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Journal of the Physical Society of Japan

Canonical Ensemble Statistics of Weakly Interacting Composite Boson Condensation: Thermodynamic Properties

Ming‐Tsung Lee

Published Oct 4, 2013• 0 citations
Publisher

Abstract

We present a canonical ensemble approach for weakly interacting composite boson systems, which is confined with a finite number of particles. Composite bosons, made of fermion pairs, correlate to each other through the Coulomb interaction associated with the fermion exchange, and are treated perturbatively. We study Bose–Einstein condensation of composite bosons according to the specific heat capacity, the composite-boson distribution function, and the condensate fraction up to the first-order perturbations. As an example, we investigate two-dimensional confined excitons. The study shows that the fermion exchange tends to assist the exciton condensation by raising the temperature. This stabilization of the exciton condensation owing to the fermion exchange is characterized by a small trough in the specific heat capacity profile. As the temperature becomes higher than the transition temperature, the effective Coulomb interaction reduces the thermal fluctuation of excitons, leading to a negative correction to the specific heat capacity in the normal phase. Weakly interacting excitons stay in a more stable thermodynamic state in comparison with ideal exciton gases.

DOI
10.7566/jpsj.82.104003

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We present a canonical ensemble approach for weakly interacting composite boson systems, which is confined with a finite number of particles. Composite bosons, made of fermion pairs, correlate to each other through the Coulomb interaction associated with the fermion exchange, and are treated perturbatively. We study Bose–Einstein condensation of composite bosons according to the specific heat capacity, the composite-boson distribution function, and the condensate fraction up to the first-order perturbations. As an example, we investigate two-dimensional confined excitons. The study shows that the fermion exchange tends to assist the exciton condensation by raising the temperature. This stabilization of the exciton condensation owing to the fermion exchange is characterized by a small trough in the specific heat capacity profile. As the temperature becomes higher than the transition temperature, the effective Coulomb interaction reduces the thermal fluctuation of excitons, leading to a negative correction to the specific heat capacity in the normal phase. Weakly interacting excitons stay in a more stable thermodynamic state in comparison with ideal exciton gases.
In this research, we develop a many-body Green's function approach for interacting composite bosons in the Bose–Einstein condensed phase. The resummation scheme dealing with the carrier–carrier exchange between many-body composite bosons in cold condensation is manifested as a diagrammatic technique through the introduction of a set of Dyson equations for composite bosons. We diagrammatically calculate the specific heat capacity of neutral composite bosons, made of mass-imbalanced fermions with opposite charges, up to the first orders of the Pauli exchange and Coulomb interaction. Condensate phenomena due to the effect of the carrier–carrier exchange are discussed. In comparison with the approximation that composite bosons behave like interacting fundamental bosons, our result shows that the energy reduction from the carrier–carrier exchange stabilizes composite-boson condensates.
The thermodynamic properties and finite-size scaling of weakly interacting Bose gases with a finite number of particles confined in a cubic box of volume L3{L}^{3}L3 with periodic boundary conditions are investigated theoretically based on an exact recursion relation for the canonical-ensemble partition function. In a similar manner to the case of an ideal Bose gas, the recursive scheme of the interacting finite system at any temperatures is established in the framework of the Bogoliubov theory. The temperature dependence of the condensate fraction and specific heat with different particles and interactions is obtained numerically. By defining two different transition temperatures of the finite systems, the effect of interactions on the transition temperatures yields the different and nonmonotonous behaviors. The finite-size scaling condensate fraction at the transition temperature for the systems is also discussed. It is shown that the calculated finite-size scaling is universally independent on the various system sizes and transition temperatures.
We study condensate statistics and thermodynamics of weakly interacting Bose gas with a fixed total number NNN of particles in a cubic box. We find the exact recursion relation for the canonical ensemble partition function. Using this relation, we calculate the distribution function of condensate particles for N=200N=200N=200. We also calculate the distribution function based on multinomial expansion of the characteristic function. Similar to the ideal gas, both approaches give exact statistical moments for all temperatures in the framework of Bogoliubov model. We compare them with the results of unconstraint canonical ensemble quasiparticle formalism and the hybrid master equation approach. The present recursion relation can be used for any external potential and boundary conditions. We investigate the temperature dependence of the first few statistical moments of condensate fluctuations as well as thermodynamic potentials and heat capacity analytically and numerically in the whole temperature range.
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.
Grand-canonical fluctuations of Bose-Einstein condensates of light are accessible to state-of-the-art experiments [J. Schmitt et al., Phys. Rev. Lett. 112, 030401 (2014).PRLTAO0031-900710.1103/PhysRevLett.112.030401]. We phenomenologically describe these fluctuations by using the grand-canonical ensemble for a weakly interacting Bose gas at thermal equilibrium. For a two-dimensional harmonic trap, we use two models for which the canonical partition functions of the weakly interacting Bose gas are given by exact recurrence relations. We find that the grand-canonical condensate fluctuations for weakly interacting Bose gases vanish at zero temperature, thus behaving qualitatively similarly to an ideal gas in the canonical ensemble (or microcanonical ensemble) rather than the grand-canonical ensemble. For low but finite temperatures, the fluctuations remain considerably higher than for the canonical ensemble, as predicted by the ideal gas in the grand-canonical ensemble, thus clearly showing that we are not in a regime in which the ensembles are equivalent.
We present a new method of calculating the distribution function and fluctuations for a Bose-Einstein condensate (BEC) of N interacting atoms. The present formulation combines our previous master equation and canonical ensemble quasiparticle techniques. It is applicable both for ideal and interacting Bogoliubov BEC and yields remarkable accuracy at all temperatures. For the interacting gas of 200 bosons in a box we plot the temperature dependence of the first four central moments of the condensate particle number and compare the results with the ideal gas. For the interacting mesoscopic BEC, as with the ideal gas, we find a smooth transition for the condensate particle number as we pass through the critical temperature.