Progress of Theoretical Physics

On Hugenholtz-Pines' Theory about Ground State Energy of Interacting Bosons

節夫三沢

Published Dec 1, 1960• 2 citations

Abstract

Hugenholtz and Pines have developed a new method evaluate the ground state energy of a system of interacting bosons, the merit of which consists in the elimination of ' the zero-momentam state and accordingly the applicability of the linked cluster expansion. Their theory, however, does not give correct results because of the undue treatment of the zero-momentum state; this is proved in a simple example. Up to the third order coupling parameter of the ground state enengy, the result of the usual RayleighSchrodinger perturbation method are compared with that obtained according to Hugenholtz-Pines' theory. The latter does not agree with the former. (auth)

DOI: 10.1143/ptp.24.1224

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Progress of Theoretical Physics

On Hugenholtz-Pines' Theory about Ground State Energy of Interacting Bosons

節夫三沢

Published Dec 1, 1960• 2 citations

Publisher

Abstract

DOI: 10.1143/ptp.24.1224

Related papers

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Physical Review AOct 1, 197129 citations

The ground-state energy per particle

E/N

and condensate density

n_0

of a dilute Bose gas are studied with a self-consistent perturbation expansion satisfying the Hugenholtz-Pines theorem and conservation laws simultaneously. A class of Feynman diagrams for the self-energy, which has escaped consideration so far, is shown to add an extra constant

c_{ip}\sim O(1)

to the expressions reported by Lee, Huang, and Yang [Phys. Rev. 106, 1135 (1957) ] as

E/N=(2π\hbar^2 an/m)[1+(128/15\sqrt{π}+16c_{ip}/5)\sqrt{a^3n}]

and

n_0/n=1-(8/3\sqrt{π}+c_{ip})\sqrt{a^3n}

, where

a

n

, and

m

are are the s-wave scattering length, particle density, and particle mass, respectively. We present a couple of estimates for

c_{ip}

; the third-order perturbation expansion yields

c_{ip}=0.412

g(r)

is the radial distribution function and

{N}_{0}

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.