Semi-Deterministic and Stochastic Sampling of Feynman Diagrams with 1/N${}_f$ Expansions

We introduce a family of (semi) bold-line series, assisted with $1\mathrm{/}{\text{N}}_f$ expansions, with ${\text{N}}_f$ being the number of fermion flavours. If there is no additional ${\mathrm{N}}_f$ cut, the series reduces to the RPA series in the density-density channel, complementary to the particle-hole and particle-particle channels introduced in [Phys. Rev. B $\mathbf{\text{102}}$, 195122 (2020)]. We performed extensive benchmarks for density, energy and pressure with $\mathrm{t}-{\mathrm{t}}^{\mathrm{\prime }}$ $\mathrm{S}\mathrm{U}({\mathrm{N}}_f)$ Hubbard model on square lattice and honeycomb lattices over a wide range of numerical methods. For benchmark purposes, we also implement bare-$\mathrm{U}$ symmetry-broken perturbation series for the 2D SU(2) Hubbard model on the honeycomb lattice, where we found encouraging results from weak to intermediate couplings.