Revisiting Bloch electrons in magnetic field: Hofstadter physics via hybrid Wannier states

Abstract

We revisit the Hofstadter butterfly for a subset of topologically trivial Bloch bands arising from a continuum free electron Hamiltonian in a periodic lattice potential. We employ the recently developed procedure -- which was previously used to analyze the case of topologically non-trivial bands [\href{https://journals.aps.org/prb/abstract/10.1103/PhysRevB.106.L121111}{Phys. Rev. B \textbf{106}, L121111 (2022)}] -- to construct the finite field Hilbert space from the zero-field hybrid Wannier basis states. Such states are Bloch extended along one direction and exponentially localized along the other. The method is illustrated for square and triangular lattice potentials and is shown to reproduce all the main features of the Hofstadter spectrum obtained from a numerically exact Landau level expansion method. In the regime when magnetic length is much longer than the spatial extent of the hybrid Wannier state in the localized direction we recover the well known Harper equation. Because the method applies to both topologically trivial and non-trivial bands, it provides an alternative and efficient approach to moir\'e materials in magnetic field.