Correlated density matrix theory of normal quantum fluids

Abstract

A variational description of the normal phase of a strongly interacting N-boson fluid at finite temperature is formulated in terms of trial density matrices designed to incorporate the essential dynamical and statistical correlations. The approach followed is a natural extension, to the noncondensed phase, of the familiar Jastrow-Feenberg variational approximation of correlated-basis functions theory, which has provided a firm foundation for microscopic calculations of the properties of liquid 4He at zero temperature. The N-body density matrix elements of the homogeneous fluid are represented in the general form W(R, R′) = J −1 φ(R) Q(R, R′) φ(R′) involving a normalization integral J = ∫ dR φ 2(R) Q(R, R), where R denotes a point in the configuration space of the system. This form is specialized by assuming φ( R) to be a wave function of Jastrow type, i.e., a product of two-body dynamical correlation functions f(r) = exp[ u(r) 2 ] , and taking Q( R, R′) to be a permanent part of two-body statistical functions Γ( r). It is in the latter assumption, which tailors the treatment to the normal phase, that the present approach departs from earlier work within variational density matrix theory, which focuses instead on the condensed or superfluid phase. The internal energy, the spatial distribution functions, and the one-body density matrix of the system may all be constructed from variational derivatives of the normalization integral J (or of a generalized version of J ), carried out with respect to the dynamical or statistical correlation functions entering the theory. The required normalization integrals are susceptible to analysis in terms of generalized Ursell-Mayer cluster diagrams; systematic diagrammatic resummations based on the hypernetted-chain (HNC) classification scheme are then performed following standard prescriptions. Neglecting elementary diagrams, one arrives at explicit expressions for the indicated equilibrium properties as functionals of the trial correlations u( r) and Γ( r). Making use of the replica technique, it is possible to treat the entropy by the same procedures. With appropriate simplifications to facilitate the associated analytic continuation, one arrives at an entropy expression equivalent to that of a system of noninteracting bosons occupying single-particle momentum states k with an occupation number n cc ( k) determined by the solution of the HNC equations arising in the treatment of the internal energy. In the culminating step of the formal development, an explicit functional expression for the Helmholtz free energy is assembled from the results obtained for the internal energy and entropy, and the Gibbs-Delbrück-Molière minimum principle is invoked. Functional variation of the free energy with respect to the correlation function u( r) and a renormalized statistical correlation function Γ cc ( r) leads to coupled Euler-Lagrange equations which are the analogs of the paired-phonon equation for a Bose superfluid at zero temperature and of the Feynman eigenvalue equation determining the elementary excitations of the superfluid phase. The solutions provide the ingredients of a condition that signals the occurrence of a Bose-Einstein transition in the correlated system. Specific attention is given to the inclusion of phonon effects in the ansatz for the correlated density matrix, a necessary refinement for quantitative treatment of the gas-liquid spinodal line and critical point. An analogous correlated density matrix formalism is developed for the normal phase of a strongly interacting fluid of spinless fermions at finite temperature. In the limiting case of zero temperature, this approach reproduces the well-known Fermi hypernetted-chain theory of the Jastrow-correlated Fermi sea.