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Physical Review

Ground State of Liquid Helium

Ira L. Karp

Published Jul 15, 1959• 49 citations
Publisher

Abstract

A calculation scheme suitable for studying the ground state of many-body Bose quantum systems with strong forces is developed. The calculation scheme is based on the nodeless property of the ground-state wave function and is capable of yielding in some cases upper and lower bounds on the ground-state energy and corresponding wave functions. From a treatment of liquid helium in the ground state, including a short-range interaction form, it is concluded that the calculation scheme is suitable for problems with involved interactions and is capable of reducing the problem to the mathematical one of calculation of the energy and lattice structure of a classical crystal at absolute zero of temperature.

DOI
10.1103/physrev.115.223

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A calculation scheme suitable for studying the ground state of many-body Bose quantum systems with strong forces is developed. The calculation scheme is based on the nodeless property of the ground-state wave function and is capable of yielding in some cases upper and lower bounds on the ground-state energy and corresponding wave functions. From a treatment of liquid helium in the ground state, including a short-range interaction form, it is concluded that the calculation scheme is suitable for problems with involved interactions and is capable of reducing the problem to the mathematical one of calculation of the energy and lattice structure of a classical crystal at absolute zero of temperature.
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.
It is usually assumed that the Bose crystal at T=0T=0T=0 corresponds to the genuine ground state of a Bose system, i.e., this state is non-degenerate and is described by the wave function without nodes. By means of symmetry analysis we show that the ground state of a Bose system of any density should correspond to a liquid or gas, but not to a crystal. The main point is that any anisotropic state of a system of spinless bosons is degenerate. We prove this for an infinite three-dimensional (3D) system and a finite ball-shaped 3D system. One can expect that it is true also for a finite system of any form. Therefore, the anisotropic state cannot be the genuine ground state. Hence, a zero-temperature natural 3D crystal should correspond to an excited state of a Bose system. The wave function Ψ0cΨ^{c}_{0}Ψ0c​ of a zero-temperature 3D Bose crystal is proposed for zero boundary conditions. Apparently, such Ψ0cΨ^{c}_{0}Ψ0c​ corresponds to a local minimum of energy (absolute minimum corresponds to a liquid). Those properties yield the possibility of existence of superfluid liquid H2_{2}2​, Ne, Ar, and other inert elements. We propose several possible experimental ways of obtaining them.
We present a variational ansatz for the ground state of a strongly correlated Bose system. This ansatz goes beyond the Jastrow-Feenberg functional form and explicitly enforces coordination shells in the structure of the wave function. We apply this ansatz to liquid helium-4 with a simple three-variable parametrization of the pair functions. The optimized wave function is found to give an excellent description of the mid-range correlations in the fluid. We also demonstrate the possibility to use this ansatz to study inhomogeneous systems. The phase separation and free surface emerge naturally in this wave function, even though it is constructed of short-range two-body functions and does not contain one-body terms. Because no explicit description of the surface is necessary, this provides a powerful description tool for cluster states.
The mathematical description of B.E. (Bose-Einstein) condensation is generalized so as to be applicable to a system of interacting particles. B.E. condensation is said to be present whenever the largest eigenvalue of the one-particle reduced density matrix is an extensive rather than an intensive quantity. Some transformations facilitating the practical use of this definition are given.An argument based on first principles is given, indicating that liquid belium II in equilibrium shows B.E. condensation. For absolute zero, the argument is based on properties of the ground-state wave function derived from the assumption that there is no "long-range configurational order." A crude estimate indicates that roughly 8% of the atoms are "condensed" (note that the fraction of condensed particles need not be identified with $\frac{{\ensuremath{\rho}}_{s}}{\ensuremath{\rho}}$). Conversely, it is shown why one would not expect B.E. condensation in a solid. For finite temperatures Feynman's theory of the lambda-transition is applied: Feynman's approximations are shown to imply that our criterion of B.E. condensation is satisfied below the lambda-transition but not above it.
In the interest of polishing the parquet approach to microscopic calculations of ground-state properties of zero-temperature quantum fluids, we have developed a perturbative scheme for the systematic improvement of the four-point function which retains the correct behavior of the parquet diagrams in both the small-r and small-k limits. Numerical implementation of this scheme indicates significantly improved agreement with binding energies calculated with Green's-function Monte Carlo techniques (i.e., differences in liquid 4He^{4}\mathrm{He}4He are less than 1.5%). The charged Bose gas and the hard-sphere system are also considered. Remaining difficulties in the calculations of the liquid structure function are noted and discussed.
A new variational wave function is proposed for the ground state of Bose liquids. It consists of a Jastrow function multiplied by exponentials of three-particle functions. The form of the wave function is deduced from a formal analysis carried out for the weakly interacting Bose gas characterized by a strength parameter $\ensuremath{\lambda}$. By using the wave function proposed here, the energy obtained is exact to $O({\ensuremath{\lambda}}^{4})$. Several useful and straightforward applications of the new method to liquid helium and Coulomb gases are suggested.