Hofstadter's Butterfly in Quantum Geometry

Abstract

We point out that the recent conjectural solution to the spectral problem for the Hamiltonian $H=e^x+e^{-x}+e^p+e^{-p}$ in terms of the refined topological invariants of a local Calabi-Yau geometry has an intimate relation with two-dimensional non-interacting electrons moving in a periodic potential under a uniform magnetic field. In particular, we find that the quantum A-period, determining the relation between the energy eigenvalue and the Kahler modulus of the Calabi-Yau, can be found explicitly when the quantum parameter $q=e^{i\hslash }$ is a root of unity, that its branch cuts are given by Hofstadter's butterfly, and that its imaginary part counts the number of states of the Hofstadter Hamiltonian. The modular double operation, exchanging $\hslash$ and $4{\pi }^2/\hslash$, plays an important role.

arXiv comment24 pages, 3 figures, v2: references added

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• Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations
• Hyperbolic Band Theory under Magnetic Field and Dirac Cones on a Higher Genus Surface
• Resurgence Matches Quantization
• Topologically quantized current in quasiperiodic Thouless pumps
• Exact Quantization Conditions, Toric Calabi-Yau and Nonperturbative Topological String
• Exact eigenfunctions and the open topological string
• Calabi-Yau geometry and electrons on 2d lattices
• BPS relations from spectral problems and blowup equations
• Turbulence Hierarchy and Multifractality in the Integer Quantum Hall Transition
• Hofstadter's Butterfly and Langlands Duality
• Instantons in the Hofstadter butterfly: difference equation, resurgence and quantum mirror curves
• Quantum Hall Effect and Langlands Program
• Fractal defect states in the Hofstadter butterfly
• Perturbative/nonperturbative aspects of Bloch electrons in a honeycomb lattice
• Quantum Periods and Spectra in Dimer Models and Calabi-Yau Geometries
• Double Inozemtsev Limits of the Quantum DELL System
• Fermionic fractional quantum Hall states: A modern approach to systems with bulk-edge correspondence
• Floquet engineering the Hofstadter butterfly in the square lattice and its effective Hamiltonian
• Bloch electrons on honeycomb lattice and toric Calabi-Yau geometry
• Bilayer graphene in periodic and quasiperiodic magnetic superlattices
• Quantum mirror curve of periodic chain geometry
• Hofstadter-Toda spectral duality and quantum groups
• Universal Computation with Quantum Fields
• Quantum curves as quantum distributions
• Hofstadter point spectrum trace and the Almost Mathieu operator
• Superlinear Hall angle and carrier mobility from non-Boltzmann magnetotransport in the spatially disordered Yukawa-Sachdev-Ye-Kitaev model on a square lattice
• On Thouless bandwidth formula in the Hofstadter model
• Gauge fields induced by curved spacetime
• Lattice random walks and quantum A-period conjecture
• Construction of Entangled Many-body States via the Higgs Mechanism
• Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux