Doping evolution of the magnetic excitations in the monolayer CuO₂

In this paper, we study theoretically the doping evolution behaviors of the magnetic excitations (MEs) in the monolayer CuO₂grown on Bi₂Sr₂CaCu₂O_8+δsubstrate. For the undoped system, the MEs exhibit the low energy commensurate behavior around (π,π). They turn to be incommensurate when the system is slightly hole-doped. In the intermediate doping regime, the low energy MEs diminish gradually. They turn to be dominated by the high energy modes. With further doping, an exotic structure transition of the MEs occurs in the heavily hole-doped regime which is directly related to the Lifshitz transition. Distinct MEs are separated by the transition point around which the low energy MEs exhibit the ring-like structure around (0, 0). Before the transition, the MEs are dominated by the broad particle-hole continuum at very high energies. In contrast, across the transition point, two new low energy modes develop around (0, 0) and (π,π) attributing to the intrapocket and interpocket particle-hole scatterings, respectively.