Anisotropic three-dimensional quantum Hall effect in topological nodal-line semimetals

The three-dimensional (3D) quantum Hall effect (QHE) has been intensely studied in Weyl semimetals and Dirac semimetals in recent years. Here, we study the 3D QHE in nodal-line semimetal through calculating the Hall conductivity numerically in a nodal-line semimetal slab. It is found that the Hall conductivity in nodal-line semimetal is anisotropic with respect to the magnetic field, Fermi energy, and sample thickness when the magnetic field is perpendicular to or parallel to the nodal line. The Hall conductivity is symmetric with respect to the energy of the nodal line in both cases. When the Fermi energy deviates from the nodal line, a Hall plateau appears and the plateau is wider and higher for the magnetic field parallel to the nodal line than perpendicular to the nodal line. For magnetic field dependence, the Hall conductivity follows $1/B$ dependence when the magnetic field is perpendicular to the nodal line, while it is independent on the magnetic field parallel to the nodal line. Moreover, the Hall conductivity as a function of sample thickness is independent of the sample thickness for the magnetic field perpendicular to the nodal line, while it is a increasing function for the magnetic field parallel to the nodal line. We give a possible explanation for these anisotropies from the energy spectrum of Landau bands under the magnetic field. This unique anisotropic 3D QHE can be a transport feature to recognize the topological nodal-line semimetals.