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Physical Review A

Three-particle correlations in the ground state of a Bose fluid

Paul Berdahl

Published Dec 1, 1974• 18 citations
Publisher

Abstract

The two- and three-particle distribution functions of the weakly interacting Bose gas in its ground state are studied via the calculation of their Fourier transforms. The form obtained for the three-particle distribution function shows that the familiar convolution approximation is not correct even in the lowest nontrivial order of perturbation theory. For the special case of the charged Bose gas at high density, the two-particle distribution function is obtained to second order in the expansion parameter ${r}_{s}^{\frac{3}{4}}$, a result which disagrees with a recent calculation. The discrepancy is traced to a use of the convolution approximation for the three-particle distribution function. Finally, Landau's theory of quantum hydrodynamics is employed to relate the long-range part of the three-particle distribution function to the Gr\"uneisen constant. These results emphasize the need to treat properly the dynamic three-particle correlations in any quantitative theory of liquid helium.

DOI
10.1103/physreva.10.2378

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The two- and three-particle distribution functions of the weakly interacting Bose gas in its ground state are studied via the calculation of their Fourier transforms. The form obtained for the three-particle distribution function shows that the familiar convolution approximation is not correct even in the lowest nontrivial order of perturbation theory. For the special case of the charged Bose gas at high density, the two-particle distribution function is obtained to second order in the expansion parameter rs34{r}_{s}^{\frac{3}{4}}rs43​​, a result which disagrees with a recent calculation. The discrepancy is traced to a use of the convolution approximation for the three-particle distribution function. Finally, Landau's theory of quantum hydrodynamics is employed to relate the long-range part of the three-particle distribution function to the Gr"uneisen constant. These results emphasize the need to treat properly the dynamic three-particle correlations in any quantitative theory of liquid helium.
In order to obtain a quantitative estimate of the filling of the Bose condensate in the ground state of a Bose liquid the authors formulate a self-consistent perturbation theory with respect to the exactly solvable hard-spheres model. This has made it possible to take into account the contribution of the two-particle correlations to the erosion of the condensate and obtain a simple expression for the relative number of condensate atoms in terms of the liquid structure factor, which is taken as the initial information. The authors use the Bogolyubov-Zubarev expression for the ground-state energy of a many-boson system in the approximation of two sums over the wave vector in order to calculate the contributions of the three- and four-particle correlations. For this, they calculate the kinetic energy and from it recover the single particle density matrix. The results of the theory are used to calculate the filling of the Bose condensate in the ground state of liquid He/sup 4/ as a function of the liquid density.
This Thesis discusses approaches to calculating static and dynamic correlation functions in onedimensional gases of interacting bosons. The first part of the Thesis deals with the momentummomentum correlation function in the weakly interacting 1D Bose gas. In the regime of phase fluctuating quasicondensate this correlator is found to differ qualitatively from the form predicted by Bogoliubov theory of the true condensate in that correlations between any two values of momentum become finite. Linear hydrodynamics used to calculate the quasicondensate correlation function prove to be adequate. A classical field approximation is used to smoothly interpolate between the quasicondensate and strongly degenerate gas regimes. In the second part the focus is shifted to the dynamical structure factor and it is shown that hydrodynamics is generally inapplicable to calculating the dynamical correlators. The hydrodynamic treatment is enhanced using impurity theory and an exact model-dependent boundary is obtained for the region of applicability of hydrodynamics on the momentum-energy plane. Numerical estimates show that non-hydrodynamic behavior should be observable for the currently available values of experimental parameters.
An exact expression for the single-particle reduced density matrix of a translationally invariant system of bosons is obtained, as an average of a functional of the density and current operators. An approximation scheme, based on small fluctuations in density and current, is used in its lowest order, to approximate the single-particle reduced density matrix and the condensate density by a functional of the structure factor, which is a measurable quantity. This is done for the ground state and for finite temperatures. The resulting expressions for the condensate density are discussed. For the ideal gas the result is correct below the condensation temperature. For the case of an interacting liquid, Bose-Einstein condensation exists, in the ground states for two- and three-dimensional systems, and for finite temperatures it does exist only in three dimensions. The present experimental knowledge of the structure factor enables only a crude estimate of the condensed fraction which is found to be 0.1-0.2.
