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We study competing orders of spinless fermions in the triangular-latticeHubbard model with nearest-neighbor interaction. We calculate the effective,momentum-resolved two-particle vertex in an unbiased way in terms of thefunctional renormalization group method and compare two different schemes forthe momentum discretization, one based on dividing the Fermi surface intopatches and one based on a channel decomposition. We study attractive andrepulsive nearest-neighbor interaction and find a competition of pairing andcharge instabilities. In the attractive case, a Pomeranchuk instability occursat Van Hove filling and $f$-wave and $p$-wave pairing emerge when the fillingis reduced. In the repulsive case, we obtain a charge density wave at Van Hovefilling and extended $p$-wave pairing with reduced filling. The $p$-wavepairing solution is doubly degenerate and can realize chiral $p+ip$superconductivity with different Chern numbers in the ground state. We discussimplications for strongly correlated spin-orbit coupled hexagonal electronsystems such as moir\'e heterostructures.

Functional renormalization of spinless triangular-lattice fermions: $N$-patch vs. truncated-unity scheme