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Spinless fermions on the honeycomb lattice with repulsive nearest-neighborinteractions are known to harbour a quantum critical point at half-filling,with critical behaviour in the Gross-Neveu (chiral Ising) universality class.The critical interaction strength separates a weak-coupling semimetallic regimefrom a commensurate charge-density-wave phase. The phase diagram of this basicmodel of correlated fermions on the honeycomb lattice beyond half-filling is,however, less well established. Here, we perform an analysis of its many-bodyinstabilities using the functional renormalization group method with a basicFermi surface patching scheme, which allows us to treat instabilities incompeting channels on equal footing also away from half-filling. Betweenhalf-filling and the van-Hove filling, the free Fermi surface is hole-like andwe again find a charge-density wave instability to be dominant at largeinteractions. Moreover, its characteristics are those of the half-filled case.Directly at the van-Hove filling the nesting property of the free Fermi surfacestabilizes a dimerized bond-order phase. At lower filling the free Fermisurface becomes electron-like and a superconducting instability with $f$-wavesymmetry is found to emerge from the interplay of intra-unitcell repulsion andcollective fluctuations in the proximity to the charge-density waveinstability. We estimate the extent of the various phases and extract thecorresponding order parameters from the effective low-energy Hamiltonians.

Bond-ordered states and $f$-wave pairing of spinless fermions on the honeycomb lattice