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We investigate the interplay of electronic correlations and spin-orbitcoupling (SOC) for a one-band and a two-band honeycomb lattice model. The maindifference between the two models concerning SOC is that in the one-band casethe SOC is a purely non-local term in the basis of the $p_z$ orbitals, whereasin the two-band case with $p_x$ and $p_y$ as basis functions it is purelylocal. In order to grasp the correlation effects on non-local spin-orbitcoupling, we apply the TRILEX approach that allows to calculate non-localcontributions to the self-energy approximatively. For the two-band case weapply dynamical mean-field theory. In agreement with previous studies, we findthat for all parameter values in our study, the effect of correlations on thespin-orbit coupling strength is that the bare effective SOC parameter isincreased. However, this increase is much weaker in the non-local than in thelocal SOC case. Concerning the TRILEX method, we introduce the necessaryformulas for calculations with broken SU(2) symmetry.

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