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Physical Review

Ground State and Low Excited States of a Boson Liquid with Applications to the Charged Boson System

Deok Kyo Lee, Eugene Feenberg

Published Feb 1, 1965• 79 citations
Publisher

Abstract

The maximum deviation of the radial distribution function $g(r)$ from its asymptotic value, $g(\ensuremath{\infty})=1$, is used as an expansion parameter in calculating the properties of a degenerate boson system. The uniform limit, defined by the condition $1\ensuremath{-}g(0)\ensuremath{\ll}1$, holds at low densities under appropriate constraints on the Fourier transforms of the interaction potential and also at high densities for the charged-boson system. In the uniform limit a procedure based on (1) a Jastrow-type trial function, (2) the Wu-Feenberg functional for the kinetic energy, (3) the Kirkwood superposition approximation, or the more accurate Abe form for the three-particle distribution function, yields the Bogoliubov formulas for the ground-state energy and the excitation energies of a degenerate or nearly degenerate boson system. Results for the ground state of the high-density charged-boson system include $\frac{〈\mathrm{KE}〉}{N}=\ensuremath{-}\frac{1}{5}\frac{〈\mathrm{PE}〉}{N}+O({{r}_{s}}^{0}),$ $\frac{E}{N}\ensuremath{\leqq}\ensuremath{-}\frac{0.8031}{{{r}_{s}}^{\frac{3}{4}}}+0.028.$ Numerical results are also computed at intermediate and moderately low densities ($0.01\ensuremath{\leqq}{r}_{s}\ensuremath{\leqq}100$).

DOI
10.1103/physrev.137.a731

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The ground state of a many-boson system is studied within the range of the Bijl-Dingle-Jastrow-type description when the radial distribution function $g(\mathcal{r})$ differs little from its asymptotic value. The treatment of the problem is based on the development of power series in $\ensuremath{\alpha}=1\ensuremath{-}g(0)$ for all physical quantities which depend on the particle density. The $n$-particle distribution functions ${p}^{(n)}$ are evaluated to order ${\ensuremath{\alpha}}^{4}$ as functionals in the $g(\mathcal{r})$ function for $n=3 \mathrm{and} 4$ using the cluster-expansion procedure outlined by Abe. These results are used in connection with the improvement of the ground-state description when the wave function is not the optimum choice. Using ${p}^{(3)}$ function obtained, the Bogoliubov-Born-Green-Kirkwood-Yvon equation is solved, also to order ${\ensuremath{\alpha}}^{4}$, for the two-particle correlation function $\mathcal{U}(\mathcal{r})$, and the first two leading corrections to the hypernetted-chain (HNC) approximation are obtained. The variational calculation along with the series expansion for $\mathcal{U}(\mathcal{r})$ yields formulas for the ground-state properties, including some corrections to known results. For a charged boson gas, numerical values of $\mathcal{U}(\mathcal{r})$, ${p}^{(3)}$, ${p}^{(4)}$, and the ground-state energy are computed using the Gaussian approximation for $g(\mathcal{r})$, and the results show that the errors associated with the HNC approximation are small. A brief discussion is presented on the method of determining the general expansion coefficients of the correlation functions of ${p}^{(n)}$ in terms of $g(\mathcal{r})$.

