Comparing the effective enhancement of local and nonlocal spin-orbit couplings on honeycomb lattices due to strong electronic correlations

We investigate the interplay of electronic correlations and spin-orbit coupling (SOC) for a one-band and two-band honeycomb lattice models. The main difference between the two models concerning SOC is that in the one-band case the SOC is a purely nonlocal term in the basis of the ${p}_{z}$ orbitals, whereas in the two-band case with ${p}_{x}$ and ${p}_{y}$ as basis functions, it is purely local. In order to grasp the correlation effects on nonlocal spin-orbit coupling, we apply the TRILEX approach that allows to calculate nonlocal contributions to the self-energy approximately. For the two-band case, we apply dynamical mean-field theory. In agreement with previous studies, we find that for all parameter values in our study, the effect of correlations on the spin-orbit coupling strength is that the bare effective SOC parameter is increased. However, this increase is much weaker in the nonlocal than in the local SOC case. Concerning the TRILEX method, we introduce the necessary formulas for calculations with broken SU(2) symmetry.