Hofstadter's Butterfly in Quantum Geometry

TL;DR

This paper reveals a deep connection between the spectral problem of a specific quantum Hamiltonian, topological invariants of Calabi-Yau geometries, and the Hofstadter butterfly pattern in condensed matter physics.

Contribution

It explicitly relates the quantum A-period to Hofstadter's butterfly, showing the branch cuts correspond to the butterfly pattern and linking the imaginary part to the number of states.

Findings

Quantum A-period branch cuts form Hofstadter's butterfly

Imaginary part counts the Hofstadter Hamiltonian states

Explicit solution when quantum parameter is a root of unity

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 $q=e^{i\hbar}$ 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 $\hbar$ and $4\pi^2/\hbar$, plays an important role.