Random-phase-approximation approach to collective modes around inhomogeneous Hartree-Fock states: One-dimensional doped Hubbard model

All the linear excitations are calculated in the random-phase approximation around a doped inhomogeneous Hartree-Fock state as well as the undoped uniform antiferromagnetic state in the one-dimensional Hubbard model. Upon doping, shape modes appear, localized around an introduced hole, as well as some ‘‘shake-off’’ branches related to combinations of local and extended Hartree-Fock orbitals. These modes are in addition to spin wave and ultragap branches in the pure antiferromagnetic state. Some characteristics of the shape modes are similar to bare processes in the t-t’-J model derived from the Hubbard model at strong coupling. Spectral weights in different susceptibility channels are calculated and shown to have peaks in low-energy regions, arising from specific shape modes.