Inhomogeneous quantum liquids: Statics, dynamics, and thermodynamics

Abstract

We review the variational theory for inhomogeneous Bose and Fermi fluids. Euler-Lagrange equations are derived for the one- and two-body correlations and systematic approximation methods are physically as well as formally motivated. For bosons, it is shown that the optimized variational method corresponds to a self-consistent approximate summation of ladder and ring diagrams. The connection between time-dependent Hartree theory and the equations of motion for strongly correlated Bose liquids is highlighted.

The finite-temperature extension of the boson theory is discussed. Drawing connections to the random phase approximation and linear response theory at finite temperatures, it is shown that present implementations of a finite-temperature version of variational theories are equivalent to a relatively simple quasiparticle description of the thermodynamics.

Building on the optimized ground states, we next develop the equations of motion method for the study of excitations and linear response in inhomogeneous Bose systems. For weakly interacting systems, the theory is equivalent to the Feynman theory of collective excitations in quantum liquids, or to the random phase approximation. When the interactions are strong, the theory amounts to a theory of effective interactions in which the strongly interacting system is mapped onto a weakly interacting one.

Finally, we turn to a systematic study of Fermi systems. It is shown how the choice of an optimal single-particle basis facilitates the formulation of Euler equations and improves the convergence of cluster expansion and integral equation methods. To interpret the Fermi theory, we develop a version of time-dependent Hartree-Fock theory for strongly interacting systems. Unlike time-dependent Hartree theory, new effects, in particular the particle-hole continuum, enter the description of the excited states. We highlight the close relationship between diagrams of the FHNC variational theory and the random phase approximation, and how the time-dependent theory can serve to improve upon the FHNC description of the ground state.