Duality and integer quantum Hall effect in isotropic 3D crystals

TL;DR

This paper demonstrates the existence of energy gaps and integer quantum Hall effects in isotropic 3D crystals under magnetic fields, revealing surprising conductivity quantizations and introducing a duality concept to explain high-field spectra.

Contribution

It introduces a duality framework connecting strong and weak magnetic field regimes to explain 3D quantum Hall phenomena in isotropic crystals.

Findings

Energy gaps similar to Hofstadter's butterfly appear in isotropic 3D systems.

Integer quantum Hall conductivities can take quantized values along different axes.

A duality links high magnetic field behavior to a weak field problem, explaining spectral features.

Abstract

We show here a series of energy gaps as in Hofstadter's butterfly, which have been shown to exist by Koshino et al [Phys. Rev. Lett. 86, 1062 (2001)] for anisotropic three-dimensional (3D) periodic systems in magnetic fields $\Vec{B}$, also arise in the isotropic case unless $\Vec{B}$ points in high-symmetry directions. Accompanying integer quantum Hall conductivities $(\sigma_{xy}, \sigma_{yz}, \sigma_{zx})$ can, surprisingly, take values $\propto (1,0,0), (0,1,0), (0,0,1)$ even for a fixed direction of $\Vec{B}$ unlike in the anisotropic case. We can intuitively explain the high-magnetic field spectra and the 3D QHE in terms of quantum mechanical hopping by introducing a ``duality'', which connects the 3D system in a strong $\Vec{B}$ with another problem in a weak magnetic field $(\propto 1/B)$.