Orbital gravitomagnetoelectric response and orbital magnetic quadrupole moment correction

We propose a fully quantum-mechanical formalism for the temperature-gradient-induced orbital magnetoelectric effect in systems without spatial inversion symmetry, P, and time-reversal symmetry T. The effect consists of two parts, i.e., an extrinsic part and an intrinsic part. We demonstrate that for the intrinsic part, a correction from the orbital magnetic quadrupole moment besides the usual Kubo formula, is necessary to avoid an unphysical divergence at zero temperature and to satisfy the Mott relation. Furthermore, we show the classification table with the magnetic point group for the intrinsic and extrinsic effects. Finally, we analyze the intrinsic part in a PT-symmetric model exhibiting an orbital magnetization order, i.e., a loop-current order, and demonstrate the enhancement near Dirac points. We believe these results will contribute to the detection and application of orbital magnetic moments beyond spin moments.