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

We introduce a family of (semi) bold-line series, assisted with $1/\text{N}_{f}$ 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 $\textbf{102}$, 195122 (2020)]. We performed extensive benchmarks for density, energy and pressure with $\mathrm{t}-\mathrm{t}'$ $\mathrm{SU}(\mathrm{N}_{f})$ 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.