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Crossed responses of spin and orbital magnetism in topological insulators

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