Functional renormalization group analysis of spinless fermions on the honeycomb lattice beyond half filling

We study spinless fermions on a honeycomb lattice with repulsive interactions (spinless $t-V$ model), which for the case of half filling is known to feature a Gross-Neveu chiral Ising quantum critical point at finite interaction strength. The phase diagram of this model for a finite chemical potential however remains elusive. We therefore perform an instability analysis using the functional renormalization group method with a basic Fermi surface patching scheme, which allows us to treat instabilities in different channels on an equal footing. The chemical potential is fixed at the beginning of the flow and defines the filling in terms of the free system. Below half filling but above the van-Hove filling the free Fermi surface is hole-like and we find a commensurate charge-density-wave to be dominant. Its characteristics are those of the half-filled case, which indicates phase separation at large interaction strength. Directly at the van-Hove filling the nesting property of the free Fermi surface stabilizes a dimerized charge-bond order phase. At lower fillings the free Fermi surface becomes electron-like and a superconducting instability with $f$-wave symmetry emerges. Finally we estimate the extend of the various phases and give expressions for their order parameters.