Crossover from Cooper pairs to composite bosons: A generalized RPA analysis of collective excitations

We study the evolution of the ground state and the excitation spectrum of the two- and three-dimensional attractive (negative-U) Hubbard model as the system evolves from a Cooper-pair regime for U much-lt t, to a composite boson regime for U much-gt t. Our work is motivated by the observation that the high-temperature superconductors, with their short coherence lengths and unusual normal-state properties, may be in an intermediate coupling regime between these two limits. A mean-field analysis of pairing, suitably generalized to account for a shift in the chemical potential, is known to be able to describe the ground-state crossover as a function of U/t. We compute the collective-mode spectrum using a generalized random-phase-approximation analysis within the equations-of-motion formalism. We find a smooth evolution of the Anderson mode for weak coupling into the Bogoliubov sound mode for hard-core bosons. We then include a long-range Coulomb interaction and show that it leads to a plasmon which again evolves smoothly from weak to strong coupling