Open momentum space method for the Hofstadter butterfly and the quantized Lorentz susceptibility

We develop a generic $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ open momentum space method for calculating the Hofstadter butterfly of both continuum (moir\'e) 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 moir\'e 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 (${t}_{\ensuremath{\nu}},{s}_{\ensuremath{\nu}}$) of a gap $\ensuremath{\nu}$ satisfying the Diophantine equation. The spectral flows can also be removed to obtain a clear Hofstadter butterfly. While ${t}_{\ensuremath{\nu}}$ is known as the Chern number, our theory identifies ${s}_{\ensuremath{\nu}}$ as a dual Chern number for the momentum space, which corresponds to a quantized Lorentz susceptibility ${\ensuremath{\gamma}}_xy=eB{s}_{\ensuremath{\nu}}$.