## Abstract

We develop a generic k·p open momentum space method for calculating the Hofstadter butterfly of both continuum (moiré) models and tight-binding models, where the quasimomentum is directly substituted by the Landau level (LL) operators. By taking a LL cutoff (and a reciprocal lattice cutoff for continuum models), one obtains the Hofstadter butterfly with in-gap spectral flows. For continuum models such as the moiré model for twisted bilayer graphene, our method gives a sparse Hamiltonian, making it much more efficient than existing methods. The spectral flows in the Hofstadter gaps can be understood as edge states on a momentum space boundary, from which one can determine the two integers (tν,sν) of a gap ν satisfying the Diophantine equation. The spectral flows can also be removed to obtain a clear Hofstadter butterfly. While tν is known as the Chern number, our theory identifies sν as a dual Chern number for the momentum space, which corresponds to a quantized Lorentz susceptibility γxy=eBsν.

Original language | English (US) |
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Article number | L161405 |

Journal | Physical Review B |

Volume | 103 |

Issue number | 16 |

DOIs | |

State | Published - Apr 20 2021 |

## All Science Journal Classification (ASJC) codes

- Electronic, Optical and Magnetic Materials
- Condensed Matter Physics