Entanglement spectrum and Wannier center flow of the Hofstadter problem

We examine the quantum entanglement spectra and Wannier functions of the square lattice Hofstadter model. Consistent with previous work on entanglement spectra of topological band structures, we find that the entanglement levels exhibit a spectral flow similar to that of the full system's energy spectrum. While the energy spectra are continuous, with cylindrical boundary conditions the entanglement spectra exhibit discontinuities associated with the passage of an energy edge state through the Fermi level. We show how the entanglement spectrum can be understood by examining the band projectors of the full system and their behavior under adiabatic pumping. In so doing we make connections with the original work by Thouless, Kohmoto, Nightingale, and den Nijs (TKNN) [Phys. Rev. Lett. 49, 405 (1982)] on topological two-dimensional band structures and their Chern numbers. Finally, we consider Wannier states and their adiabatic flows and draw connections to the entanglement properties.