Open Momentum Space Method for Hofstadter Butterfly: Application in Moire Models

We develop a generic open momentum space method for calculating the Hofstadter butterfly of both continuum models (for Moire superlattices, etc) 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 model for twisted bilayer graphene, our method gives a sparse Hamiltonian, making it much more efficient to calculate the spectrum. The spectral flows can be understood as edge states on a momentum space boundary, from which one can determine the two integers (tv, sv) of a gap v satisfying the Diophantine equation. The spectral flows can also be removed to plot a clear Hofstadter butterfly. While tv is known as the Chern number, our picture identifies sv as a dual Chern number in the momentum space, which corresponds to a quantized Lorentz susceptibility γxy = eBsv.