In our previous publication [Ozaki and Ogata, Phys. Rev. Res. 3, 013058 (2021)], the quantization of the orbital-Zeeman (OZ) cross term in the magnetic susceptibility, or the cross term of spin Zeeman and orbital effect, was shown for the Kane-Mele model using the expansion around the Dirac points. In the present study, we accurately evaluate the orbital, spin-Zeeman, and OZ cross term of the Kane-Mele model using a recently developed formulation. This method uses the explicit real-space Bloch wave functions and enables us to accurately evaluate each contribution taking into account an infinite number of high-energy bands and core electron contribution at arbitrary chemical potential. We find that the effect beyond the previous formulations does not change the quantization of the OZ cross term. We also study the OZ cross term in the metalic case, and the sum rule of the OZ cross term with respect to the chemical potential is found. The possibility of experimental detection of the quantization is also discussed.

• Received 21 April 2022
• Accepted 12 December 2022

DOI:https://doi.org/10.1103/PhysRevB.107.085201