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Physical review. B, Condensed matter

Diagrammatic derivation of Bose condensate fractions

Tilo Wettig B, Andrew David Jackson

Published Jan 1, 1996• 3 citations
Publisher

Abstract

We describe a diagrammatic calculation of the condensate fraction of dense and strongly interacting Bose systems at zero temperature using the framework of parquet theory. Starting from the proper self-energy of parquet theory we perform a major rearrangement of diagrams to ensure proper counting and to express the final result in terms of well-behaved quantities. A number of interesting limiting cases are considered, in particular the hard sphere Bose gas at low densities where analytical results can be obtained. The present formalism is well-suited for extension to finite temperatures, and we consider possible strategies in that direction. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.818

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We describe a diagrammatic calculation of the condensate fraction of dense and strongly interacting Bose systems at zero temperature using the framework of parquet theory. Starting from the proper self-energy of parquet theory we perform a major rearrangement of diagrams to ensure proper counting and to express the final result in terms of well-behaved quantities. A number of interesting limiting cases are considered, in particular the hard sphere Bose gas at low densities where analytical results can be obtained. The present formalism is well-suited for extension to finite temperatures, and we consider possible strategies in that direction. \textcopyright{} 1996 The American Physical Society.
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Using the energy-momentum relation derived by Brueckner and Sawada, the low-temperature thermodynamical properties of a Bose gas with hard-sphere interaction are investigated. The two cases, (1) when this energy spectrum is similar to that observed in liquid helium II and (2) when it becomes similar to that derived by Lee, Huang, and Yang, are investigated separately. In the first case (where the spectrum exhibits a nonmonotonic character), the analytical derivation of the expressions for the various thermodynamical properties is rendered possible by dividing the momentum range into three suitable intervals such that within each interval the epsilon -p relation can be approximated to a simple algebraic one. The numerical magnitudes of the various parameters characterizing the algebraic relations are calculated for the case of a gas with density equal to that of liquid helium(II) and for three plausible values of the hard-sphere diameter. The calculated results agree qualitatively with those obtained from the experimental study of this liquid. For the second case, expressions are derived for the various macroscopic properties of the assembly in terms of suitable mathematical functions and then are written down in the form of series expansions for the two particular cases of interest. (auth)
In the interest of polishing the parquet approach to microscopic calculations of ground-state properties of zero-temperature quantum fluids, we have developed a perturbative scheme for the systematic improvement of the four-point function which retains the correct behavior of the parquet diagrams in both the small-r and small-k limits. Numerical implementation of this scheme indicates significantly improved agreement with binding energies calculated with Green's-function Monte Carlo techniques (i.e., differences in liquid 4He^{4}\mathrm{He}4He are less than 1.5%). The charged Bose gas and the hard-sphere system are also considered. Remaining difficulties in the calculations of the liquid structure function are noted and discussed.
We consider the nonrelativistic interacting Bose gas at zero and finite temperature. Using the 1/N-expansion, we derive expressions for the free energy density and the number density to next-to-leading order in 1/N. Outside the critical region and at weak coupling, our calculations reduce to the well-known mean-field results for the dilute Bose gas. We also rederive the nonperturbative critical density for Bose condensation which was first calculated by Baym, Blaizot, Holzmann, Lalöe, and Vautherin.
Through exact numerical solutions we show Bose-Einstein condensation (BEC) for the one-dimensional (1D) bosons with repulsive short-range interactions at zero temperature by taking a particular large size limit. Following the Penrose-Onsager criterion of BEC, we define condensate fraction by the fraction of the largest eigenvalue of the one-particle reduced density matrix. % We show the finite-size scaling behavior such that condensate fraction is given by a scaling function of one-variable: interaction parameter multiplied by a power of particle number. Condensate fraction is nonzero and constant for any large value of particle number or system size, if the interaction parameter is proportional to the negative power of particle number. %Here the interaction parameter is defined by the coupling constant of the delta-function potentials devided by the density. %With the scaling behavior we derive various themodynamic limits where condensate fraction is constant for any large system size; for instance, it is the case even in the system of a finite particle number.