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Affordable Five-Orbital Dynamical Mean-Field Theory for Layered Iridates and Rhodates

Full $d$-manifold DMFT with numerically exact solvers has remained computationally prohibitive for spin-orbit materials due their scaling and severe sign problem, forcing the community to rely on simplified one- and three-band models that omit the $e_g$ states despite their proximity with the $t_{2g}$ orbitals. We present the first full five-orbital Dynamical Mean-Field Theory (DMFT) calculations including spin-orbit coupling for the layered iridates and rhodates \bio~and \bro, revealing that the correlation effects shift significantly the $e_g$ states through static mean-field corrections rather than dynamical fluctuations. Motivated by this insight, we introduce hybrid-DMFT (hDMFT), which treats these orbitals and their coupling to the low-energy manifold at the mean-field level while maintaining near quantitative accuracy at a drastically reduced computational cost. These calculation establish hDMFT as a practical and accurate method for full $d$-manifold studies of layered iridates and rhodates, enabling systematic investigations of temperature, doping and pressure dependence that were previously computationally intractable.