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Physical review. B, Condensed matter

Self-consistent-field approximation method for a one-dimensional many-boson system

Oscar Hipólito, R. Lobo

Published Feb 1, 1978• 1 citation
Publisher

Abstract

The method of self-consistent-field approximation is applied for the study of the ground-state properties of a one-dimensional system of bosons, of density $\ensuremath{\rho}$, first interacting through finite rigid rods, of length $l$, and again via a repulsive $\ensuremath{\delta}$-function potential. We obtain the energy spectrum and the sound velocity. The results are close to the exact one for $\ensuremath{\rho}l<0.5$. The energy spectrum obtained shows a rotonlike region which is qualitatively similar to the two- and three-dimensional versions. For a Bose gas interacting through a repulsive $\ensuremath{\epsilon}$ potential of strength $2c$, we derive the energy spectrum, the sound velocity, and the ground-state energy. The results are compared with Girardeau and Lieb's exact expressions. In the small-coupling limit our calculations show the same results as obtained by Bogoliubov's perturbation theory. The ground-state energy obtained for small $\ensuremath{\gamma}=\frac{c}{\ensuremath{\rho}}$ is found to agree closely with several different works found in the literature. Unfortunately our theory diverges for the ground-state energy in the limit $\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\infty}$. The sound velocity in this limit is finite but twice as small as from Girardeau's exact results.

DOI
10.1103/physrevb.17.1433

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The method of self-consistent-field approximation is applied for the study of the ground-state properties of a one-dimensional system of bosons, of density \ensuremathρ\ensuremath{\rho}\ensuremathρ, first interacting through finite rigid rods, of length lll, and again via a repulsive \ensuremathδ\ensuremath{\delta}\ensuremathδ-function potential. We obtain the energy spectrum and the sound velocity. The results are close to the exact one for \ensuremathρl<0.5\ensuremath{\rho}l<0.5\ensuremathρl<0.5. The energy spectrum obtained shows a rotonlike region which is qualitatively similar to the two- and three-dimensional versions. For a Bose gas interacting through a repulsive \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ potential of strength 2c2c2c, we derive the energy spectrum, the sound velocity, and the ground-state energy. The results are compared with Girardeau and Lieb's exact expressions. In the small-coupling limit our calculations show the same results as obtained by Bogoliubov's perturbation theory. The ground-state energy obtained for small \ensuremathγ=c\ensuremathρ\ensuremath{\gamma}=\frac{c}{\ensuremath{\rho}}\ensuremathγ=\ensuremathρc​ is found to agree closely with several different works found in the literature. Unfortunately our theory diverges for the ground-state energy in the limit \ensuremathγ\ensuremath→\ensuremath∞\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\infty}\ensuremathγ\ensuremath→\ensuremath∞. The sound velocity in this limit is finite but twice as small as from Girardeau's exact results.
The self-consistent-field approximation is applied for the study of the energy spectrum, sound velocity, and correlation function on a one-dimensional, spin-1/2 Fermi system, interacting as rigid rods of length lll. Two very distinct regions are observed, which are named as low- and high-density behaviors, corresponding to small and large Landau's F0{F}_{0}F0​, respectively. The results are very similar to those obtained for a boson system with the same type of interaction, as one might expect. For the very high densities the energy spectrum approaches the E(q)=0E(q)=0E(q)=0 line, indicating the existence of an instability, a liquid to crystal transition, which is very neatly followed by observing the behavior of the pair-correlation function as density increases.
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.
We investigate ground-state and excitation spectrum of a system of non-relativistic bosons in one-dimension interacting through repulsive, two-body contact interactions in a self-consistent Gaussian mean-field approximation. The method consists in writing the variationally determined density operator as the most general Gaussian functional of the quantized field operators. There are mainly two advantages in working with one-dimension. First, the existence of an exact solution for the ground-state and excitation energies. Second, neither in the perturbative results nor in the Gaussian approximation itself we do not have to deal with the three-dimensional patologies of the contact interaction . So that this scheme provides a clear comparison between these three different results.
The approximate description of a many-boson system by its density fluctuations is considered. The representation of the chemical potential by a scattering matrix is compared with an exact solution in one dimension. It is found to be valid exactly in the low coupling limit and valid to within 10% for intermediate coupling strengths. In the case treated, the ground-state properties are given quite accurately for intermediate coupling strengths. The excitation spectrum is described less accurately. The velocity of sound obtained from the slope of the phonon spectrum is formally different from that obtained using the compressibility. Here, the sound velocities obtained from the two methods are found to agree numerically. Data for actual He4{\mathrm{He}}^{4}He4 and a simple linear approximation to the density dependence of the scattering matrix are used. The sound velocity obtained is in rough agreement with the experimentally observed velocity in liquid He4{\mathrm{He}}^{4}He4.
We study the correlation effects in a one-dimensional Bose gas with repulsive delta-function interaction. The correlation effects are described by a local-field correction which takes into account the short-range correlations. We find that the ground state energy is in good agreement with the exact result up to intermediate coupling strengths, showing an improvement over the Bogoliubov and other related approximations. The velocity of sound, the static structure factor, and pair-correlation function are also calculated within the present approximation.
A gas of one-dimensional Bose particles interacting via a repulsive delta-function potential has been solved exactly. All the eigenfunctions can be found explicitly and the energies are given by the solutions of a transcendental equation. The problem has one nontrivial coupling constant, $\ensuremath{\gamma}$. When $\ensuremath{\gamma}$ is small, Bogoliubov's perturbation theory is seen to be valid. In this paper, we explicitly calculate the ground-state energy as a function of $\ensuremath{\gamma}$ and show that it is analytic for all $\ensuremath{\gamma}$, except $\ensuremath{\gamma}=0$. In Part II, we discuss the excitation spectrum and show that it is most convenient to regard it as a double spectrum---not one as is ordinarily supposed.