We calculate the superfluid transition temperature of homogeneous interacting Bose gases in three and two spatial dimensions using large-scale path integral Monte Carlo simulations (with up to N=10;{5} particles). In 3D we investigate the limits of the universal critical behavior in terms of the scattering length alone by using different models for the interatomic potential. We find that this type of universality sets in at small values of the gas parameter na3 < or approximately 10(-4). This value is different from the estimate na3 < or approximately 10(-6) for the validity of the asymptotic expansion in the limit of vanishing na3. In 2D we study the Berezinskii-Kosterlitz-Thouless transition of a gas with hard-core interactions. For this system we find good agreement with the classical lattice |psi|4 model up to very large densities. We also explain the origin of the existing discrepancy between previous studies of the same problem.
Using renormalization-group arguments we show that the low-temperature thermodynamics of a three- or two-dimensional dilute Bose gas is fully determined by a universal scaling function ${\mathcal{F}}_{d}(\ensuremath{\mu}/{k}_{B}T,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(T))$ once the mass
m and the
s-wave scattering length
ad of the bosons are known (
d is the space dimension). Here $\ensuremath{\mu}$ and
T denote the chemical potential and temperature of the gas, and the temperature-dependent dimensionless interaction constant $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(T)$ is a function of $m{a}_{d}^{2}{k}_{B}T/{\ensuremath{\hbar}}^{2}$. We compute the scaling function
F2 using a nonperturbative renormalization-group approach and find that both the $\ensuremath{\mu}/{k}_{B}T$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(T)$ dependencies are in very good agreement with recent experimental data obtained for a quasi-two-dimensional Bose gas with or without optical lattice. We also show that the nonperturbative renormalization-group estimate of the Berezinskii-Kosterlitz-Thouless transition temperature compares well with the result obtained from a quantum Monte Carlo simulation of an effective classical field theory.
The weakly interacting Bose gas in two dimensions is considered in the dilute limit
n1/2a\ensuremath{\ll}1, where n is the particle density and a is the range of the potential. The standard many-body perturbation theory for this system has two separate divergences: the first, associated with classical phase fluctuations, is responsible for the vanishing of the long-range order; the second is quantum mechanical and is connected with the vanishing of the scattering t matrix at long wavelengths and low energies. An earlier diagrammatic theory of Popov, which provides a consistent description of the system in the dilute limit, is rederived heuristically from a quasiparticle picture, and also using the renormalization group. It is shown that the superfluid transition temperature is
Tc\ensuremath{\approxeq}4\ensuremath{\pi}(${\ensuremath{\Elzxh}}^{2}$/2m)n/[ln ln(1/
na2)], and the condition of validity of the dilute limit is ln ln(1/
na2)\ensuremath{\gg}1. The connection to the dilute Bose gas in dimensions d>2 and the universal behavior beyond the extreme asymptotic low-density domain are also discussed.
We revised the large- N expansion for a three-dimensional Bose system with short-range repulsion in normal phase. Particularly, for the model potential that is characterised only by the s -wave scattering length a the full numerical calculations of the critical temperature in the 1 / N -approximation as a function of the gas parameter a n 1 / 3 are performed. Additionally to the well-known result in the dilute limit we estimated analytically the leading-order strong-coupling behavior of the Bose-Einstein condensation transition temperature. It is shown that the critical temperature shift of the non-ideal Bose gas grows at small a n 1 / 3 , reaches some maximal value and then falls down becoming negative.
We use the nonperturbative linear $\ensuremath{\delta}$ expansion method to evaluate analytically the coefficients
c1 and ${c}_{2}^{\ensuremath{''}}$ that appear in the expansion for the transition temperature for a dilute, homogeneous, three-dimensional Bose gas given by ${T}_{c}{=T}_{0}({1+c}_{1}{\mathrm{an}}^{1/3}+[{c}_{2}^{\ensuremath{'}}\mathrm{ln}{(an}^{1/3}{)+c}_{2}^{\ensuremath{''}}]{a}^{2}{n}^{2/3}{+O(a}^{3}n)),$ where
T0 is the result for an ideal gas, a is the s-wave scattering length, and n is the number density. In a previous work the same method has been used to evaluate
c1 to order ${\ensuremath{\delta}}^{2}$ with the result
c1=3.06. Here, we push the calculation to the next two orders obtaining
c1=2.45 at order ${\ensuremath{\delta}}^{3}$ and
c1=1.48 at order ${\ensuremath{\delta}}^{4}.$ Analyzing the topology of the graphs involved we discuss how our results relate to other nonperturbative analytical methods such as the self-consistent resummation and the
1/N approximations. At the same orders we obtain ${c}_{2}^{\ensuremath{''}}=101.4,$ ${c}_{2}^{\ensuremath{''}}=98.2,$ and ${c}_{2}^{\ensuremath{''}}=82.9.$ Our analytical results seem to support the recent Monte Carlo estimates ${c}_{1}=1.32\ifmmode\pm\else\textpm\fi{}0.02$ and ${c}_{2}^{\ensuremath{''}}=75.7\ifmmode\pm\else\textpm\fi{}0.4.$
The physical properties of a homogeneous system at the thermodynamic equilibrium are strongly constrained by its dimensionality. In this respect, the two dimensional Bose gas constitutes a particularly interesting system: while it is impossible to observe a long range order at non-zero temperatures, there exists nevertheless a phase transition to a superfluid state at low temperatures. Furthermore, owing to the reduced dimensionality, the equation of state of the weakly interacting two-dimensional Bose gas is scale invariant. In this thesis, we present an experimental study of the two-dimensional Bose gas. We measure its equation of state with two different methods, and find a good agreement with analytic and numerical predictions. These results confirm as well the scale invariance of the system. Furthermore, one of the methods allows for a determination of the equation of state with a single adjustable parameter. We then characterize the superfluid response of the system., by showing evidence for a dissipationless response of the system to a moving perturbation. Finally, we analyze the fluctuations dynamics of the 2D Bose gas, which confirms both the suppression of density fluctuations in the superfluid phase, and the dominating contribution of phonons to the phase fluctuations.