# Advanced First-Principle Modeling of Relativistic Ruddlesden--Popper Strontium Iridates
## 5 Conclusions  
In summary, we reviewed the fundamental properties of RP series of iridates Sr\({}_{n+1}\)Ir\({}_{n}\)O\({}_{3n+1}\) (\(n=1,2\) and \(\infty\)) using an array of advanced methodological approaches: fully relativistic noncollinear LSDA+\(U\), cRPA, \(GW\)+SOC, and BSE+SOC. Besides collecting the results that we obtained in the last five years in a thematically focused context, we also included some original results, specifically a complete \(U\)/SOC phase diagram and an analysis of the band structures disentangling the role of \(U\) and SOC. In addition, we extended the quantification of the most relevant energy scales of all three compounds. For three compounds, the estimated crystal field energy (about 4 eV) represents the dominant energy scale, and therefore, the low-energy physics can be described by using the low-lying \(t_{2g}\) states of Ir only. The SOC strength is relatively small (\(\approx\)0.5 eV) but comparable to the strength of the Hubbard \(U\) (\(\approx\)1.5 eV) quantified by cRPA. The MIT transition with respect to the intermingled interactions of SOC and \(U\) provide clear evidence for the relativistic Mott-Hubbard character of the insulating state in the \(n=1\) and \(n=2\) phases. A comparative study between LSDA+\(U\)+SOC and \(GW\)+SOC shows that both methods yield similar band structures except that \(GW\) pushes down the O-\(2p\) states by about 0.5 eV and demonstrates that the predicted band gaps for Sr\({}_{2}\)IrO\({}_{4}\) and Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\) are in excellent agreement with the measurements. However, both methods fail to describe the correlated semi-metallic state of SrIrO\({}_{3}\), incorrectly predicting a pseudogap at the Fermi level, and \(GW\) finds a mass enhancement of only 1.64, largely underestimated compared to experimental estimations. Because of similar quasi-2D layered crystal structures, Sr\({}_{2}\)IrO\({}_{4}\) and Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\) exhibit overall similar properties, in particular in terms of the MIT phase diagrams, electronic band structures, and optical conductivity spectra, whereas their magnetic properties are radically different. The formation of an \(ab\)-canted AFM state in Sr\({}_{2}\)IrO\({}_{4}\) is a result of intermingled competition between the isotropic exchange, single-ion anisotropy, and DM interactions, and new magnetic phases (in-plane collinear AFM ordering and \(c\)-collinear AFM ordering) can be developed by tuning the balance of these magnetic interactions through octahedral distortions (octahedral rotations and tetragonal distortions). The \(c\)-collinear AFM ground state of Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\) is very robust due to strong interlayer magnetic coupling. The spin phase transition to the \(ab\)-canted AFM phase can only be achieved by increasing the interlayer distance to a very large and experimentally unobtainable value. The computed \(GW\)+BSE optical conductivity spectra of three iridates show strong excitonic effects and reproduce the experimentally observed double-peak structure very well, in particular for Sr\({}_{2}\)IrO\({}_{4}\) and Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\). However, for SrIrO\({}_{3}\), the description is less satisfactory, since \(GW\) does not account well for the correlated metallic state of SrIrO\({}_{3}\) and going beyond \(GW\) is necessary. Our analysis describes well the progressive redshift of the main optical peaks as a function of dimensionality \(n\), which is correlated with the gradual decrease in electronic correlation \(U\) and gradual closing of the band gaps towards the metallic \(n=\infty\) limit. The major physics of the bulk phases of RP iridates are presently very well understood. Some discrepancies in the interpretation of the optical and electronic properties of the 3 Dend member remain, which will hopefully be resolved in forthcoming studies.