Effect of the tilt on the three-dimensional quantum Hall effect in tilted Weyl semimetals

Abstract

Three-dimensional (3D) quantum Hall effect (QHE) in Weyl semimetal has attracted intense interests recent years. In this paper, we theoretically investigate the 3D QHE in tilted Weyl semimetals from bulk states for magnetic field parallel to the separation of Weyl nodes. According to the correspondence between bulk Chern number and Hall conductivity, we analytically calculate the Hall conductivity as a function of Fermi energy and the tilted terms in a thin film of a tilted Weyl semimetal using periodic boundary condition. The width of the Hall plateaus as a function of Fermi energy is influenced by the tilted terms and the energy gap between the $n=-1$ and $n=1$ Landau levels (LLs). When the two Weyl nodes tilt to the same direction and tilt away from each others, the plateau width decreases as a quadratic function and linear function respectively as the tilt increases. When the two Weyl nodes tilt towards to each other, the plateau width increases as a linear function. However, the width is limited by the energy gap between the $n=-1$ and $n=1$ LLs which decreases as the tilt increases. We also numerically calculate the Hall conductivity using open boundary condition and the numerical results show well consistent with the analytical results. Our study can be significant for understanding the effects of tilt on the 3D QHE in tilted Weyl semimetals.