Pair excitations and vertex corrections in fermi fluids and the dynamic structure function of two-dimensional ³He

We use the equations-of-motion approach for time-dependent pair correlations in strongly interacting Fermi liquids to develop a theory of the excitation spectrum and the single-particle self energy in such systems. We present here the fully correlated equations and their approximate solutions for ₃He. Our theory has the following properties: It reduces to both, i) the "correlated" random-phase approximation (RPA) for strongly interacting fermions if the two-particle-two-hole correlations are ignored, and, ii) to the correlated Brillouin-Wigner perturbation theory for boson quantum fluids in the appropriate limit, iii) It preserves the two first energy-weighted sum rules, and systematically improves upon higher ones, iv) A familiar problem of the standard RPA is that it predicts a roton energy that lies more than a factor of two higher than what is found in experiments. A popular cure for this is to introduce an effective mass in the Lindhard function. No such ad-hoc assumption is invoked in our work. We demonstrate that the inclusion of correlated pair-excitations improves the dispersion relation significantly. Finally, a novel form of the density response function is derived that arises from vertex corrections in the proper polarization.