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 superexchange interactions, whose competition is controllable by tuning the Hubbard 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.

Figure 1

Schematic pictures of (a) a tetrahedron of four V ions (blue circles), and (b) the splitting of the $t_{2g}$ orbitals under the trigonal-distortion potential, ${\mathrm{\Delta }}_{\mathrm{tri}}$.

Figure 2

Schematic pictures of a V-O-V bond (a) with and (b) without the inversion center. Blue and orange circles denote V and O ions, respectively. This angle deviation breaks the mirror symmetry about the plane including the V ions.

Figure 3

Schematic pictures of (a) the even-mirror hoppings of ${\widehat{H}}_0$ and (b) the odd-mirror hoppings of ${\widehat{H}}_{\mathrm{odd}}$ for the nearest-neighbor V ions between sublattices 1 and 2 and the corresponding values. The color differences represent the sign differences of the wave function.

Figure 4

Schematic pictures of examples of the processes of the second-order perturbation by (a) using ${\widehat{H}}_0$ and (b) using ${\widehat{H}}_{\mathrm{odd}}$. Yellow circles denote the electrons, and gray arrows denote the perturbations.

Figure 5

Relation between $\frac{J_{\mathrm{H}}}{U}$ and the sign of $J^{\mathrm{FM}}-J^{\mathrm{AF}}$ in the four limiting cases without the $t_{\mathrm{odd}}^2$ terms within the $O\left(\frac{J_{\mathrm{H}}}{U}\right)$ terms for $J^{\prime }=J_{\mathrm{H}}$ and $U^{\prime }=U-2J_{\mathrm{H}}$. $J^{\mathrm{FM}}-J^{\mathrm{AF}}$ becomes AF or FM in the red-line region or the blue-line region, respectively. Since $J^{\mathrm{FM}}-J^{\mathrm{AF}}$ is expressed by the power series of $\frac{J_{\mathrm{H}}}{U}$, the results are meaningful only for $\frac{J_{\mathrm{H}}}{U}<1$.

Figure 6

Schematic pictures of examples of the processes of (a) ${\widehat{H}}_{3\mathrm{rd}}^{\mathrm{KE}-\mathrm{KE}-LS}$, (b) ${\widehat{H}}_{3\mathrm{rd}}^{LS-\mathrm{KE}-\mathrm{KE}}$, and (c) ${\widehat{H}}_{3\mathrm{rd}}^{\mathrm{KE}-LS-\mathrm{KE}}$. Yellow circles denote the electrons, and gray arrows denote the perturbations.

Figure 7

$\frac{D}{J^{\mathrm{FM}}-J^{\mathrm{AF}}}$ as a function of $\frac{t_{\mathrm{odd}}}{A}$ in (a) $A\gg B$ or $\frac{t_{\mathrm{odd}}}{B}$ in (b) $A\ll B$ or (c) $A\sim B$ or (d) $A\sim -B$ for $\frac{J_{\mathrm{H}}}{U}=0.1$ (red lines) and 0.3 (blue lines). $\frac{D}{J^{\mathrm{FM}}-J^{\mathrm{AF}}}$ were estimated by considering the $O\left(\frac{J_{\mathrm{H}}}{U}\right)$ terms at $J^{\prime }=J_{\mathrm{H}}$ and $U^{\prime }=U-2J_{\mathrm{H}}$ and setting ${\lambda }_{LS}=0.03$ eV and ${\mathrm{\Delta }}_{\mathrm{tri}}=1$ eV in Eqs. (D3, D4, D5, D6). Those four cases are different from the four limiting cases considered in Figs. 5 and 8.

Figure 8

$\frac{{\mathrm{\Delta }}_{\mathrm{tri}}}{U}$ dependence of $\frac{D^{\prime }}{D}$ in the four limiting cases considered in Fig. 5 for $\frac{J_{\mathrm{H}}}{U}=0.1$ (red lines) and 0.3 (blue lines).