Entanglement spectra and entanglement thermodynamics of Hofstadter bilayers

We study Hofstadter bilayers, i.e. coupled hopping models on two-dimensional square lattices in a perpendicular magnetic field. Upon tracing out one of the layers, we find an explicit expression for the resulting entanglement spectrum in terms of the energy eigenvalues of the underlying monolayer system. For strongly coupled layers, the entanglement Hamiltonian is proportional to the energetic Hamiltonian of the monolayer system. The proportionality factor, however, cannot be interpreted as the inverse thermodynamic temperature, but represents a phenomenological temperature scale. We derive an explicit relationship between both temperature scales which is in close analogy to a standard result of classic thermodynamics. In the limit of vanishing temperature, thermodynamic quantities such as entropy and inner energy approach their ground-state values, but show a fractal structure as a function of magnetic flux