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Journal de Physique IV (Proceedings)

Properties of4He in one dimension

E. Krotscheck, Michael David Miller

Published May 1, 2000• 0 citations
Publisher

Abstract

It has been suggested recently that 4 He can be prepared and studied as a quasi one-dimensional quantum fluid. We calculate the static and dynamic properties of one-dimensional 4 He using variational methods based upon the Jastrow-Feenberg wavefunction and its extension to dynamic systems with time-dependent, correlated wavefunctions. The zero temperature equation of state shows that in one-dimension 4 He is just barely self-bound with a binding energy of 0.002 K at a density of 0.036 A -1 . We also calculate the excitation spectrum including corrections that contain multi-phonon processes and study the density dependence of the roton feature as well as the static response function. In addition we demonstrate the presence of strong anomalous dispersion in the phonon regime. Finally, we introduce a 3 He impurity and calculate the zero concentration chemical potential as a function of He linear density. We also compute the 3 He- 3 He effective interaction in the 4 He background and compute the energy of dimerization.

DOI
10.1051/jp4:2000733

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It has been suggested recently that 4 He can be prepared and studied as a quasi one-dimensional quantum fluid. We calculate the static and dynamic properties of one-dimensional 4 He using variational methods based upon the Jastrow-Feenberg wavefunction and its extension to dynamic systems with time-dependent, correlated wavefunctions. The zero temperature equation of state shows that in one-dimension 4 He is just barely self-bound with a binding energy of 0.002 K at a density of 0.036 A -1 . We also calculate the excitation spectrum including corrections that contain multi-phonon processes and study the density dependence of the roton feature as well as the static response function. In addition we demonstrate the presence of strong anomalous dispersion in the phonon regime. Finally, we introduce a 3 He impurity and calculate the zero concentration chemical potential as a function of He linear density. We also compute the 3 He- 3 He effective interaction in the 4 He background and compute the energy of dimerization.
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Dynamic Many-Body Theory: The dynamics of atomic impurities in 4^44He
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"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."
Ground-state properties of two-dimensional ${}^{3}\mathrm{He}\ensuremath{-}{}^{4}\mathrm{He}$ mixtures are studied at zero temperature. A general argument based on the long-ranged attraction of the phonon exchange is given for the existence of 3He{}^{3}\mathrm{He}3He dimers in low-concentration mixtures with 4He{}^{4}\mathrm{He}4He. The binding energy of dimers ranges from milli- to microkelvins with increasing 4He{}^{4}\mathrm{He}4He density. By comparing the 3He{}^{3}\mathrm{He}3He impurity chemical potential in 4He{}^{4}\mathrm{He}4He with the one in pure 3He{}^{3}\mathrm{He}3He we conclude that at finite pressures 3He{}^{3}\mathrm{He}3He dimers form a mixture with 4He{}^{4}\mathrm{He}4He with a maximum solubility of $\ensuremath{\approx}3%$.
For one-dimensional many-boson systems there exists a considerable body of analytical treatments of model systems. The existence of some exactly known properties of these systems and the fact that some of these properties differ from those of 4He{}^{4}\mathrm{He}4He in higher dimensions are a strong impetus to apply microscopic many-body methods to quasi-one-dimensional systems with realistic interactions. The results on the model systems can help as a guide to understanding those systems, where due to the nature of the interaction, exact results are not available. In this paper, we examine three model systems: hard rods, the Morse potential, and the Lennard-Jones 6-12 potential using the nonperturbative variational approach based on Feenberg's method of correlated basis functions. In the hard rod system we can compare our equation of state with the exact result and discuss the low-lying excitations. We examine both finite-size rods and the point rod limit. For the Morse system, the existence of a two-body bound state, dimerization, can be determined analytically. We examine the effect of dimerization on the many-body equation of state and argue that dimerization is manifested in the many-boson system as the appearance of a many-body bound state (i.e., a zero pressure, negative energy system with nonzero density). For the Lennard-Jones system we numerically determine the onset of dimerization and, as with the Morse system, we show that in one dimension the many-body bound state only exists for those systems which dimerize. For each of these systems we track high-density instabilities in the equations of state which we tentatively identify as zero-temperature liquid-solid transitions.
We develop a variational theory for the determination of the ground-state properties of an inhomogeneous Bose fluid at zero temperature. Euler-Lagrange equations are derived for the one- and two-body correlations, and systematic approximation methods are physically motivated. It is shown that the optimized variational method provides a self-consistent approximate summation of ladder and ring diagrams. Numerical applications are presented for the density profile, distribution functions, energy, and chemical potential of films of 4He^{4}\mathrm{He}4He atoms. Both the bulk and surface energies are calculated and found to be in fair agreement with experiments.