Causal versus local GW+EDMFT scheme and application to the triangular-lattice extended Hubbard model

Using the triangular-lattice extended Hubbard model as a test system, we compare $GW+\mathrm{EDMFT}$ results for the recently proposed self-consistency scheme with causal auxiliary fields to those obtained from the standard implementation which identifies the impurity Green's functions with the corresponding local lattice Green's functions. Both for short- and long-ranged interactions we find similar results, but the causal scheme yields slightly stronger correlation effects at half-filling. We use the two implementations of $GW+\mathrm{EDMFT}$ to compute spectral functions and dynamically screened interactions in the parameter regime relevant for $1T\text{\ensuremath{-}}{\mathrm{TaS}}₂$. These simulations allow us to clarify if the sample used in a recent photoemission study [M. Ligges et al., Phys. Rev. Lett. 120, 166401 (2018)] was half-filled or hole doped.