Electron interactions, spin-orbit coupling, and intersite correlations in pyrochlore iridates

We perform combined density functional and dynamical mean-field calculations to study the pyrochlore iridates ${\mathrm{Lu}}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$, ${\mathrm{Y}}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$, and ${\mathrm{Eu}}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$. Both single-site and cluster dynamical mean-field calculations are performed and spin-orbit coupling is included. Paramagnetic metallic phases, antiferromagnetic metallic phases with tilted Weyl cones, and antiferromagnetic insulating phases are found. The magnetic phases display all-in/all-out magnetic ordering, consistent with previous studies. Unusually for electronically three-dimensional materials, the single-site dynamical mean-field approximation fails to reproduce qualitative material trends, predicting in particular that the paramagnetic phase properties of ${\mathrm{Y}}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ and ${\mathrm{Eu}}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ are almost identical, although in experiments the Y compound has a much higher resistance than the Eu compound. This qualitative failure is attributed to the importance of intersite magnetic correlations in the physics of these materials.

Figure 1

(a) Crystal structure of pyrochlore iridates. The large dark sphere (blue online) is the $R$ site; the large light sphere (grey online) is Ir; the small one (red online) is O. (b) Band structure of Y-227 within paramagnetic DFT+$U$+SO. The energy window is chosen to include Ir $t_{2g}$ bands, which split into the higher-energy $J_{\mathrm{eff}}=1/2$ [dark (blue online) solid lines] and lower-energy $J_{\mathrm{eff}}=3/2$ [dark (blue online) dashed lines] manifold in the presence of SOC. The light (red online) bands are irrelevant noniridium $d$ states. (c) Brillouin zone of the fcc lattice.

Figure 2

(a) Generic ground-state phase diagram for pyrochlore iridates obtained with DFT+CDMFT calculations as described in the main text. The shaded region indicates the interaction strengths for which the paramagnetic phases are locally stable, but higher in energy compared than the antiferromagnetic phases. Abbreviations: PM-M, paramagnetic metal; AF-M, antiferromagnetic metal with AIAO magnetic ordering and Weyl cones; AF-I, antiferromagnetic insulator with AIAO ordering but without Weyl cones. (b) Energy as a function of the interaction strength for all the states obtained with DFT+CDMFT for Y-227. The small filled circles indicate the energy of the paramagnetic phase. The open squares represent the antiferromagnetic insulating phase and the open circles represent the antiferromagnetic metallic phase. The precise behavior near $U\sim 0.5–0.6\mathrm{eV}$ is not resolved due to finite bath size effects.

Figure 3

$\langle S\rangle$ as a function of the interaction strength for both Y-227 and Eu-227. The results are obtained within CDMFT. The solid lines identifies the evolution of the ground state while the dashed lines identifies the evolution of the metastable phases. The open symbols represent the antiferromagnetic phases which are obtained by decreasing $U$. The filled symbols represent the paramagnetic phases which are obtained by increasing $U$. The inset of this figure shows the $U$-dependent behavior of the double occupancy for the ground state, which is computed as $\frac{1}{U}\frac{1}{\beta }{\sum }_n{\sum }_k\frac{1}{N_c}Tr\frac{1}{2}\mathrm{\Sigma }\left(i{\omega }_n\right)\mathbf{G}(k,i{\omega }_n)$. We will leave the discussion of the material dependence to the next section.

Figure 4

Spectral function for Y-227 in (a) paramagnetic metallic phase (b) antiferromagnetic metallic phase and (c) antiferromagnetic insulating phase. The dashed circle of (b) highlights the region where the Weyl crossing occurs. The zero of energy is the chemical potential.

Figure 5

Gap size as a function of the interaction strength obtained within paramagnetic sDMFT. For all compounds, the filled symbols represent the metallic phase which is obtained by increasing $U$ while the open symbols represent the insulating phase which is obtained by decreasing $U$. The solid lines identify the evolution of the ground state while the dashed lines represent the metastable phases.

Figure 6

Gap size as a function of the interaction strength obtained within (a) magnetic CDMFT (six bath orbitals) and (b) paramagnetic CDMFT (eight bath orbitals). Convergence difficulties related to the finite bath size prevent us determining the gap size in the vicinity of the metal-insulator transition; the critical $U$ is estimated using a linear extrapolation (dotted lines). The two panels share the same ordinate. In (a), the insulating phase is shown by open symbols; in (b), the filled symbols represent the metallic phase, which is obtained by increasing $U$, while the open symbols represent the insulating phase, which is obtained by decreasing $U$. The solid lines identify the ground state, while the dashed lines represent metastable phases.

Figure 7

Total density of states in $J_{\mathrm{eff}}=1/2$ manifold obtained within paramagnetic DFT+$U$+SO calculations for Lu-227, Y-227, and Eu-227. The dashed line denotes the fermi energy.

Figure 8

On-site (a) and intersite (b) contributions to the imaginary part of the bare hybridization function $\mathbf{\Delta }\left(\omega \right)$ (defined in the main text). The two panels share the same ordinate. The inset of (b) shows an expanded view of the range $-0.3\lesssim \omega \lesssim 0$ highlighting the shift of the peak.

Figure 9

Projection of the bare hybridization function $\mathbf{\Delta }\left(\omega \right)$ onto the irreducible representations ${\mathrm{\Gamma }}_6$, ${\mathrm{\Gamma }}_7$, and ${\mathrm{\Gamma }}_8$ of the tetrahedral point group. The three panels share the same horizontal axis. (b) shows that the material-dependent peak appears only in the ${\mathrm{\Gamma }}_7$ representation.

Figure 10

Material difference of the hybridization function for Y-227 and Eu-227 in the case of $U=1.1$ eV and ${\omega }_0=0.001$ eV in (a) antiferromagnetic state and (b) paramagnetic state. The color for a given matrix element in a given solution is defined as $\left|\right|{\mathrm{\Delta }}_{ij}^{\text{Y}}\left(i{\omega }_0\right)|-|{\mathrm{\Delta }}_{ij}^{\text{Eu}}\left(i{\omega }_0\right)\left|\right|$.

Figure 11

Gap size as a function of the interaction strength with (a) $N_b=6$ and (b) $N_b=8$. The two panels share the same ordinate.