Nonperturbative description of the butterfly diagram of energy spectra for materials immersed in a magnetic field

We propose a nonperturbative method to calculate the butterfly diagram of energy spectra for materials immersed in a magnetic field. We apply the proposed method to a crystalline silicon immersed in a magnetic field. It is shown that the conventional Hofstadter butterfly diagram is of low accuracy not only in the high magnetic field region of the diagram but also even in the experimentally available magnetic field region. This means that the present butterfly diagram is regarded as a replacement for the Hofstadter butterfly diagram. We also show that the correction to the Hofstadter buttery diagram would be observed under the ultrahigh magnetic field that is available in experiments.