We develop a generic $\mathbf{k}·\mathbf{p}$ 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 ($t_{\nu },s_{\nu }$) of a gap $\nu$ satisfying the Diophantine equation. The spectral flows can also be removed to obtain a clear Hofstadter butterfly. While $t_{\nu }$ is known as the Chern number, our theory identifies $s_{\nu }$ as a dual Chern number for the momentum space, which corresponds to a quantized Lorentz susceptibility ${\gamma }_{xy}=eB{s}_{\nu }$.

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• Electronic structure
• Landau levels
• 2-dimensional systems
• Two-dimensional electron system
• Band structure methods
• Free-electron model
• Lattice models in condensed matter
• Linear response theory