Charge density wave and pairing order in the Holstein model on the honeycomb lattice away from half-filling

Recent studies have revealed that the Holstein model on a honeycomb lattice generates charge density wave (CDW) order at half-filling if the electron-phonon interaction strength surpasses a critical threshold and this transition occurs at a finite temperature. Here we consider the honeycomb-lattice Holstein model away from half-filling and employ the determinant quantum Monte Carlo method to explore the fate of CDW and possible superconductivity. We find that CDW order persists under light doping conditions, albeit with a reduced CDW region compared to the half-filling scenario. Similar to half-filling, the transition from semimetal order to CDW order also occurs at a finite temperature. Additionally, we investigate antiferromagnetic order and s-wave pairing order in the presence of doping.