Chern number and Hall conductivity in three-dimensional quantum Hall effect in Weyl semimetals

The three-dimensional (3D) quantum Hall effect (QHE) in topological semimetals has attracted much interest in recent years. We study the 3D QHE in Weyl semimetals combining the Chern number calculated from Landau levels and the Hall conductivity calculated using the Kubo formula from the bulk-edge correspondence. We derive the Chern numbers under magnetic field using topological analysis. We get the magnetic field and Fermi energy dependence of the Hall conductivity according to the correspondence between the Chern number and Hall conductivity in a Weyl semimetal slab with the periodic boundary condition from the perspective of bulk states. We numerically calculate the Hall conductivity using the Kubo formula in a Weyl semimetal slab with the open boundary condition. The results of the Hall conductivity using the periodic boundary condition and open boundary condition are consistent. Our study demonstrates the 3D QHE in Weyl semimetals from both the bulk states and edge states through the bulk-edge correspondence.