arXiv:hep-th/0207019 · Physical Review A

ζ -function regularization and the interacting Bose gas at low temperature

David J. Toms

Published Jul 30, 2002• 4 citations

Abstract

$\ensuremath{\zeta}$-function methods are used to study the properties of the nonrelativistic interacting Bose gas at finite temperature and density. Results for the ground-state energy and pressure are obtained at both zero and finite temperatures. The method used does not restrict the form of the interaction, which can be completely general. A similar procedure is then applied to evaluate the ground-state energy of a binary mixture consisting of different bosons. Analytical results are obtained in the case where the two species have the same mass.

DOI: 10.1103/physreva.66.013619

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arXiv:hep-th/0207019 · Physical Review A

ζ -function regularization and the interacting Bose gas at low temperature

David J. Toms

Published Jul 30, 2002• 4 citations

arXiv HTML PDF

Abstract

\ensuremath{\zeta}

-function methods are used to study the properties of the nonrelativistic interacting Bose gas at finite temperature and density. Results for the ground-state energy and pressure are obtained at both zero and finite temperatures. The method used does not restrict the form of the interaction, which can be completely general. A similar procedure is then applied to evaluate the ground-state energy of a binary mixture consisting of different bosons. Analytical results are obtained in the case where the two species have the same mass.

DOI: 10.1103/physreva.66.013619

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zeta-function methods are used to study the properties of the nonrelativistic interacting Bose gas at finite temperature and density. Results for the ground-state energy and pressure are obtained at both zero and finite temperatures. The method used does not restrict the form of the interaction, which can be completely general. A similar procedure is then applied to evaluate the ground-state energy of a binary mixture consisting of different bosons. Analytical results are obtained in the case where the two species have the same mass.zeta-function methods are used to study the properties of the nonrelativistic interacting Bose gas at finite temperature and density. Results for the ground-state energy and pressure are obtained at both zero and finite temperatures. The method used does not restrict the form of the interaction, which can be completely general. A similar procedure is then applied to evaluate the ground-state energy of a binary mixture consisting of different bosons. Analytical results are obtained in the case where the two species have the same mass.

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\ensuremath{\zeta}

s

-wave scattering length. The results obtained for the energy and the pressure are compared to the virial expansion for temperatures larger than the critical temperature. At very low temperatures we find agreement with the ground-state energy calculated using the diffusion Monte Carlo method.

E(\mathrm{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

n

is the density and

{f}_{0}

is the scattering length for the assumed two-body interaction between the bosons. The coefficients

a

and

b

are independent of the shape of the interaction, and are the only terms thus far calculated. The coefficient

b

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.