Convergence of energy-dependent incommensurate antiferromagnetic neutron scattering peaks to commensurate resonance in underdoped bilayer cuprates

The recently discovered coexistence of incommensurate antiferromagnetic neutron scattering peaks and commensurate resonance in underdoped YBa2Cu3O6+x is calling for an explanation. Within the t-J model, the doping and energy dependence of the spin dynamics of the underdoped bilayer cuprates in the normal state is studied based on the fermion-spin theory by considering the bilayer interactions. Incommensurate peaks are found at [(1 ± δ)π, π] and [π, (1 ± δ)π] at low energies with δ initially increasing with doping at low dopings and then saturating at higher dopings. These incommensurate peaks are suppressed, and the parameter δ is reduced with