The spin-orbit Mott insulator Sr${}_2$IrO${}_4$ has attracted a lot of interest in recent years from theory and experiment due to its close connection to isostructural high-temperature copper oxide superconductors. Despite of not being superconducting its spectral features closely resemble those of the cuprates, including Fermi surface and pseudogap properties. In this article, we review and extend recent work in the theoretical description of the spectral function of pure and electron-doped Sr${}_2$IrO${}_4$ based on a cluster extension of dynamical mean-field theory (oriented-cluster DMFT) and compare it to available angle-resolved photoemission data. Current theories provide surprisingly good agreement for pure and electron-doped Sr${}_2$IrO${}_4$, both in the paramagnetic and antiferromagnetic phases. Most notably, one obtains simple explanations for the experimentally observed steep feature around the $M$ point and the pseudo-gap-like spectral feature in electron-doped Sr${}_2$IrO${}_4$.
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