M. L. Ristig

Long-range Jastrow correlations

Abstract We develop a promising many-body method to evaluate the equation of state for dense neutron matter and liquid helium. The ground state of the Fermi fluid is described by a conventional Jastrow ansatz. We admit the presence of short- and long-range correlations. Under this assumption we study the generating function which has been introduced by Wu and Feenberg. We employ a graphic formulation and develop the diagrammatic expansion of the generating function and the radial distribution function. If long-range correlations are assumed, the diagrams have singular parts. We give a proof that the total contribution of such diagrams to the generating function which contain two, three, and four correlation lines is of finite value. The same property is shown for a selected class of singular diagrams containing α correlation lines (α>4). To verify the cancellation phenomenon we introduce a two-body function which serves graphically as an insertion into selected singular diagrams. For the remaining classes of diagrams we need three-, four-, ⋯, n-body insertions. The result is cast into the form of a theorem. The cancellation rests on the exclusion principle and does not depend on the special shape of the correlation function. Finally, a generalized hypernetted-chain summation of diagrams which represent the radial distribution function is executed. The procedure includes exchange contributions and can be employed if short-and/or long-range correlations are present.

Hypernetted-chain approximation for dense Fermi fluids

Abstract The ground-state of a Fermi fluid is approximated by a Jastrow-ansatz. The factorized Iwamoto-Yamada expansion of the radial distribution function is suitably rearranged. Thereupon a hypernetted-chain summation of this expansion is executed which includes exchange contributions.

Studies in the method of correlated basis functions

Abstract The method of correlated basis functions provides a framework for quantitative treatments of infinitely extended nuclear matter, finite nuclei and the helium liquids. The essential formal apparatus of this approach is surveyed. A perturbation theory is described which provides a means of systematic solution of the matrix eigenvalue problem in a non-orthogonal correlated basis. General cluster-expansion algorithms are offered for evaluation of the required matrix elements in the correlated representation. Specific applications to infinite nuclear matter with non-central interactions and to the odd-parity levels of the 16 O nucleus are presented.

The Iwamoto-Yamada cluster expansion: Its structure and renormalization

Abstract We analyze in detail the structure of the Iwamoto-Yamada expansion of the correlated energy expectation value for a homogeneous Fermi medium. By the introduction of induced two-, … n -body potentials, the n -body cluster, 2 ⩽ n ≦ A , is decomposed into a proper n -body part and m -body combination terms, 2 ≦ m ≦ n . Explicit formulae are presented through the five-body case. Examination of these structural results leads to a factorization theorem for the total weight function associated with a given (diagonal) induced-potential matrix element and then to a renormalization of the complete energy expansion and a renormalization of the singleorbital weight function. The definition of the latter function is extended in such a way as to exploit previous findings by Wu and Feenberg; thereupon a functional of this generalized distribution analogous to the grand-canonical partition function of classical statistical mechanics is brought into play and the renormalized theory recast in a compact (and possibly more general) variational form. Specific renormalized theories emerge upon specification of the two-, … n -body correlation operators, interesting prescriptions being supplied by the Jastrow, unitary operator and the Brueckncr-Bethe-Goldstone schemes.

Bose-Einstein condensation and the λ transition in liquid helium

Integrating seminal ideas of London, Feynman, Uhlenbeck, Bloch, Bardeen, and other illustrious antecessors, this paper continues the development of an ab initio theory of the λ transition in liquid 4He. The theory is based upon variational determination of a correlated density matrix suitable for description of both normal and superfluid phases, within an approach that extends to finite temperatures the very successful correlated wave-functions theory of the ground state and elementary excitations at zero temperature. We present the results of a full optimization of a correlated trial form for the density matrix that includes the effects both of temperature-dependent dynamical correlations and of statistical correlations corresponding to thermal phonon/roton and quasiparticle/hole excitations—all at the level of two-point descriptors. The optimization process involves constrained functional minimization of the associated free energy through solution of a set of Euler–Lagrange equations, consisting of a generalized paired-phonon equation for the structure function, an analogous equation for the Fourier transform of the statistical exchange function, and a Feynman equation for the dispersion law of the collective excitations. Violation of particle-hole exchange symmetry emerges as an important aspect of the transition, along with broken gauge symmetry. In conjunction with a semi-phenomenological study in which renormalized masses are introduced for quasiparticle/hole and collective excitations, the results suggest that a quantitative description of the λ transition and associated thermodynamic quantities can be achieved once the trial density matrix is modified—notably through the addition of three-point descriptors—to include backflow effects and allow for ab initio treatment of important variations in effective masses.

Correlated one-body density matrix of boson superfluids

The correlated density matrix theory is employed and further developed to analyze the one-body density matrix ρ1(|r1-r2|) of the normal and superfluid phases of a strongly interacting Bose system at non-zero temperature. The approach continues the formal development described in an earlier article and is based on a suitable trial ansatz for the many-body density matrixW(R, R′)∼Φ(R) Q(R, R′) Φ(R′) with the wave function Φ and incoherence factorQ incorporating the essential statistical and dynamical correlations. Special attention is given to the appearance of off-diagonal long-range order in function ρ1(|r1-r2|) and its relation to the condensation strength Bcc characterizing the degree of coherence in the superfluid phase. We derive a number of structural relations that have counterparts in known results on ρ1 in the Jastrow variational theory of the Bose ground state. We discuss Bose-Einstein condensation and make contact to Landaus phenomenological theory of continuous phase transitions. Numerical estimates are presented on the condensation strength and the condensate fraction of liquid4He as functions of the temperature.

Equivalent nonlocal nucleon-nucleon potentials

The local nucleon-nucleon potential which contains a strong repulsive core is often replaced by an equivalent weak but non-local potential. Such potentials can be obtained by unitary transformations of the two-body Hamiltonian. Necessary and sufficient conditions are derived which must be fulfilled by the unitary transformations in order to give equivalent potentials. Special examples of unitary transformations with this property are investigated.

Condensate fraction and momentum distribution of liquid helium

Abstract An expansion is presented which allows practical evaluation of the one-particle density matrix of the ground state of a Bose fluid in terms of its two-body, three-body, … spatial distribution functions.

Ab initio treatments of the Ising model in a transverse field

In this article new results are presented for the zero-temperature ground-state properties of the spin-half transverse Ising model on various lattices using three different approximate techniques. These are, respectively, the coupled cluster method, the correlated basis function method, and the variational quantum Monte Carlo method. The methods, at different levels of approximation, are used to study the ground-state properties of these systems, and the results are found to be in excellent agreement both with each other and with results of exact calculations for the linear chain and results of exact cumulant series expansions for lattices of higher spatial dimension. The different techniques used are compared and contrasted in the light of these results, and the constructions of the approximate ground- state wave functions are especially discussed.

Pairing energy of liquid 4He

Abstract The two-particle density matrix of a Bose system described by a Jastrow wave function displays off-diagonal long-range order associated with strong correlations between pairs of bosons with non-zero momenta ħ q , -ħ q . In conjunction with the zero-momentum condensate, these correlations give rise to a finite contribution to the energy expectation value per particle, which is calculated for liquid 4 He at two values of the density.