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Bose condensate in liquid He/sup 4/

I. O. Vakarchuk

Published May 1, 1986• 0 citations
OpenAlex

Abstract

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.

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Physical Review A•Dec 1, 1974•18 citations
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.
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 ground state of a system of interacting Bose particles is examined by a pair-correlation variational method based on exact eigenstates of the total particle number, without assuming macroscopic occupation of the single-particle state k = 0 (``simple'' Bose condensation). Known results are recovered in the case of repulsive interactions of weak and intermediate strengths, where the ground state does in fact exhibit simple condensation. But it is shown that in the case of weak attractive interactions the ground state exhibits ``generalized'' condensation, in which a finite fraction of particles have momenta which are macroscopically negligible, but no one single-particle state is macroscopically occupied. A weak-coupling expansion is derived for the ground-state energy in this case, the leading term of which is three times as low as that of the state one would obtain in the case of simple condensation. By the use of a pseudopotential, this result is adapted to the case of a strongly interacting system with a small negative scattering length α and fixed low density ρ, giving E0/n=−(6π ℏ2 ρ | α | /m)[1−(128/45) π−1/2(ρ | α | 3)1/2+…]. The relationships between equilibrium density, ground-state energy, and sound velocity predicted by this expression are in order-of-magnitude agreement with those obtaining for liquid He4.
One considers the superfluid (SF) state of a Bose liquid with a strong repulsion between bosons, in which at T=0, along with a weak single-particle Bose-Einstein condensate (BEC), there exists an intensive pair coherent condensate (PCC), analogous to the Cooper condensate in a Fermi liquid with an attraction between the fermions. Such a PCC emerges in a system of bosons due to an oscillating sign-changing momentum dependence of the Fourier component of the pair interaction potential, which is characteristic of a certain family of repulsion potentials. In such cases, the Fourier component is negative in some domain of nonzero momentum transfer, which corresponds to an effective attraction. The collective effects of renormalization (``screening'') of the initial interaction lead to a suppression of the repulsion and an enhancement of the effective attraction. It is the ratio of the BEC density to the full density of the liquid n0/n≪1n_0/n\ll 1n0​/n≪1 that is used as a small parameter---unlike in the Bogolyubov theory for a quasi-ideal Bose gas, in which the small parameter is the ratio of the number of supracondensate excitations to the number of particles in an intensive BEC, (n−n0)/n0≪1(n-n_0)/n_0\ll 1(n−n0​)/n0​≪1. A closed system of nonlinear integral equations for the normal and anomalous self-energy parts is obtained, in the framework of a renormalized perturbation theory built on combined hydrodynamic and field variables. In the framework of the hard-spheres model, a spectrum of quasiparticles is obtained, which is in good accordance with the experimental spectrum of elementary excitations in superfluid 4^44He. The question of applicability of the Landau criterion in the absence of quantum vortices is discussed.
A self-consistent model of the superfluid (SF) state of a Bose liquid with strong interaction between bosons is considered, in which at T=0, along with a weak single-particle Bose-Einstein condensate (BEC), there exists an intensive pair coherent condensate (PCC) of bosons, analogous to the Cooper pair condensate of fermions. Such a PCC emerges due to an effective attraction between bosons in some regions of momentum space, which results from an oscillating sign-changing momentum dependence of the Fourier component V(p) of the interaction potential. The collective many-body effects of renormalization ("screening") of the initial interaction, which are described by the bosonic polarization operator Π(p,ω), lead to a suppression of the repulsion [V(p)>0] and an enhancement of the effective attraction [V(p)<0] in the respective domains of nonzero momentum transfer. The ratio of the BEC density n_0 to the total particle density n of the Bose liquid is used as a small parameter of the model. A closed system of nonlinear integral equations for the normal and anomalous self-energy parts is obtained with account for the terms of first order in the BEC density. In the framework of the ``soft spheres'' model with the single fitting parameter--the value of the repulsion potential at r=0, a theoretical quasiparticle spectrum E(p) is obtained, which is in good accordance with the experimental spectrum E_{exp}(p) of elementary excitations in superfluid 4^44He.
The ground state of a charged Bose gas is studied throughout a large range of densities in a self-consistent-field approximation formalism which includes the short-range correlations between bosons through a local-field correction. Numerical self-consistent calculations have been carried out in order to determine the static structure factor $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$. Hence, from $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ we have obtained the pair-correlation functions, the ground-state energy which essentially is the correlation energy, the pressure of the gas, and, from Feynman's relation, the energy spectrum of elementary excitations. The response of the system to a static impurity charge has also been investigated. In the high-density limit our calculations show the same results as those given by Bogoliubov's perturbation theory. In the intermediate-density region, corresponding to the strongly coupled system, our results are in very good agreement with recent calculations based on the hypernetted-chain integral equation as well as Monte Carlo computations.
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.