Open Momentum Space Method for Hofstadter Butterfly and the Quantized Lorentz Susceptibility

Abstract

We develop a generic $\mathbf{k}\cdot \mathbf{p}$ 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 ${\gamma }_{xy}=eB{s}_{\nu }$.

5+30 pages, 3+4 figures

References in corpus (2)

• Landau levels in twisted bilayer graphene and semiclassical orbits
• Revealing Hofstadter Spectrum for Graphene in a Periodic Potential

Cited by in corpus (6)

• Heavy fermion representation for twisted bilayer graphene systems
• Reentrant Correlated Insulators in Twisted Bilayer Graphene at 25T ($2\pi$ Flux)
• Hofstadter Topology with Real Space Invariants and Reentrant Projective Symmetries
• Magnetic Bloch Theorem and Reentrant Flat Bands in Twisted Bilayer Graphene at $2\pi$ Flux
• Atomistic theory of moiré Hofstadter's butterfly in magic-angle graphene
• Revisiting Bloch electrons in magnetic field: Hofstadter physics via hybrid Wannier states