The contribution of order-parameter fluctuations to the static and dynamic properties of a weakly interacting Bose gas near the Bose-Einstein condensation temperature Tc{T}_{c}Tc​ is investigated via a novel approach. Self-consistent approximations for the propagator of order-parameter fluctuations GGG are generated in successive temperature ranges closer and closer to Tc{T}_{c}Tc​. Mean-field theory corresponds to assuming that the correlation of fluctuations G2{G}_{2}G2​ is given by G2(1)=GG{{G}_{2}}^{(1)}=GGG2​(1)=GG. This first approximation is valid in the range ${\ensuremath{\tau}}_{1}<\ensuremath{\tau}=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}<1$, where at $\ensuremath{\tau}={\ensuremath{\tau}}_{1}$ the neglected terms are comparable to those included. In this paper we consider the second approximation G2(2)=GGVGG{{G}_{2}}^{(2)}=GGVGGG2​(2)=GGVGG, which involves the first-order interaction between the fluctuations. Two independent criteria for the range of validity ${\ensuremath{\tau}}_{2}<\ensuremath{\tau}<{\ensuremath{\tau}}_{1}$ of the second approximation are established. It is conjectured that the qualitative results of the present model calculation are applicable to liquid helium in the presently accessible temperature range. The predicted temperature dependences of the specific heat, superfluid density, and fourth sound are in agreement with the scaling hypothesis and available experiments in helium. The condensate density is predicted to vanish linearly with $\ensuremath{\tau}$. It is shown that the contribution of order-parameter fluctuations to the thermal conductivity diverges as ${\ensuremath{\tau}}^{\frac{\ensuremath{-}1}{3}}$ above Tc{T}_{c}Tc​, and it is suggested that the absence of logarithmic factors is due to the inconsistent treatment of entropy fluctuations.
In 1963, a simplified approach was developed to study the ground-state energy of an interacting Bose gas with a purely repulsive potential. It consists in the derivation of an equation, which is not based on perturbation theory and which gives the exact expansion of the energy at low densities. This equation is expressed directly in the thermodynamic limit and only involves functions of three variables, rather than 3N3N3N. Here, we revisit this approach, introduce two more equations, and show that these yields accurate predictions for various observables for all densities for repulsive potentials with positive Fourier transform. Specifically, in addition to the ground-state energy, we have shown that the simplified approach gives predictions for the condensate fraction, two-point correlation function, and momentum distribution. We have carried out a variety of tests by comparing the predictions of the equations with quantum Monte Carlo calculations for exponential interaction potentials as well as a different, finite range potential of positive type, and have found remarkable agreement. We thus show that the simplified approach provides an alternative theoretical tool to understand the behavior of the many-body Bose gas, not only in the small and large density ranges, which have been studied before, but also in the range of intermediate density, for which much less is known.
This article reviews recent advances in the theory of the three-dimensional dilute homogeneous Bose gas at zero and finite temperature. Effective-field-theory methods are used to formulate a systematic perturbative framework that can be used to calculate the properties of a system at T=0.T=0.T=0. The perturbative expansion of these properties is essentially an expansion in the gas parameter na3,\sqrt{{\mathrm{na}}^{3}},na3​, where aaa is the sss-wave scattering length and nnn is the number density. In particular, the leading quantum corrections to the ground-state energy density, the condensate depletion, and long-wavelength collective excitations are rederived in an efficient and economical manner. Nonuniversal effects are also discussed. These effects are higher-order corrections that depend on properties of the interatomic potential other than the scattering length, such as the effective range. The article critically examines various approaches to the dilute Bose gas in equilibrium at finite temperature. These include the Bogoliubov approximation, the Popov approximation, the Hartree-Fock-Bogoliubov approximation, the \ensuremath{\Phi}-derivable approach, optimized perturbation theory, and renormalization-group techniques. The article ends with a look at recent calculations of the critical temperature of the dilute Bose gas, which include 1/N1/N1/N techniques, lattice simulations, self-consistent calculations, and variational perturbation theory.