J-freezing and Hund's rules in spin-orbit-coupled multiorbital Hubbard models

We investigate the phase diagram of the spin-orbit-coupled three orbital Hubbard model at arbitrary filling by means of dynamical mean-field theory combined with continuous-time quantum Monte Carlo. We find that the spin-freezing crossover occurring in the metallic phase of the non-relativistic multiorbital Hubbard model can be generalized to a $\mathbf{J}$-freezing crossover, with $\mathbf{J}=\mathbf{L}+\mathbf{S}$, in the spin-orbit-coupled case. In the $\mathbf{J}$-frozen regime the correlated electrons exhibit a non-trivial flavor selectivity and energy dependence. Furthermore, in the regions near $n=2$ and $n=4$ the metallic states are qualitatively different from each other, which reflects the atomic Hund's third rule. Finally, we explore the appearance of magnetic order from exciton condensation at $n=4$ and discuss the relevance of our results for real materials.