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We investigate the superconducting (SC), charge-density wave (CDW), and antiferromagnetic (AFM) phases in the extended Hubbard model at zero temperature and half-filling. We employ the iterated equations of motion approach to compute the two-particle Green's functions and their spectral densities. This renders a comprehensive analysis of the behavior of collective excitations possible as the model's parameters are tuned across phase transitions. We identify the well-known amplitude (Higgs) and phase (Anderson-Bogoliubov) modes within the superconducting phase and observe a similar excitation (cooperon) in the CDW phase which shifts towards zero energy as the system approaches the phase transition to the SC phase. In the CDW phase, close to the phase transition to the AFM phase, we find a collective mode, an exciton, that does not change significantly and another mode, a longitudinal magnon, that emerges from the two-particle continuum as the system approaches the phase transition to the AFM phase. It becomes identical with the former at the transition. In the AFM phase, their roles are reversed. Additionally, we find a transversal Goldstone magnon located at zero energy.
Althüser et al. (Wed,) studied this question.