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

TL;DR

This paper demonstrates that Hofstadter's butterfly and quantum Hall effects can occur in three-dimensional metals with weak external periodic modulations, revealing fractal energy gaps and quantized Hall tensors due to interference effects.

Contribution

It introduces a new physical system—3D metals with external periodic modulations—for realizing Hofstadter's butterfly and quantum Hall effects, expanding beyond tight-binding models.

Findings

Fractal energy gaps depend on magnetic field tilt angle.

Quantized Hall tensors are computed for the system.

The phenomenon is mathematically similar to 3D tight-binding systems.

Abstract

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 {\it et al.} [Phys. Rev. Lett. {\bf 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 $\Vec{B}$, resulting generally in fractal energy gaps as a function of the tilting angle of $\Vec{B}$, 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.