Magnetic excitation spectra in pyrochlore iridates

Metal-insulator transitions in pyrochlore iridates (A${}_2$Ir${}_2$O${}_7$) are believed to occur due to subtle interplay of spin-orbit coupling, geometric frustration, and electron-electron interactions. In particular, the nature of magnetic ordering of iridium ions in the insulating phase is crucial for the understanding of several exotic phases recently proposed for these materials. We study the spectrum of magnetic excitations in the intermediate-coupling regime for the so-called all-in/all-out magnetic state in pyrochlore iridates with nonmagnetic $A$-site ions ($A$ $=$ Eu,Y). This state was found to be preferred in previous theoretical studies. We find that the effect of charge fluctuations on the magnetic excitations in the intermediate-coupling regime leads to a strong departure from the lowest-order spin-wave calculations based on spin models obtained in the strong-coupling limit. We discuss the characteristic features of the magnetic excitation spectrum that can lead to conclusive identification of the magnetic order in future resonant inelastic x-ray (or neutron) scattering experiments. This knowledge about the nature of magnetic order and its low-energy features may also provide useful information on the accompanying metal-insulator transition.

Figure 1

The top left tetrahedron indicates the sublattice indices used in our work. The red vector in the top right tetrahedron shows the nearest-neighbor ${\mathbf{d}}_{32}$. The gold vectors ${\mathbf{p}}_{14}$ and ${\mathbf{q}}_{14}$ are perpendicular to the $C_2$ axis and span the plane in which next-nearest-neighbor ${\mathbf{v}}_{\langle \langle 14\rangle \rangle }$ must lie. The bottom right tetrahedron with the green vectors shows the all-in/all-out configuration.

Figure 2

Mean-field phase diagram with hopping amplitudes ${\theta }^{\prime }=\theta$, $t^{\prime }/t=0.1$, $t=1$, and ${\varphi }^{\prime }=5\pi /6$. First-order transitions are indicated by dashed lines, while second-order transitions are indicated by solid black lines. Lines A ($\theta =1.45$) and B ($\theta =1.70$) represent the two cuts for which the RPA dynamical structure factors are calculated (Fig. 3). The phases indicated are strong topological insulator (STI), metallic (M), semimetallic (SM), topological Weyl semimetal (TWS), metallic all-in/all-out (mAIAO), rotated all-in/all-out (AIAO${}^{\prime }$), and all-in/all-out (AIAO).

Figure 3

RPA dynamical structure factors for (a) $\theta =1.45$ and (b) $\theta =1.70$ for various values of $U$; $\omega$ is in units of $t=1$. These correspond to cuts A and B in Fig. 2, respectively. Sharp dispersions can be seen at larger values of $U$, while lighter intensity and broadened spectra are seen due to Landau damping in parts of the Brillouin zone at lower $U$. Degeneracies at $\Gamma$, $X$, and $W$ are symmetry protected and are robust features of the AIAO state.

Figure 4

Evolution of the Holstein-Primakoff spin-wave spectra as angle $\theta$ is varied between $({\theta }_t/2\approx 0.96)<\theta$ and $\theta <(\pi -{\theta }_t/2\approx 2.19)$; $\omega$ is in units of $t^2/U$. $\theta$ and $\pi -\theta$ yield identical spin-wave spectra, and this redundancy is reflected by each plot having two $\theta$ values. Like the RPA results in Fig. 3, degeneracies at $\Gamma$, $X$, and $W$ are symmetry protected.

Figure 5

RPA results at $U=6.0$ for cut B (blue and yellow color map) overlaid with a fit with the NN linear spin-wave spectrum (red); $\omega$ is in units of $t=1$. Although some features along the $\Gamma$-$X$-$W$ line can be fitted, other high-symmetry lines show a larger discrepancy.