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Crossed magnetic responses between spin and orbital angular momentum arestudied in time-reversal symmetric topological insulators. Due to spin-orbitcoupling in the quantum spin Hall systems and three-dimensional topologicalinsulators, the magnetic susceptibility has crossed (intersectional) componentsbetween spin and orbital part of magnetism. In this study, the crossedsusceptibility for the orbital magnetization is studied in two- andthree-dimensional topological insulator models, in which an external magneticfield interacts with the electron spin by Zeeman coupling via distinctg-factors for conduction and valence energy bands. The crossed susceptibilityin two-dimensional quantum spin Hall insulators shows a quantized signature ofthe $\mathbb{Z}_2$ topological phase in response to Zeeman coupling via anaveraged g-factor, and the quantization persists even when$\sigma^z$-conservation of electrons is broken by a tilted magnetic field. Thebulk orbital magnetization is interpreted by the persistent edge currentattributed to the chiral anomaly at the (1+1)-dimensional boundary. Inthree-dimensional topological insulators, we found that the crossedsusceptibility is proportional to the difference of g-factors of conduction andvalence electrons, which is qualitatively different from the two-dimensionalcase. Steep changes of the crossed susceptibility in three dimensions at thephase transition points are explained by the surface Dirac fermion theory.Finally, dependence of the crossed susceptibility on g-factors in two- andthree-dimensional cases is discussed from the viewpoint of time-reversal andparticle-hole symmetries.

Crossed responses of spin and orbital magnetism in topological insulators