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

David J. Toms

Published Dec 31, 2002• 0 citations
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Abstract

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
10.25405/ncl.34648569

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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.
$\ensuremath{\zeta}$-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.
By using exact path-integral Monte Carlo methods we calculate the equation of state of an interacting Bose gas as a function of temperature both below and above the superfluid transition. The universal character of the equation of state for dilute systems and low temperatures is investigated by modeling the interatomic interactions using different repulsive potentials corresponding to the same sss-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.
Abstract: The one-loop effective potential for non-relativistic bosons with a delta function repulsive potential is calculated for a given chemical potential using functional methods. After renormalization and at zero temperature it reproduces the standard ground state energy and pressure as function of the particle density. At finite temperatures it is found necessary to include ring corrections to the one-loop result in order to satisfy the Goldstone theorem. It is natural to introduce an effective chemical potential directly related to the order parameter and which uniformly decreases with increasing temperatures. This is in contrast to the the ordinary chemical potential which peaks at the critical temperature. The resulting thermodynamics in the condensed phase at very low temperatures is found to be the same as in the Bogoliubov approximation where the degrees of freedom are given by the Goldstone bosons. At higher temperatures the ring corrections dominate and result in a critical temperature unaffected by the interaction. 1
A dilute homogeneous two-dimensional (2D) Bose gas at zero temperature is studied with the method developed earlier by the authors. This method allows for considering renormalization of an arbitrary pairwise potential in a self-consistent manner, without the pseudopotential delta-function representation. Low-density expansions are derived for the chemical potential, ground-state energy, pair distribution function, kinetic and interaction energies. The expansion parameter is found to be a dimensionless in-medium scattering amplitude u obeying the equation 1/u+ln u=-ln(na(2)pi)-2gamma, where na(2) and gamma are the gas parameter and the Euler constant, respectively. It is shown that the ground-state energy is mostly kinetic in the low-density limit. This result does not depend on a specific form of the pairwise interaction potential, contrary to the 3D case. A new form of the 2D Gross-Pitaevskii equation is proposed within our scheme.
We investigate the properties of a dilute boson-fermion mixture at zero temperature. The ground-state energy of the system is calculated to second order in the Bose-Fermi coupling constant. The Green's function formalism is applied to obtain results for the effect of Bose-Fermi interactions on the spectrum of phonon excitations, the momentum distribution, the condensate fraction, and the superfluid density of the bosonic component. A quantitative discussion of these effects is presented in the case of mixtures with attractive interspecies interaction.