  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.
