Theory of orbital magnetic quadrupole moment and magnetoelectric susceptibility

We derive a quantum-mechanical formula of the orbital magnetic quadrupole moment (MQM) in periodic systems by using the gauge-covariant gradient expansion. This formula is valid for insulators and metals at zero and nonzero temperature. We also prove a direct relation between the MQM and magnetoelectric (ME) susceptibility for insulators at zero temperature. It indicates that the MQM is a microscopic origin of the ME effect. Using the formula, we quantitatively estimate these quantities for room-temperature antiferromagnetic semiconductors BaMn₂As₂ and CeMn₂Ge_2−xSi_x . We find that the orbital contribution to the ME susceptibility is comparable with or even dominant over the spin contribution.

[1] A. Shitade, H. Watanabe, and Y. Yanase, Phys. Rev. B 98, 020407(R) (2018).