Magnetic orders and topological phases from $f$-$d$ exchange in pyrochlore iridates

We study theoretically the effects of $f$-$d$ magnetic exchange interaction in the R${}_2$Ir${}_2$O${}_7$ pyrochlore iridates. The R${}^{3+}$ $f$ electrons form localized Kramers or non-Kramers doublets, while the Ir${}^{4+}$$d$ electrons are more itinerant and feel a strong spin-orbit coupling. We construct and analyze a minimal model capturing this physics, treating the Ir subsystem using a Hubbard-type model. First neglecting the Hubbard interaction, we find Weyl semimetal and Axion insulator phases induced by the $f$-$d$ exchange. Next, we find that $f$-$d$ exchange can cooperate with the Hubbard interaction to stabilize the Weyl semimetal over a larger region of parameter space than when it is induced by $d$-electron correlations alone. Applications to experiments are discussed.

Figure 1

A projective view of R${}_2$Ir${}_2$O${}_7$ in the (111) plane. Left: The neighboring Ir (in dark red) and R (in light blue) tetrahedra. “1,2,3,4” label the four sublattices. Ir/R atoms are marked with big/small (red/blue) circles. Empty/dark/light circles indicate that the atoms are below/above/in the (111) plane. Right: An IrR${}_6$ complex singled out from the left figure.

Figure 2

(a) Phase diagrams of the STI with $t_{\sigma }=-2t$ and (b) the semimetal phase with $t_{\sigma }=-t$ after including $f$-$d$ exchange. See text for further discussion of the phase diagrams. (c) and (d) Curie-Weiss temperature ${\Theta }_{\mathrm{CW}}$ (solid line) and the mean-field ordering temperature $T_c$ (dashed/dotted line) of the truncated exchange model (up to fourth neighbor for R system) are plotted against $\Phi$. $T_c$ is obtained by restricting to the $\mathbf{q}=\mathbf{0}$ magnetic ordering (see Appendix B). $T_c$ for the all-in all-out (two-in two-out) state is the dashed (dotted) curve. (e) The $c_2$-$t_{\sigma }$ phase diagram for Ir subsystem where the magnetic order is all-in all-out. In the figure, NI $=$ normal insulator.

Figure 3

The Ir electron band structure with the all-in all-out magnetic order. The energy unit is set to $t$ and $c_1=0,c_2=0.05t$. The dashed (red) line is $E_F$. (a) AI phase with $t_{\sigma }=-2t$. (b) WSM phase with $t_{\sigma }=-t$.[39]

Figure 4

The mean-field phase diagram at $t_{\sigma }=-t$ and $c_1=0$. The (red) thick line is the narrow WSM phase induced by Ir correlation. The dashed line is the nonmagnetic Ir semimetal phase. The green region is the broad WSM phase after including $f$-$d$ exchange. All-in all-out magnetic order is present throughout the phase diagram.

Figure 5

The dependence of RKKY exchanges on the angle $\Phi$. The results are obtained for a finite system with $10\times 10\times 10$ unit cells. $J_i$ is the $i$th neighbor exchange. $J_3$ is the exchange on third neighbor bonds that have one lattice site at the midpoint of the bonds. $J_3^{\prime }$ is the exchange on third neighbor bonds that have no lattice site at the midpoint of the bonds. $J_7$ is the exchange on seventh neighbor bonds that have two additional lattice sites along the bonds. $J_7^{\prime }$ is the exchange on seventh neighbor bonds that have no lattice sites along the bonds. Upper: STI phase with $t_{\sigma }=-2t$. Lower: Semimetal phase with $t_{\sigma }=-t$.