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Physical Review A

Ground State of Liquid He4

William Porter Francis, Geoffrey V. Chester, L. Reatto

Published Jan 1, 1970• 70 citations
Publisher

Abstract

A trial wave function describing the ground state of a quantum system of $N$ interacting bosons is written in the Jastrow form, a product of pair functions. With the interaction potential chosen to represent liquid ${\mathrm{He}}^{4}$, 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.

DOI
10.1103/physreva.1.86

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In this paper we suggest a new ground-state wave function and low-temperature density matrix for a strongly interacting system of bosons. Our basic assumption is that the system can support long-wavelength phonons and that these can propagate independently of any other mode of motion. We therefore write the ground-state function as the product of two factors. One factor arises from the zero-point motion of the phonons, and we show that it has the form ${\ensuremath{\Pi}}_{i<j}f({r}_{\mathrm{ij}})$, where $\mathrm{ln}f(r)$ has an infinite range. The other factor is assumed to have this same form but with $\mathrm{ln}f(r)$ of finite range; it takes into account the short-range correlations arising from the strong repulsive part of the interparticle potential. The function we have chosen to represent the short-range correlations is not new; functions of this kind were first introduced by Bijl and later by Jastrow. At finite temperatures we use a density matrix for an ensemble of excited phonon states. We find that for small wave vectors $k$, the structure factor $S(k)$ is equal to $\frac{\ensuremath{\hbar}k}{2mc}$, a result that was first derived by Feynman. At a finite temperature $T$, $S(k)$ tends to the constant value $\frac{{k}_{B}T}{m{c}^{2}}$, where $m$ is the mass of the particles and $c$ the velocity of propagation of the phonons. The momentum distribution ${n}_{k}$ has a ${k}^{\ensuremath{-}1}$ singularity at absolute zero and a stronger, ${k}^{\ensuremath{-}2}$ singularity at finite temperatures. The small-$k$ behavior of both $S(k)$ and ${n}_{k}$ is completely controlled by the correlations introduced by the phonons. Both the ground-state function and the density matrix imply that there is a finite fraction of particles in the zero-momentum state in three dimensions; this fraction does not seem to be appreciably affected by the infinite-range correlations introduced by the phonons. We find, however, that these correlations imply that a one-dimensional Bose system does not exhibit Bose-Einstein condensation at any temperature, while a two-dimensional system exhibits Bose-Einstein condensation only at absolute zero.

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•Aug 1, 1979•0 citations
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.
In this paper we suggest a new ground-state wave function and low-temperature density matrix for a strongly interacting system of bosons. Our basic assumption is that the system can support long-wavelength phonons and that these can propagate independently of any other mode of motion. We therefore write the ground-state function as the product of two factors. One factor arises from the zero-point motion of the phonons, and we show that it has the form ${\ensuremath{\Pi}}_{i<j}f({r}_{\mathrm{ij}})$, where lnf(r)\mathrm{ln}f(r)lnf(r) has an infinite range. The other factor is assumed to have this same form but with lnf(r)\mathrm{ln}f(r)lnf(r) of finite range; it takes into account the short-range correlations arising from the strong repulsive part of the interparticle potential. The function we have chosen to represent the short-range correlations is not new; functions of this kind were first introduced by Bijl and later by Jastrow. At finite temperatures we use a density matrix for an ensemble of excited phonon states. We find that for small wave vectors kkk, the structure factor S(k)S(k)S(k) is equal to $\frac{\ensuremath{\hbar}k}{2mc}$, a result that was first derived by Feynman. At a finite temperature TTT, S(k)S(k)S(k) tends to the constant value kBTmc2\frac{{k}_{B}T}{m{c}^{2}}mc2kB​T​, where mmm is the mass of the particles and ccc the velocity of propagation of the phonons. The momentum distribution nk{n}_{k}nk​ has a ${k}^{\ensuremath{-}1}$ singularity at absolute zero and a stronger, ${k}^{\ensuremath{-}2}$ singularity at finite temperatures. The small-kkk behavior of both S(k)S(k)S(k) and nk{n}_{k}nk​ is completely controlled by the correlations introduced by the phonons. Both the ground-state function and the density matrix imply that there is a finite fraction of particles in the zero-momentum state in three dimensions; this fraction does not seem to be appreciably affected by the infinite-range correlations introduced by the phonons. We find, however, that these correlations imply that a one-dimensional Bose system does not exhibit Bose-Einstein condensation at any temperature, while a two-dimensional system exhibits Bose-Einstein condensation only at absolute zero.
Within the framework of the variational theory for strongly interacting Bose liquids, we study the optimal determination of three-body correlation factors. The Euler-Lagrange equation for the three-body correlations suggests an iterative procedure in which the three-body correlation factor is expressed as a series of diagrams containing only the pair distribution function. The long-wavelength behavior of the three-body structure function agrees formally with the prediction of quantum hydrodynamics. The simplest approximation for the optimized three-body correlation function is identical to the prediction of second-order correlated-basis-function perturbation theory. We show that this simplest approximation sums essentially the same sets of diagrams that are included in non-optimized hypernetted-chain calculations with three-body correlations. Along with the three-body correlations we include those elementary diagrams of fourth and fifth order which have a comparable topological structure. The predictions for the ground-state energy and the pair distribution function are substantially improved compared with hypernetted-chain calculations restricted to two-body correlations.
It is verified that the class of wave functions used in variational calculations for 4He^{4}\mathrm{He}4He has Bose-Einstein condensation. These trial wave functions have the form of a product of pair functions. Two cases are considered: In the first the correlations in the wave function are assumed to be short range and in the second the correlations include also the long-range part due to the zero-point motion of phonons. The one- and two-dimensional system is briefly considered.
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.
"A theory is developed to allow the systematic calculation of the properties of quantum fluids given the microscopic two-body potential. Physical arguments are used to construct a correlation operator that approximately transforms the states of a non-interacting system into the corresponding states of the interacting-system. The theory can be regarded as a microscopic realization of Landau's assumption of a one-to-one correspondence between the non-interacting and interacting states. It is variational since the correlation operator is determined by minimizing E(p) the expectation value of the ground-state energy. We first give a survey of various other many-body theories including the simpler Jastrow variational method. Differential equations are derived to calculate the correlation operator and the related Jastrow correlation"" function. A comparison of results in several Bose systems with various methods is given. The Fermi hypernetted chain integral equations are generalized to allow the calculation of E(p) in Fermi systems, using the correlation operator wave function. The method is applied to liquid 3He and the v2-model of nuclear matter. In 3He the correlation operator gives results in much closer agreement with experiment than that with the Jastrow wave function. A new set of hypernetted chain equations are then derived to allow the calculation of the excited st;ate energies in Bose systems. The theory is then applied to the calculation of the single quasi-particle excitation spectrum of liquid 4He. The spectrum is found to be a stringent test on the form of the correlation operator, deficiencies in the correlation operator giving corresponding errors in the spectrum. Good results are obtained for the density dependence of the roton energy. A variational principle for the free energy is used to generalize the correlation operator technique to finite temperatures. Results are given for the finite temperature properties of the v8-model of nuclear matter."