  In the quest of new exotic phases of matter due to the interplay of various
interactions, iridates hosting a spin-orbit entangled $j_{\mathrm{eff}}=1/2$
ground state have been in the spotlight in recent years. Also in view of
parallels with the low-energy physics of high-temperature superconducting
cuprates, the validity of a single- or few-band picture in terms of the
$j_{\mathrm{eff}}$ states is key. However, in particular for its structurally
simple member Ba$_2$IrO$_4$, such a systematic construction and subsequent
analysis of minimal low-energy models are still missing. Here we show by means
of a combination of different ab initio techniques with dynamical mean-field
theory that a three-band model in terms of Ir-$j_{\mathrm{eff}}$ states fully
retains the low-energy physics of the system as compared to a full Ir-$5d$
model. Providing a detailed study of the three-band model in terms of
spin-orbit coupling, Hund's coupling and Coulomb interactions, we map out a
rich phase diagram and identify a region of effective one-band metal-insulator
transition relevant to Ba$_2$IrO$_4$. Compared to available angle-resolved
photoemission spectra, we find good agreement of salient aspects of the
calculated spectral function and identify features which require the inclusion
of non-local fluctuations. In a broader context, we envisage the three- and
five-band models developed in this study to be relevant for the study of doped
Ba$_2$IrO$_4$ and to clarify further the similarities and differences with
cuprates.