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Physical Review A•Jul 1, 1971•7 citations
The maximum deviation of the radial distribution function g(r)g(r)g(r) from its asymptotic value, g(\ensuremath∞)=1g(\ensuremath{\infty})=1g(\ensuremath∞)=1, is used as an expansion parameter in calculating the properties of a degenerate boson system. The uniform limit, defined by the condition 1\ensuremath−g(0)\ensuremath≪11\ensuremath{-}g(0)\ensuremath{\ll}11\ensuremath−g(0)\ensuremath≪1, holds at low densities under appropriate constraints on the Fourier transforms of the interaction potential and also at high densities for the charged-boson system. In the uniform limit a procedure based on (1) a Jastrow-type trial function, (2) the Wu-Feenberg functional for the kinetic energy, (3) the Kirkwood superposition approximation, or the more accurate Abe form for the three-particle distribution function, yields the Bogoliubov formulas for the ground-state energy and the excitation energies of a degenerate or nearly degenerate boson system. Results for the ground state of the high-density charged-boson system include 〈KE〉N=\ensuremath−15〈PE〉N+O(rs0),\frac{〈\mathrm{KE}〉}{N}=\ensuremath{-}\frac{1}{5}\frac{〈\mathrm{PE}〉}{N}+O({{r}_{s}}^{0}),N〈KE〉​=\ensuremath−51​N〈PE〉​+O(rs​0), EN\ensuremath≦\ensuremath−0.8031rs34+0.028.\frac{E}{N}\ensuremath{\leqq}\ensuremath{-}\frac{0.8031}{{{r}_{s}}^{\frac{3}{4}}}+0.028.NE​\ensuremath≦\ensuremath−rs​43​0.8031​+0.028. Numerical results are also computed at intermediate and moderately low densities (0.01\ensuremath≦rs\ensuremath≦1000.01\ensuremath{\leqq}{r}_{s}\ensuremath{\leqq}1000.01\ensuremath≦rs​\ensuremath≦100).
The ground state of a many-boson system is studied within the range of the Bijl-Dingle-Jastrow-type description when the radial distribution function g(r)g(\mathcal{r})g(r) differs little from its asymptotic value. The treatment of the problem is based on the development of power series in $\ensuremath{\alpha}=1\ensuremath{-}g(0)$ for all physical quantities which depend on the particle density. The nnn-particle distribution functions p(n){p}^{(n)}p(n) are evaluated to order ${\ensuremath{\alpha}}^{4}$ as functionals in the g(r)g(\mathcal{r})g(r) function for n=3and4n=3 \mathrm{and} 4n=3and4 using the cluster-expansion procedure outlined by Abe. These results are used in connection with the improvement of the ground-state description when the wave function is not the optimum choice. Using p(3){p}^{(3)}p(3) function obtained, the Bogoliubov-Born-Green-Kirkwood-Yvon equation is solved, also to order ${\ensuremath{\alpha}}^{4}$, for the two-particle correlation function U(r)\mathcal{U}(\mathcal{r})U(r), and the first two leading corrections to the hypernetted-chain (HNC) approximation are obtained. The variational calculation along with the series expansion for U(r)\mathcal{U}(\mathcal{r})U(r) yields formulas for the ground-state properties, including some corrections to known results. For a charged boson gas, numerical values of U(r)\mathcal{U}(\mathcal{r})U(r), p(3){p}^{(3)}p(3), p(4){p}^{(4)}p(4), and the ground-state energy are computed using the Gaussian approximation for g(r)g(\mathcal{r})g(r), and the results show that the errors associated with the HNC approximation are small. A brief discussion is presented on the method of determining the general expansion coefficients of the correlation functions of p(n){p}^{(n)}p(n) in terms of g(r)g(\mathcal{r})g(r).
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 simple model of a boson system is considered for which the exact Schr\"odinger equation reduces to a difference equation which can be solved numerically. Application of the Bogoliubov method and comparison of its results with the numerical results then yield some insight into the validity of the Bogoliubov approximations. The simple model considered has, in the absence of interactions, only a zero energy state and two states of unit energy available to each boson. Initially there is assumed to be no interaction between bosons in the degenerate excited states and all existing interactions are assumed to be repulsive. The ground-state energy is calculated in the Bogoliubov approximation, with and without the depletion effect considered, and in another numerical approximation. These results and the results of the numerical solution of the exact Schr\"odinger equation converge to the Bogoliubov result with depletion ignored when the number of particles in the system approaches infinity. It is surprising to note that, for the interaction strengths considered, the Bogoliubov result is within 3% of the numerical result for as few as 32 particles in the system, and within 10% for only 4 particles. A modification of the above system is considered in which there exists an additional two-body interaction between particles in the degenerate excited states which may be attractive or repulsive. It is shown that the ground-state energy, with this additional interaction present, deviates from the Bogoliubov value linearly with the strength of the added interaction, i.e., $W={W}_{\mathrm{Bog}}+\ensuremath{\alpha}F$ (with FFF the strength parameter of the interaction). The factor $\ensuremath{\alpha}$ is found to be small and constant over a wide range of FFF and in this range the deviation of WWW from the Bogoliubov value is small. There is, however, a transition point FT{F}_{T}FT​, beyond which $\ensuremath{\alpha}$ is constant and large and the usual Bogoliubov approximation is invalid for interactions more attractive than that characterized by FT{F}_{T}FT​. A Bogoliubov-like approximation is shown to be quite accurate in this region. It is then shown that $F_{T}^{\ensuremath{-}1}$ is a linear function of the number of particles in the system.
The ground state energy and elementary excitations of a charged gas of bosons at high densities are examined by use of the method developed by Bogoliubov for boson gases. It is conjectured, but not herein established, that this method yields exact results in the high-density limit analogous to those obtained by Gell-Mann and Brueckner, and by Sawada, in the corresponding case of a charged fermion gas. The ground state energy is essentially correlation energy, and is therefore negative, and its magnitude varies as the one-fourth power of the density at high densities. The elementary excitations have for low momenta the energy appropriate to plasma waves, and for high momenta the energy appropriate to single-particle excitation. There is therefore an energy gap, suggesting that the gas is both a superfluid and a superconductor at low temperatures. At low densities the behavior of a charged gas is independent of statistics; hence, such a gap must disappear as the system is expanded.
In this paper properties of a boson gas at zero temperature are investigated by means of field-theoretic methods. Difficulties arising from the depletion of the ground state are resolved in a simple way by the elimination of the zero-momentum state. The result of this procedure when applied to the calculation of the Green's functions of the system is identical to that of Beliaev. It is then shown generally that for a repulsive interaction the energy E(k)E(\mathrm{k})E(k) of a phonon of momentum k, which is found as the pole of a one-particle Green's function, approaches zero for zero momentum, which means that the phonon spectrum does not exhibit an energy gap.The Green's function method is applied to the calculation of the properties of a low-density boson gas. The next order term beyond that calculated by Lee and Yang, and Beliaev for the ground-state energy is obtained and the general form of the series expansion is found to be $(\frac{{E}_{0}}{\ensuremath{\Omega}})=\frac{1}{2}{n}^{2}{f}_{0}[1+a{(n{{f}_{0}}^{3})}^{\frac{1}{2}}+b(n{{f}_{0}}^{3})\mathrm{ln}n{{f}_{0}}^{3}+c(n{{f}_{0}}^{3})+d{(n{{f}_{0}}^{3})}^{\frac{3}{2}}\mathrm{ln}(n{{f}_{0}}^{3})+\ensuremath{\cdots}],$ where nnn is the density and f0{f}_{0}f0​ is the scattering length for the assumed two-body interaction between the bosons. The coefficients aaa and bbb are independent of the shape of the interaction, and are the only terms thus far calculated. The coefficient bbb is in agreement with the hard-sphere gas calculations of Wu and of Sawada.A discussion is given of the intermediate-density calculation of Brueckner and Sawada, and certain possible improvements in the method of summing a selected set of higher-order terms are proposed.