A path-integral quantum Monte Carlo method is used to calculate finite temperature properties of up to 1000 hard-core bosons in a two-dimensional isotropic harmonic-oscillator potential. If the interatomic repulsions are sufficiently short range, an abrupt increase in the condensate fraction and a hump in the specific heat occur close to the critical temperature of ideal bosons. The critical temperature and the condensate fraction are in general lowered by an increase in the hard-core radius a. If a is decreased below a certain level, the condensate fraction becomes indistinguishable from the corresponding value of the ideal bosons. For up to 1000 particles, this occurs when ${\mathrm{ln}}^{\ensuremath{-}1}{(1/na}^{2})\ensuremath{\lesssim}0.1,$ where n is the average particle density.
We examine the possibility of Bose-Einstein condensation in one-dimensional interacting Bose gas subjected to confining potentials of the form ${V}_{\mathrm{ext}}{(x)=V}_{0}(|x|{/a)}^{\ensuremath{\gamma}},$ in which $\ensuremath{\gamma}<2,$ by solving the Gross-Pitaevskii equation within the semiclassical two-fluid model. The condensate fraction, chemical potential, ground state energy, and specific heat of the system are calculated for various values of interaction strengths. Our results show that a significant fraction of the particles is in the lowest energy state for a finite number of particles at low temperature, indicating a phase transition for weakly interacting systems.
Bose-Einstein condensation has been experimentally found to take place in finite trapped systems when one of the confining frequencies is increased until the gas becomes effectively two-dimensional (2D). We confirm the plausibility of this result by performing path-integral Monte Carlo (PIMC) simulations of trapped Bose gases of increasing anisotropy and comparing them to the predictions of finite-temperature many-body theory. PIMC simulations provide an essentially exact description of these systems; they yield the density profile directly and provide two different estimates for the condensate fraction. For the ideal gas, we find that the PIMC column density of the squeezed gas corresponds quite accurately to that of the exact analytic solution and, moreover, is well mimicked by the density of a 2D gas at the same temperature; the two estimates for the condensate fraction bracket the exact result. For the interacting case, we find 2D Hartree-Fock solutions whose density profiles coincide quite well with the PIMC column densities and whose predictions for the condensate fraction are again bracketed by the PIMC estimates.
The ground-state properties of a system with a small number of interacting bosons over a wide range of densities are investigated. The system is confined in a two-dimensional isotropic harmonic trap, where the interaction between bosons is treated as a hard-core potential. By using variational Monte Carlo method, we diagonalize the one-body density matrix of the system to obtain the ground-state energy, condensate wavefunction and the condensate fraction. We find that in the dilute limit the depletion of central condensate in the 2D system is larger than in a 3D system for the same interaction strength; however as the density increases, the depletion at the centre of 2D trap will be equal to or even lower than that at the centre of 3D trap, which is in agreement with the anticipated in Thomas–Fermi approximation. In addition, in the 2D system the total condensate depletion is still larger than in a 3D system for the same scattering length.
We prove the following results. (i) One-dimensional Bose gases which interact via unscaled integrable pair interactions and are confined in an external potential increasing faster than quadratically undergo a complete generalized Bose-Einstein condensation (BEC) at any temperature, in the sense that a macroscopic number of particles are distributed on a o(N) number of one-particle states. (ii) In a one dimensional harmonic trap the replacement of the oscillator frequency ω by ω ln N/N gives rise to a phase transition at a ≡ �ωβ = 1 in the noninteracting gas. For a < 1 the limit distribution of n0/N a is exponential and 〈n0〉/N a → 1. For a> 1 there is BEC with a condensate density 〈n0〉/N → 1 − a −1. For a ≥ 1, (lnN/N)(n0 − 〈n0〉) is asymptotically distributed following Gumbel’s law. For any a> 0 the free energy is −(π 2 /6βa)N / lnN + o(N/ln N), with no singularity at a = 1. (iii) In Model (ii) both above and below the critical temperature the gas undergoes a complete generalized BEC, thus providing a coexistence of ordinary and generalized condensates below the critical point. (iv) Adding an interaction 〈UN 〉 = o(N ln N) to Model (ii) we prove that a complete generalized BEC occurs for any β> 0.
Bose-Einstein condensation (BEC) of an ideal gas is investigated, beyond the thermodynamic limit, for a finite number
N of particles trapped in a generic three-dimensional power-law potential. We derive an analytical expression for the condensation temperature
Tc in terms of a power series in ${x}_{0}={\ensuremath{\varepsilon}}_{0}/{k}_{B}{T}_{c}$, where ${\ensuremath{\varepsilon}}_{0}$ denotes the zero-point energy of the trapping potential. This expression, which applies in Cartesian, cylindrical, and spherical power-law traps, is given analytically at infinite order. It is also given numerically for specific potential shapes as an expansion in powers of
x0 up to the second order. We show that, for a harmonic trap, the well-known first-order shift of the critical temperature $\ensuremath{\Delta}{T}_{c}/{T}_{c}\ensuremath{\propto}{N}^{\ensuremath{-}1/3}$ is inaccurate when $N\ensuremath{\leqslant}{10}^{5}$, the next order (proportional to ${N}^{\ensuremath{-}1/2}$) being significant. We also show that finite-size effects on the condensation temperature cancel out in a cubic trapping potential, e.g., $V(r)\ensuremath{\propto}{r}^{3}$. Finally, we show that in a generic power-law potential of higher order, e.g., $V(r)\ensuremath{\propto}{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}>3$, the shift of the critical temperature becomes positive. This effect provides a large increase of
Tc for relatively small atom numbers. For instance, an increase of about $+$40% is expected with
104 atoms in a $V(r)\ensuremath{\propto}{r}^{12}$ trapping potential.