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arXiv:cond-mat/0603479 · International Journal of Modern Physics B

GROUND-STATE ENERGY DENSITY OF A DILUTE BOSE GAS IN THE CANONICAL TRANSFORMATION

Sang‐Hoon Kim, CHUL KU KIM, Mukunda P. Das

Published Dec 30, 2007• 0 citations
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Abstract

The ground-state energy density of an interacting dilute Bose gas system is studied in the canonical transformation scheme. It is shown that the transformation scheme enables us to calculate a higher order correction of order na3 in the particle depletion and ground-state energy density of a dilute Bose gas system, which corresponds to the density fluctuation resulting from the excited states. Considering a two-body interaction only, the coefficient of the na3 term is shown to be 2(π-8/3) for the particle depletion, and 16(π-8/3) for the ground-state energy density.

DOI
10.1142/s0217979207038423

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The ground-state energy density of an interacting dilute Bose gas system is studied in the canonical transformation scheme. It is shown that the transformation scheme enables us to calculate a higher order correction of order na3 in the particle depletion and ground-state energy density of a dilute Bose gas system, which corresponds to the density fluctuation resulting from the excited states. Considering a two-body interaction only, the coefficient of the na3 term is shown to be 2(π-8/3) for the particle depletion, and 16(π-8/3) for the ground-state energy density.
The ground-state energy of a system of interacting N-bosons is investigated in a collective formalism. Ultraviolet divergences in perturbation calculations are eliminated by the introduction of quasi-particle operators which are related by a nonlinear canonical transformation to the collective variables, i.e., the density fluctuation operators and their canonical conjugates. The transformation is presented in terms of a series in powers of N−1/2. The ground-state energy appears as a series in powers of N−1. The perturbation contributions of order N−2 to the ground-state energy are calculated. Most of these contributions can be incorporated into the lower order ones which agree with the previously obtained results.
The textbook calculation of the ground-state energy of a dilute gas of Bose particles is examined in detail, and certain mathematical inconsistencies are pointed out. On the basis of this analysis, a refined approach suitable for soft interaction potentials which lend themselves to a low-order Born approximation is developed. This procedure emphasizes the low-density character of the resulting formula for the ground-state energy, and avoids all divergent expressions at intermediate stages of the computation. It is stressed that the standard Bogoliubov approximation, if not augmented by some additional device, leads to an error which manifests itself already in the lowest order of the density. - PACS 03.75.Hh, 05.30.Jp, 03.65.Nk.
A modeled Bose system consisting of N particles with two-body interaction confined within volume V under inhomogeneity of the system is investigated using the Feynman path integral approach. The two-body interaction energy is assumed to be dependent on the two-parameter interacting strength a and the correlation length l. The inhomogeneity of the system or the porosity can be represented as density [Formula: see text] with interacting strength b and correlation length L. The mean field approximation on the two-body interaction in the Feynman path integrals representation is performed to obtain the one-body interaction. This approximation is equivalent to the Hartree approximation in the many-body electron gas problem. This approximation has shown that the calculation can be reduced to the effective one-body propagator. Performing the variational calculations, we obtain analytical results of the ground state energy which is in agreement with that from Bugoliubov's approach.
According to a formula that was put forward many decades ago the ground state energy per particle of an interacting, dilute Bose gas at density ρρρ is 2πℏ2ρa/m2π\hbar^2ρa/m2πℏ2ρa/m to leading order in ρa3≪1ρa^3\ll 1ρa3≪1, where aaa is the scattering length of the interaction potential and mmm the particle mass. This result, which is important for the theoretical description of current experiments on Bose-Einstein condensation, has recently been established rigorously for the first time. We give here an account of the proof that applies to nonnegative, spherically symmetric potentials decreasing faster than 1/r31/r^31/r3 at infinity.