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Progress of Theoretical Physics

Three-Particle Correlations in the Ground State of a Bose Fluid

Shuichiro Yamasaki, Masahiko HIROOKA

Published Oct 1, 1981• 2 citations
Publisher

Abstract

The three-particle structure function in the ground state of a boson system is studied in a formalism which establishes phonon type quasi-particles coupled to each other by the residual interaction preserving the linearity of the spectrum. The leading order results agree with those from the unquantized theories of Bogoliubov-Zubarev and Feenberg, et al. The next order corrections are calculated, and one of them is found to bear a strong resemblance to the result of Lee and Lee. Possible relations of the present formalism to the correlated basis function method in the uniform limit are discussed on a phenomenological level.

DOI
10.1143/ptp.66.1200

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The three-particle structure function in the ground state of a boson system is studied in a formalism which establishes phonon type quasi-particles coupled to each other by the residual interaction preserving the linearity of the spectrum. The leading order results agree with those from the unquantized theories of Bogoliubov-Zubarev and Feenberg, et al. The next order corrections are calculated, and one of them is found to bear a strong resemblance to the result of Lee and Lee. Possible relations of the present formalism to the correlated basis function method in the uniform limit are discussed on a phenomenological level.
The ground-state energy of an interacting many-boson system obtained by Bogoliubov and Zubarev using the collective-coordinate method is shown to be equivalent through second order to the variation-perturbation energy evaluated in the uniform limit by means of the method of correlated basis functions. It is also shown that the wave function derived with a slight modification of the Bogoliubov-Zubarev approach is equivalent to that determined formally from the Rayleigh-Schr\"odinger perturbation theory in the uniform limit.
The method of correlated basis functions is used in the study of the ground-state properties of a one-dimensional system of bosons interacting through a repulsive $\ensuremath{\delta}$-function potential. Ground-state energies are determined numerically for intermediate values of the coupling constant in three steps: (1) a trial form of the radial distribution function generated by the Bijl-Dingle-Jastrow (BDJ) type of wave function is introduced to obtain approximate ground-state energies, (2) the paired-phonon analysis is applied to improve the BDJ description of the ground state, and (3) leading corrections to the improved approximate energies are evaluted from the second-order perturbation energy generated by the three-phonon vertex. The obtained energy values are found to agree closely with the exact results calculated by Lieb and Liniger.
The ground state of a many-particle boson system is studied for two closely related limits: the uniform limit and the weak coupling limit. The former is defined by $\ensuremath{\alpha}=1\ensuremath{-}g(0)\ensuremath{\ll}1$ and the latter by $\ensuremath{\beta}=\frac{(N\ensuremath{-}{N}_{0})}{N}\ensuremath{\ll}1$, where g(r)g(r)g(r) is the radial distribution function and N0{N}_{0}N0​ is the mean occupation number of the zero-momentum state. In the uniform limit the variation-perturbation approach based on (a) the method of correlated basis functions and (b) the series expansion in powers of $\ensuremath{\alpha}$ is found to be equivalent to the field-theoretic treatment given by Brueckner (for the charged-boson gas) in the weak coupling limit. In particular, it is shown that the variation-perturbation energy obtained for the uniform limit in the momentum representation is identical through second order to the ground-state energy evaluated for $\ensuremath{\beta}\ensuremath{\ll}1$ by summing one- and two-ring diagrams in the Bogoliubov occupation-number representation. The charged-boson gas and the one-dimensional boson system with a $\ensuremath{\delta}$-function interaction are considered to examine some of the interesting features of the uniform-limit procedure.
A finite temperature system of degenerate bosons is studied with the aid of the extended Bogoliubov transformation which has been proposed in a preceding part I. The energies of quasi-particles and the vertex functions describing their interaction are determined in an approximation which includes the lowest order effects of the interaction between non-zero momentum bosons. The quasi-particle energy depends on the distribution of thermal excitations, and is of phonon type in the long wavelength limit. The phonon nature of the spectrum is preserved by the interaction between quasi-particles, because the vertex functions vanish in the long wavelength limit. In the limit as the condensate is depleted completely, the proposed formalism simplifies to the Hartree-Fock theory.
The ground-state energy of a one-dimensional system of many bosons interacting via a repulsive s-function potential is carried out through third-order corrections, when the coupling parameter γ of the system is small. The procedures' are basically developed following both the Bogoliubov-Zubarev and the Davison-Feenberg methods. It is seen numerically that a slight difference exists in between, at least, starting from the third-order correction; the former one gives rise to a more improved agreement with the exact results obtained by Lieb and Liniger, and also, the variational contribution in the later is still dominant even in the third-order correction.
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.