in corpus (9)

• Exact-WKB analysis for SUSY and quantum deformed potentials: Quantum mechanics with Grassmann fields and Wess-Zumino terms
• Fractal defect states in the Hofstadter butterfly
• Quantum phase transition and Resurgence: Lessons from 3d $\mathcal{N}=4$ SQED
• Quantum Periods and Spectra in Dimer Models and Calabi-Yau Geometries
• Bloch electrons on honeycomb lattice and toric Calabi-Yau geometry
• Exact WKB in all sectors I: Potentials with degenerate saddles
• Lattice random walks and quantum A-period conjecture
• Exact-WKB Analysis of Two-level Floquet Systems
• Exact WKB in all sectors II: Potentials with non-degenerate saddles