Sr$_{2}$IrO$_{4}$ has often been described via a simple, one-band pseudo-spin 1/2 model, subject to electron-electron interactions, on a square lattice, fostering analogies with cuprate superconductors, believed to be well described by a similar model. In this work we argue - based on a detailed study of the low-energy electronic structure by circularly polarized spin and angle-resolved photoemission spectroscopy combined with dynamical mean-field theory calculations - that a pseudo-spin 1/2 model fails to capture the full complexity of the system. We show instead that a realistic multi-band Hubbard Hamiltonian, accounting for the full correlated $t_{2g}$ manifold, provides a detailed description of the interplay between spin-orbital entanglement and electron-electron interactions, and yields quantitative agreement with experiments. Our analysis establishes that the $j_{3/2}$ states make up a substantial percentage of the low energy spectral weight, i.e. approximately 74% as determined from the integration of the $j$-resolved spectral function in the $0$ to $-1.64$ eV energy range. The results in our work are not only of relevance to iridium based materials, but more generally to the study of multi-orbital materials with closely spaced energy scales.

We discuss the efficient implementation of general impurity solvers for dynamical mean-field theory. We show that both Lanczos and quantum Monte Carlo in different flavors (Hirsch-Fye, continuous-time hybridization- and interaction-expansion) exhibit excellent scaling on massively parallel supercomputers. We apply these algorithms to simulate realistic model Hamiltonians including the full Coulomb vertex, crystal-field splitting, and spin-orbit interaction. We discuss how to remove the sign problem in the presence of non-diagonal crystal-field and hybridization matrices. We show how to extract the physically observable quantities from imaginary time data, in particular correlation functions and susceptibilities. Finally, we present benchmarks and applications for representative correlated systems.