PHYSICAL REVIEW A 82, 063618 (2010) Application of the static ﬂuctuation approximation to the computation of the thermodynamic properties of an interacting trapped two-dimensional hard-sphere Bose gas Asaad R. Sakhel, 1 Saleem I. Qashou, 3 Roger R. Sakhel, 2 and Humam B. Ghassib 4 1 Al-Balqa Applied University, Faculty of Engineering Technology, Amman 11134, Jordan 2 Department of Basic Sciences, Faculty of Information Technology, Al-Isra University, Amman 11622, Jordan 3 Department of Physics, Faculty of Science and Information Technology, Zarqa Private University, Zarqa 13132, Jordan 4 Department of Physics, The University of Jordan, Amman, Jordan (Received 19 June 2010; published 14 December 2010) The static ﬂuctuation approximation (SFA) is applied to compute the thermodynamic properties of a trapped two-dimensional (2D) interacting hard-sphere (HS) Bose gas in the weakly and strongly interacting regime. A mean-ﬁeld approach involving a variational wave function is used to compute the mean-ﬁeld energy as a function of temperature for each harmonic oscillator (HO) state plugged into the SFA technique. In the variational approach, a parameter α is introduced into the harmonic oscillator wave function in order to take into account the changes in the width when the repulsive interactions between the bosons are increased. In the weakly interacting regime, below the critical temperature, the total energy of all HO states (evaluated by our model) matches the non- interacting result very well. However, beyond the critical temperature, we “ﬁt” our energies to the classical limit for 2D bosons in a trap by using a suitably proposed weighting function. We compare our results to earlier results of mean-ﬁeld theory. Further, we evaluate the density matrix arising from correlations between the HO orbitals. DOI: 10.1103/PhysRevA.82.063618 PACS number(s): 67.85.−d, 05.70.Ce, 03.75.Hh I. INTRODUCTION The trapped two-dimensional (2D) interacting Bose gas presents many challenges. The evaluation of the thermody- namic properties represents just one of these challenges. It is not a straightforward matter and usually requires path- integral Monte Carlo (PIMC) calculations. In addition, many properties of 2D trapped Bose gases have yet to be explored both experimentally and theoretically [1]; various problems remain open [2], such as the estimation of the BEC transition temperature in a 2D interacting system and the explicit rela- tionship between superﬂuidity and the quasicondensate. Two- dimensional systems have generally become very interesting, thanks to the interplay between BEC and Kosterlitz-Thouless transitions and other issues [3]. It is therefore in order to develop methods for exploring such systems. Various methods and techniques have been used to investi- gate the trapped 2D Bose gas. Variational methods [3,4], the classical-ﬁeld simulation technique [5], path-integral Monte Carlo methods [1,6,7], as well as Hartree-Fock-Bogoliubov theory [8] have all been applied. For example, Nho and Landau [1] used a ﬁnite-temperature PIMC method and showed that BEC can form at ﬁnite temperature. In order to perform their calculations, they used only N = 27 hard-sphere bosons, because a larger number required an extensive amount of computational time. In this article, our chief goal is to demonstrate an application of the static ﬂuctuation approximation (SFA) [9,10] in an evaluation of the thermodynamic properties of a trapped, 2D hard-sphere (HS) Bose gas and to compare to the corresponding 3D properties. Comparisons are also made with corresponding 2D analytical results for the thermodynamic properties. The relative advantage of the SFA is that it can yield the thermodynamic properties of a system for a broad range of temperature in one single calculation. One can obtain easily using the SFA, the energy ﬂuctuations, number ﬂuctuations, and thermodynamic properties, all as functions of temperature. The SFA particularly invokes the role of the energy ﬂuctuations in the determination of the thermodynamic properties. It essentially corrects for any approximations done in the mean-ﬁeld evaluation of the energy via the energy ﬂuctuations. This corrected energy is then incorporated into the thermodynamic potential ln Q, where Q is the grand canonical partition function from which all the thermodynamic properties are evaluated. In addition, the SFA can handle a large number of particles up to N ∼ 1000. On the other hand, in other methods such as PIMC calculations, one can obtain the thermodynamic properties for one temperature at a time only, i.e., point by point. However, the SFA is limited in that it fails at very strong interactions g> 0.01, whereas other methods, such as PIMC calculations, are excellent in this regard as they can deal with strongly interacting systems. The SFA works very well mainly in the weakly to strongly interacting regime up to g = 0.01. Some of the properties we evaluate have been—to our knowledge—rarely addressed in the literature, such as the thermal behavior of the energy for each harmonic oscillator (HO) state m and the number ﬂuctuations 〈( ˆ N m ) 2 〉. Another goal is to shed more light on the effects of dimensionality on such systems by comparing 2D to 3D. We thus consider N hard-sphere bosons in a 2D harmonic trap in a broad range of temperature T . The interactions be- tween the bosons are modelled by a δ function pseudopotential. The energies for each HO state are evaluated using many-body mean-ﬁeld theory, and the thermodynamic properties are obtained using SFA in 2D, applied earlier [11] to the 3D trapped interacting Bose gas. It should be emphasized that, with the SFA method applied here, we are able to use a large number of particles, N = 1000. It should also be emphasized that our goal is not to provide a method for computing the energies but rather to obtain them for use in SFA. Nevertheless, the mean-ﬁeld model we present gives the energies accurately in the condensate regime below the transition temperature but 1050-2947/2010/82(6)/063618(16) 063618-1 ©2010 The American Physical Society