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Physical Review Letters

Theory of Liquid He4 at Zero Temperature

B. H. Brandow

Published Feb 3, 1969• 14 citations
Publisher

Abstract

We present a linked-cluster expansion for the ground state of a many-boson system which permits an exact treatment of particle conservation without introducing a chemical potential. We then propose a program of partial summations designed to permit accurate calculations with realistic He-He potentials. This theory treats the depletion effect in a fully consistent manner.

DOI
10.1103/physrevlett.22.173

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We present a linked-cluster expansion for the ground state of a many-boson system which permits an exact treatment of particle conservation without introducing a chemical potential. We then propose a program of partial summations designed to permit accurate calculations with realistic He-He potentials. This theory treats the depletion effect in a fully consistent manner.
The previous configuration-space cluster integral treatment of the properties of the ground state of a many-boson system is modified by including those higher order diagrams consistent with the pair excitation approximation, which were previously omitted. The cluster expansions involve a parameter, analogous to the fugacity in classical expansions, whose definition automatically accounts for the depletion of the free particle ground state. The resulting expectation value for the Hamiltonian is in agreement with that obtained by field-theoretic methods for states of pair-excitation type.
The Hartree-Fock-Bogoliubov equations for a homogeneous boson system at finite temperature are derived using linearized operator equations of motion. By a partial decoupling of two of the equations, a temperature-dependent form of the Lippmann-Schwinger equation for a local ttt matrix emerges naturally with a fully dressed energy denominator. In order to use a realistic potential the theory is phenomenologically modified by (i) redefining the chemical potential to remove an unphysical energy gap, and (ii) removing (possibly divergent) Hartree-Fock-like terms from the single-particle energy. The resulting set of coupled nonlinear integral equations is solved self-consistently for the Morse-dipole-dipole and the Frost-Musulin potentials. The excitation spectra obtained are in qualitative agreement with the observed phonon-roton spectrum, but the temperature dependence is given incorrectly. The calculations were made at a realistic density and the condensate fraction is about 60% at absolute zero.
A trial wave function describing the ground state of a quantum system of NNN interacting bosons is written in the Jastrow form, a product of pair functions. With the interaction potential chosen to represent liquid He4{\mathrm{He}}^{4}He4, and with the parametrized form of the pair function chosen to include a long-range term which has been found necessary to represent the zero-point motion of the long-wavelength density oscillations, a variational calculation has been performed using a new approximate integral equation for the pair distribution function. This equation, which can also be used for classical fluids, is found to be more accurate for repulsive potentials than the Percus-Yevick equation and comparable to (but much simpler than) the Percus-Yevick 2 equation. The essential results are that including the zero-point motion in the wave function tends to lower the energy, raises the equilibrium density, corrects the behavior of the structure function and the momentum distribution of the particles in the low-wave-number region, and slightly decreases the Bose-Einstein condensate fraction. The value of the lower limit on the wavelength of the density oscillations was determined variationally to be about three interparticle spacings.
The properties of a Bose system of particles with repulsive interactions have previously been treated using perturbation theory in the formalism of second quantization. Others have also considered this problem by dealing with the wave function in configuration space, using the theory of cluster expansions. In these latter papers, variation with respect to a parameter in a trial function for the ground state has been shown to yield a ground-state energy close to the exact asymptotic expressions obtained from perturbation theory. The connection between the two methods is not immediately obvious from these cluster expansion treatments. It is shown here that, as one might expect, the cluster integral method can be handled so that it is completely equivalent to the pair approximation in perturbation theory.
Abstract We define a formalism of a self-consistent description of the ground state of a weakly interacting Bose system, accounting for higher order terms in expansion of energy in the diluteness parameter. The approach is designed to be applied to a Bose–Bose mixture in a regime of weak collapse where quantum fluctuations lead to stabilization of the system and formation of quantum liquid droplets. The approach is based on the generalized Gross–Pitaevskii equation accounting for quantum depletion and renormalized anomalous density terms. The equation is self-consistently coupled to modified Bogoliubov equations. We derive well defined procedure to calculate the zero temperature renormalized anomalous density—the quantity needed to correctly describe the formation of quantum liquid droplet. We pay particular attention to the case of droplets harmonically confined in some directions. The method allows to determine the Lee–Huang–Yang-type contribution to the chemical potential of inhomogeneous droplets when the local density approximation fails.
In order for a boson-expansion theory to be useful for practical purposes, it must satisfy at least two requirements: It must be in the form of a linked-cluster expansion, and the pure (ideal) boson states must be usable as basis states. Previously, we constructed such a boson theory and used it successfully for many realistic calculations. This construction, however, lacked mathematical rigor. In the present paper, we develop an entirely new approach, which results in the same boson expansions obtained earlier, but now in a mathematically rigorous fashion. The achievement of the new formalism goes beyond this. Its framework is much more general and flexible than was that of the earlier formalism, and it allows us to extend the calculations beyond what had been done in the past.NUCLEAR STRUCTURE Boson expansion theories, commutator method, Marumori-Yamamura-Tokunaga method, fermion system, norm matrix, linked-cluster expansion, ideal-boson state, physical boson state, even-even nucleus, odd-AAA nucleus, nuclear collectivity.