Interplay of spin-orbit coupling, crystal field splitting and correlations: a ghost rotationally invariant slave boson treatment

We investigate the interplay of spin-orbit coupling, crystal field splittings, and electronic correlations in the $t_{2g}$ Hubbard-Kanamori model within the recently formulated ghost rotationally invariant slave-boson method (GRISB). In particular, we study a tight binding model of Sr$_2$RuO$_4$ with parameters extracted from density functional theory and linearized quasiparticle self-consistent GW (LQSGW) calculations; we study the behavior of different physical quantities as the number of ghosts increases to examine the convergence of GRISB to dynamical mean field theory (DMFT) and experimental results; and we leverage the ability of GRISB to investigate the model over a wide range of parameters at low temperature. In particular, we examine both static and dynamical observables driven by the spin orbit coupling (SOC) and study how they vary as a function of the Hubbard $U$ and Hund's coupling $J$. GRISB converges quickly for most of these observables, and the calculations reveal the following: $U$ enhances the spin-orbit coupling while $J$ suppresses it. We also study the shape of the Fermi surface within different methodologies, and examine the Lifshitz transition which takes place as a function of strain in this material.