We describe the development of a systematic theory of excitations in strongly interacting Fermi systems. Technically, we derive the equations of motion for multi–pair excitations from a stationarity principle. This method has, in Fermi systems, so far been developed only to the level of one–particle–one–hole excitations, where it leads to the (correlated) random phase approximation (RPA). We extend the analysis here to pair excitations. Our work is motivated by the fact that time–dependent pair correlations are necessary for explaining the physics of the phonon–roton spectrum in 4 He . It is therefore plausible that the same processes also have visible effects in the excitation spectrum of 3 He . Further motivation is derived from recent measurements of the dynamic structure function in two–dimensional 3 He . We first formulate the theory for a second quantized, weakly interacting Hamiltonian and then generalize the theory to a correlated ground state. We show that the inclusion of Jastrow–Feenberg type correlations leads to prescriptions for calculating weak effective interactions from a microscopic, strongly interacting Hamiltonian.
We present a method for the treatment of correlations in finite Fermi systems in a more consistent way than the random phase approximation (RPA). The main differences with respect to previous approaches are underlined and discussed. In particular, no use of renormalized operators is made. By means of the method of linearization of equations of motion, we derive a set of RPA-like equations depending only on the one-body density matrix. The latter is no more assumed to be diagonal and it is expressed in terms of the
X and
Y amplitudes of the particle-hole phonon operators. This set of nonlinear equations is solved via an iterative procedure, which allows us to calculate the energies and the wave functions of the excited states. After presenting our approach in its general formulation, we test its quality by using metal clusters as a good test laboratory for a generic many-body system. Comparison to RPA shows significant improvements. We discuss also how the present approach can be further extended beyond the particle-hole configuration space, getting an approximation scheme in which energy weighted sum rules are exactly preserved, thus solving a problem common to all extension of RPA proposed until now. The implementation of such approach will be afforded in a future work.
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 3 He . 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.