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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Universal renormalization of saddle-point integrals for condensed Bose gases

Martin Holthaus, Eva Kalinowski

Published Dec 1, 1999• 8 citations
Publisher

Abstract

When treating the ground-state contribution exactly, a variant of the saddle-point method emerges that works even for condensed Bose gases. Results thus obtained, such as canonical partition functions, differ by universal renormalization factors from those provided by the conventional but incorrect scheme. The amended method yields the statistical properties of ideal and very weakly interacting Bose gases with a fixed number of particles with particular simplicity.

DOI
10.1103/physreve.60.6534

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When treating the ground-state contribution exactly, a variant of the saddle-point method emerges that works even for condensed Bose gases. Results thus obtained, such as canonical partition functions, differ by universal renormalization factors from those provided by the conventional but incorrect scheme. The amended method yields the statistical properties of ideal and very weakly interacting Bose gases with a fixed number of particles with particular simplicity.
The renormalization group is not only a powerful method for describing universal properties of phase transitions but it is also useful for evaluating non-universal thermodynamic properties beyond meanfield theory. In this contribution we concentrate on these latter aspects of the renormalization group approach. We introduce its main underlying ideas in the familiar context of the ideal Bose gas and then apply them to the case of an interacting, confined Bose gas within the framework of the random phase approximation. We model confinement by periodic boundary conditions and demonstrate how confinement modifies the flow equations of the renormalization group changing thus the The recent experimental realization of Bose-Einstein condensation has renewed the interest in the study of trapped, dilute, weakly interacting Bose gases. Despite the long history of this subject there still exist unresolved
We consider equilibrium canonical-ensemble fluctuations of the Bose-Einstein condensate (BEC) in a trap using the particle-number conserving operator formalism of Girardeau and Arnowitt [Phys. Rev. 113, 755 (1959)] and the concept of the canonical-ensemble quasiparticles. We find explicit expressions for the characteristic function and all cumulants of the ground-state occupation statistics both for the dilute weakly interacting and ideal Bose gases. We present analytical formulas for the moments of the ground-state occupation fluctuations in the weakly interacting and ideal Bose gases in a box (``homogeneous gas'') as well as in the ideal Bose gas in an arbitrary power-law trap, in particular, in an arbitrary harmonic trap. We calculate the effect of Bogoliubov coupling between quasiparticles on suppression of the ground-state occupation fluctuations at moderate temperatures and their enhancement at very low temperatures. We clarify a crossover between ideal-gas and weakly interacting gas statistics, which is governed by a pair-correlation, squeezing mechanism. The important conclusion is that in most cases the ground-state occupation fluctuations are anomalously large and are not Gaussian even in the thermodynamic limit.
Fluctuations of the number of condensed atoms in a finite-size, weakly interacting Bose gas confined in a box potential are investigated for temperatures up to the critical region. The canonical partition functions are evaluated using a recursive scheme for smaller systems, and a saddle-point approximation for larger samples that allows to treat realistic size systems containing up to N ∼10 5 particles. We point out the importance of particle-number constraint and interactions between out of condensate atoms for the statistics near the critical region. For sufficiently large systems, the crossover from the anomalous to normal scaling of the fluctuations is observed. The excitations are described in a self-consistent way within the Bogoliubov-Popov approximation, and the interactions between thermal atoms are described by means of the Hartree-Fock method.
We develop a reliable procedure for calculating the microcanonical fluctuations of the ground state occupation number for harmonically trapped ideal Bose gases, and show that these fluctuations vanish uniformly when the temperature approaches zero. The key point is the precise determination of the number of microstates from the canonical partition sum, thus avoiding a failure of the usual saddle point method. We also demonstrate why the magnitude of the condensate fluctuations does not depend on the total particle number.
In 1963, a Simple Approach was developed to study the ground state energy of an interacting Bose gas. It consists in the derivation of an Equation, which is not based on perturbation theory, and which gives the exact expansion of the energy at low densities. This Equation is expressed directly in the thermodynamic limit, and only involves functions of 333 variables, rather than 3N3N3N. Here, we revisit this approach, and show that the Equation yields accurate predictions for various observables for all densities. Specifically, in addition to the ground state energy, we have shown that the Simple Approach gives predictions for the condensate fraction, two-point correlation function, and momentum distribution. We have carried out a variety of tests by comparing the predictions of the Equation with Quantum Monte Carlo calculations, and have found remarkable agreement. We thus show that the Simple Approach provides a new theoretical tool to understand the behavior of the many-body Bose gas, not only in the small and large density ranges, which have been studied before, but also in the range of intermediate density, for which little is known.