Integer quantum Hall effect and Hofstadter’s butterfly spectra in three-dimensional metals in external periodic modulations

We propose that Hofstadter’s butterfly accompanied by quantum Hall effect that is similar to those predicted to occur in 3D tight-binding systems by Koshino et al. [Phys. Rev. Lett. 86, 1062 (2001)] can be realized in an entirely different system—3D metals applied with weak external periodic modulations (e.g., acoustic waves). Namely, an effect of two periodic potentials interferes with Landau’s quantization due to an applied magnetic field $\mathit{B},$ resulting generally in fractal energy gaps as a function of the tilting angle of $\mathit{B},$ for which the accompanying quantized Hall tensors are computed. The phenomenon arises from the fact that, while the present system has a different physical origin for the butterfly from the 3D tight-binding systems, the mathematical forms are remarkably equivalent.