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Canadian Journal of Physics

The momentum distribution of normal and superfluid liquid 4He

David M. Ceperley, E. L. Pollock

Published Nov 1, 1987• 90 citations
Publisher

Abstract

Path-integral methods can be employed to calculate properties of boson systems at any temperature. We have performed a number of simulations of liquid helium and have calculated the momentum distribution as a function of temperature. We find that liquid helium has a condensate below 2 K but not above. The momentum distribution is found to be non-Gaussian even in normal liquid helium. We have also calculated the difference in the momentum distribution between a boson system and a distinguishable particle system.

DOI
10.1139/p87-222

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Europhysics Letters (EPL)•Nov 1, 1997•9 citations
Path-integral methods can be employed to calculate properties of boson systems at any temperature. We have performed a number of simulations of liquid helium and have calculated the momentum distribution as a function of temperature. We find that liquid helium has a condensate below 2 K but not above. The momentum distribution is found to be non-Gaussian even in normal liquid helium. We have also calculated the difference in the momentum distribution between a boson system and a distinguishable particle system.
Below Tλ = 2.17 K, bulk 4He is a superfluid and has a non-zero Bose-Einstein condensate fraction. This work will focus primarily on how phenomena such as superfluidity, Bose condensation and superfluid vortices are manifested in microscopic, inhomogeneous helium systems. Path Integral Monte Carlo is a powerful method for calculating the equilibrium properties of quantum systems at finite temperature. We have achieved linear scaling of computer time with number of particles through the use of neighbor lists, allowing us to simulate systems of several thousand atoms. We have derived a local superfluid estimator and used it to examine the microscopic superfluid response around a molecule rotating in a helium nanodroplet. We found that the first solvation layer is well-described by a two dimensional superfluid, with thermal excitations occuring at a lower temperature than in bulk helium. The effective moment of inertia of a linear impurity in a helium droplet is calculated, and compared with experimental scattering results. In addition, we calculated the vortex formation energy for both pure droplets and droplets doped with linear impurities, and found that the linear impurities had a negligible impact on the formation energy. A possible spectroscopic signature of vortices in doped helium droplets was suggested. After deriving a local estimator, we calculated the condensate fraction throughout the free helium surface of a semi-infinite slab. These results, along with densitydensity correlation functions, were used to characterize the surface excitations and calculate the extent to which ripplons are present. In addition, the ripplon dispersion relation was calculated using imaginary-time correlation functions, and found to be lll in good agreement with experimental results. Finally, we have calculated the Dey be-Waller factor in solid helium for a range of temperatures and densities, and compared the scaling behavior with the predictions of harmonic theory. The first non-Gaussian contribution to the density distribution was calculated.
Abstract In this chapter we introduce some basic concepts of superfluid 4He that we will draw on in Chapters 8-12. Liquid 4 He has a long and rich history and there are many concepts that are implicit in the field today. We begin with a sketch of the unique mass and heat flow character of superfluid 4He. The assumptions of the two-fluid model used successfully to describe these properties are noted in §7.1. We introduce Bose-Einstein condensation in a Bose gas and the extension of this to interacting superfluid 4 He in §7.2. This leads naturally to the one-body density matrix which is the intermediate scattering function in the impulse approximation,We introduce a number of basic properties such as the static structure factor, the pair correlation function, and the momentum distribution which can be calculated from first principles using Monte Carlo and path integral Monte Carlo (PIMC) methods. We conclude by noting the two current descriptions of the characteristic excitation in superfluid 4He. The first is a density excitation, based on the Landau and Feynman view, with suitable modifications at higher Q to take account of correlations among atoms. In the second, the density excitation is separated into a part due to the excitation of single particles out of the condensate to states above the condensate and a part due to regular density excitations involving only states above the condensate. In the latter picture the condensate appears explicitly in S(Q, ω).
Liquid helium is the prototypical example of a superfluid – a liquid that flows without viscosity and transfers heat without a temperature gradient. These properties are intimately related to the Bose condensation that occurs in this strongly interacting liquid. Bose condensation is most directly observed in the single particle atomic momentum distribution, where the Bose condensate appears as a delta function singularity. In this article, we discuss the experimental techniques used to observe the condensate and the current status of measurments of the Bose condensate in liquid helium. Introduction Liquid helium ( 4 He) has fascinated physicists ever since Kammerlingh-Onnes liquified the last of the so called permanent gases in 1908. However, evidence of what is without doubt the most fascinating property of this unique liquid, superfluidity, was not reported until almost 25 years later [1]. The superfluid phase, where heat is transferred without a thermal gradient and mass flows without a driving pressure, is a macroscopic manifestation of microscopic quantum effects governing the behavior of atoms [2]. Bose–Einstein condensation was first proposed by London [3] as the microscopic explanation for these fascinating phenomena. Helium is unique among condensed atomic systems since the bulk properties of the liquid are dominated by quantum effects. The most important of these are the statistical effects that arise for identical particles. For bosons, such as 4 He atoms, there is no limitation on the number of particles that can occupy a single quantum state.
The momentum distribution n(k)n(k)n(k) and condensate fraction N0N\frac{{N}_{0}}{N}NN0​​ for liquid helium and for a charged Bose gas are evaluated in perturbation theory through second order in 1N\frac{1}{\sqrt{N}}N​1​. For liquid helium at normal vapor pressure, to this order in perturbation theory, we obtain unphysical, negative values for n(k)n(k)n(k). This result will be discussed in detail.
The ground state of homogeneous, superfluid - 4 He–normal- 3 He mixtures is studied by Fixed-Node Diffusion Monte Carlo simulations. The condensate fraction of the 4 He component only depends on the 3 He concentration through its effect on the average density. Slight deviations of the structural properties from previous variational results, obtained within the Average Correlation Approximation (ACA), are discussed. The momentum distribution and effective mass of the Fermi ( 3 He) component are also calculated.