# Advanced First-Principle Modeling of Relativistic Ruddlesden--Popper Strontium Iridates
## 4 Results and Discussions
### Optical Spectra  
To conclude with the presentation of the fundamental properties of the RP iridates family, we briefly survey the optical characteristics in this section. Figure 10 shows our computed optical conductivity for the considered iridates through the solution of the BSE using the QP energies and screened interactions \(W\) derived from the \(GW\) calculations. For comparison, DMFT-calculated results [154] and experimental data [18; 31; 36] are also given. We found that all the compounds are characterized by a double-peak structure (\(\alpha\) and \(\beta\)), consistent with the experiments [18; 31; 36]. However, the agreement depends on the level of theory and is material-dependent. For instance, though both BSE and RPA are able to reproduce the two dominant peaks, a good quantitative agreement with experiments is only obtained at the BSE level for Sr\({}_{2}\)IrO\({}_{4}\) and Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\), for which the calculated and measured \(\alpha\) and \(\beta\) transitions are centered almost at the same excitation energies. As compared to RPA, the dramatic redshift of the \(\alpha\) and \(\beta\) peaks predicted by BSE indicates strong excitonic effects, but no bound exciton is discerned in our BSE calculations. A double-peak structure was also observed in SrIrO\({}_{3}\), which is qualitatively consistent with the recent experimental findings by [PERSON] [36]. However, the agreement between theory and experiment is less satisfactory compared to the \(n=1\) and \(n=2\) cases. For example, the calculated \(\alpha\) and \(\beta\) peaks are centered at higher energies than the experimental ones [36] and the Drude peak is broader and more intense. We note that, for SrIrO\({}_{3}\), the DMFT calculations [154] reproduce relatively well the \(\beta\) peak but that the \(\alpha\) peak is not detected. Figure 9: The total energy difference between the _ab_-canted AFM and _c_-collinear AFM phases as a function of the interlayer distance, which is schematically shown in the inset. The ground-state structure is marked as GS. To identify the character of the optical transitions, the BSE oscillator strengths \(S_{\rm A}\) (Equation (21)) were calculated and shown as histograms in Figure 10a-c. The oscillator strengths are associated with the dominant \(k\)-point dependent interband transitions, which are represented by arrows in the band structure plots of Figure 10d-f. We note that the width of the arrows is proportional to the corresponding amplitude of BSE eigenvectors. This analysis clearly demonstrates that the \(\alpha\) peak arises from the optical transition from the \(J_{\rm eff}=1/2\) LHB to the \(J_{\rm eff}=1/2\) UHB whereas the \(\beta\) peak comes from \(J_{\rm eff}=3/2\) to \(J_{\rm eff}=1/2\) UHB excitations, as schematically evidenced in the DOS given in Figure 10g-i. Since the \(J_{\rm eff}=1/2\) LHB and UHB are rather flat and parallel for Sr\({}_{2}\)IrO\({}_{4}\), in particular along the \(X\)-\(M\) direction, the \(\alpha\)-type excitations are particularly strong compared to the \(\beta\)-type excitations, leading to a relatively narrower width of the \(\alpha\) peak. Moving onto Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\), the LHB and UHB split and are less parallel than those in Sr\({}_{2}\)IrO\({}_{4}\). As a result, the \(\alpha\) peak becomes less intense and broader. For SrIrO\({}_{3}\), the \(J_{\rm eff}=1/2\) band topology is strongly perturbed due to the substantial hybridization between Ir-\(5d\) and O-\(2p\) orbitals originating from the underlying distorted three-dimensional orthorhombic crystal structure. Ultimately, this leads to admixing of the \(J_{\rm eff}=3/2\) states with the lower \(J_{\rm eff}=1/2\) band. As a result, the \(\alpha\) and \(\beta\) peaks are much broader. However, agreement with the experiment is not satisfactory. The reasons are twofold: First, \(GW\) does not properly account for the strong bandwidth renormalization observed experimentally [152], which drastically changes the band topology near the Fermi level and thus affects the optical excitations.