We investigate the equilibrium phase-coherence properties of Bose-condensed particle systems, focusing on their shape dependence and finite-size scaling (FSS). We consider three-dimensional (3D) homogeneous systems confined to anisotropic $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}{L}_{a}$ boxes, below the Bose-Einstein-condensate (BEC) transition temperature
Tc. We show that the phase correlations develop peculiar anisotropic FSS for any
T<Tc, in the large-
L limit keeping the ratio $\ensuremath{\lambda}\ensuremath{\equiv}{L}_{a}/{L}^{2}$ and the particle density fixed. This phenomenon is effectively described by the 3D spin-wave theory. Its universality is confirmed by quantum Monte Carlo simulations of the 3D Bose-Hubbard model in the BEC phase. The phase-coherence correlations of very elongated BEC systems, $\ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$, are characterized by the coherence length ${\ensuremath{\xi}}_{a}\ensuremath{\sim}{A}_{t}{\ensuremath{\rho}}_{s}/T$, where
At is the transverse area and ${\ensuremath{\rho}}_{s}$ is the superfluid density.
Motivated by the recent achievement of space-based Bose-Einstein condensates (BEC) with ultracold alkali-metal atoms under microgravity and by the proposal of bubble traps which confine atoms on a thin shell, we investigate the BEC thermodynamics on the surface of a sphere. We determine analytically the critical temperature and the condensate fraction of a noninteracting Bose gas. Then we consider the inclusion of a zero-range interatomic potential, extending the noninteracting results at zero and finite temperature. Both in the noninteracting and interacting cases the crucial role of the finite radius of the sphere is emphasized, showing that in the limit of infinite radius one recovers the familiar two-dimensional results. We also investigate the Berezinski-Kosterlitz-Thouless transition driven by vortical configurations on the surface of the sphere, analyzing the interplay of condensation and superfluidity in this finite-size system.
Research on the phase transition and phase diffusion between the Mott-insulator and superfluid phase has not yet been done extensively for the case of magnons. In this thesis we start from a model Hamiltonian for an easy-plane magnetic insulator and construct the phase diagram describing the transition between the Mott insulator and superfluid phase. