Thermodynamic properties of trapped, two-dimensional interacting Bose gases in Hartree-Fock approximation

Abstract

We demonstrate the importance of the consistent treatment of mutual interaction between the condensate and the thermal component by applying the semiclassical Hartree-Fock model to the 2D trapped interacting Bose system. In contrast to the case where the interaction from the thermal component is neglected, the present system shows a lowering of the critical temperature and increase of the critical chemical potential due to the short-range repulsive interaction. Recent experimental observations of Bose-Einstein condensation (BEC) [1, 2] in dilute atomic gases confined in a magnetic trap have created great theoretical interest in weakly interacting Bose gases. At zero temperature the Gross-Pitaevskii (GP) equation [3] provides satisfactory ground state properties of the condensate. For finite temperature the Hartree-Fock-Bogoliobov-Popov (HFBP) theory has been developed to describe the finite-temperature properties of BEC [4,5]. Moreover, since the elementary collective excitations do not contribute significantly to the thermodynamics [6], the HFBP theory has been further simplified to the Hartree-Fock (HF) model where the condensate is described by the GP equation combined with the Thomas-Fermi (TF) approximation, and thermal components are treated as non-interacting bosons in an effective mean potential [7]. Recently, this simple and intuitive two-fluid model has been applied to obtain a semi-analytic expression of 3D BEC by further neglecting atomic interaction from the thermal component [8]. The validity of the two-fluid model has been confirmed by comparisions with the more elaborate self-consistent calculations for the 3D case [9]. All BEC experiments on trapped Bose gases to date have been naturally performed on the 3D system. However, over the last few years, several schemes for the preparation of atomic quasi-2D gases by using various planar waveguides for atoms have been proposed [10]. Theoretical study of the 2D system is also interesting, because even though the 2D homogeneous ideal boson system does not undergo BEC, a number of studies have indicated that it might be possible if a proper confining potential is applied [11-13]. Recently, Bayindir et al showed that 2D interacting Bose gases have BEC-like behaviour similar to those of an ideal system for weak interaction by applying a simplified form of the two-fluid model which neglects the interaction from the thermal component [14]. The results of the quantum Monte Carlo (QMC) study by Pearson et al [15] also strongly support the idea that there exists BEC in two dimensions for finite N , even though, as pointed out by Krauth [16], the effect of interaction is largely underestimated.

Semiclassical Hartree-Fock theory of a rotating Bose-Einstein condensation

“…(46) obeys the third law of thermodynamics which demands a vanishing of the heat capacity at zero temperature, and above T 0 is quite linear, in very good agreement with the standard theoretical result: 3k B /N (corresponds to the Dulong-Petit law in the very high temperature limit). This interesting general shape of the heat capacity is accepted in the literature [8,12,17,27].…”

Thermodynamics of a two-dimensional interacting Bose gas trapped in a quartic potential

“…The first question needs to be answered is whether this conclusion is an artifact of the approximations done in the course of calculations. In a previous study on 2D harmonic traps [48], it is seen that there is no significant difference between the results of condensate fraction in terms of the approaches used. For the weak interactions, it is found that the condensate fraction result obtained by using a semiclassical Hartree-Fock model without neglecting the effect of atomic interaction from the thermal component is quite consistent with that obtained by the zero-order perturbation [34].…”

Cusp catastrophe of symmetry breaking in two-component Bose–Einstein condensates

“…Various numerical techniques have been applied to determine the stationary solutions of the GPEF [17][18][19]. In this study, we used the gradient projection technique.…”