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Microscopic theory of Dzyaloshinsky-Moriya interaction in pyrochlore oxides with spin-orbit coupling

Pyrochlore oxides show several fascinating phenomena, such as the formation of heavy fermions and the thermal Hall effect. Although a key to understanding some phenomena may be the Dzyaloshinsky-Moriya (DM) interaction, its microscopic origin is unclear. To clarify the microscopic origin, we constructed a $t_{2g}$-orbital model with the kinetic energy, the trigonal-distortion potential, the multiorbital Hubbard interactions, and the $LS$ coupling, and derived the low-energy effective Hamiltonian for a $d^{1}$ Mott insulator with the weak $LS$ coupling. We first show that lack of the inversion center of each nearest-neighbor V-V bond causes the odd-mirror interorbital hopping integrals. Those are qualitatively different from the even-mirror hopping integrals, existing even with the inversion center. We next show that the second-order perturbation using the kinetic terms leads to the ferromagnetic and the antiferromagnetic super exchange interactions. Then, we show the most important result: the third-order perturbation terms using the combination of the even-mirror hopping integral, the odd-mirror hopping integral, and the $LS$ coupling causes the DM interaction due to the mirror-mixing effect, where those hopping integrals are necessary to obtain the antisymmetric kinetic exchange and the $LS$ coupling is necessary to excite the orbital angular momentum at one of two sites. We also show that the magnitude and sign of the DM interaction can be controlled by changing the positions of the O ions and the strength of the Hubbard interactions. We discuss the advantages in comparison with the phenomenological theory and Moriya's microscopic theory, applicability of our mechanism, and the similarities and differences between our case and the strong-$LS$-coupling case.