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Physical Review

Variation Ground-State Energy of a Bose System

K. Sawada

Published Nov 29, 1965• 2 citations
Publisher

Abstract

An upper bound of the variational ground-state energy of a Bose system, for purely repulsive interactions and for a certain class of mainly repulsive interactions, is given using the Jastrow-type variational wave function. The upper bound obtained for a hard-sphere interaction is slightly lower than the bound obtained previously by Dyson.

DOI
10.1103/physrev.140.a1564

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Variational Ground-State Energy of a Bose System. II

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Physical Review•Aug 5, 1966•6 citations
An upper bound of the variational ground-state energy of a Bose system, for purely repulsive interactions and for a certain class of mainly repulsive interactions, is given using the Jastrow-type variational wave function. The upper bound obtained for a hard-sphere interaction is slightly lower than the bound obtained previously by Dyson.
We investigate the properties of the ground state of a system of interacting bosons on regular lattices with coordination number k≥2k\geq 2k≥2. The interaction is a pure, infinite, on-site repulsion. Our concern is to give an improved upper bound on the ground state energy per site. For a density ρρρ a trivial upper bound is known to be −kρ(1−ρ)-kρ(1-ρ)−kρ(1−ρ). We obtain a smaller variational bound within a reasonably large family of trial functions. The estimates make use of a large deviation principle for the energy of the Ising model on the same lattice.
The ground-state energy of a system of hard-sphere bosons of diameter $\ensuremath{\sigma}$ is calculated by a variational method using a Jastrow wave-function for the fluid phase and a "localized" Jastrow times a Gaussian wave function for the solid phase. The transition densities are ${\ensuremath{\rho}}_{\mathrm{fluid}}=0.23\ifmmode\pm\else\textpm\fi{}0.02$ particles/${\mathrm{\ensuremath{\sigma}}}^{3}$ and ${\ensuremath{\rho}}_{\mathrm{solid}}=0.25\ifmmode\pm\else\textpm\fi{}0.02$ particles/${\mathrm{\ensuremath{\sigma}}}^{3}$.
We consider a gas of bosons interacting through a hard-sphere potential with radius a\frak{a}a in the thermodynamic limit. We derive a simple upper bound for the ground state energy per particle at low density. Our bound captures the leading term 4πρa4πρ\frak{a}4πρa and shows that corrections are smaller than Cρa(ρa3)1/2C ρ\frak{a} (ρ\frak{a}^3)^{1/2}Cρa(ρa3)1/2, for a sufficiently large constant C>0C > 0C>0. In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy is, in fact, of the order ρa(ρa3)1/2ρ\frak{a} (ρ\frak{a}^3)^{1/2}ρa(ρa3)1/2, in agreement with the Lee-Huang-Yang prediction.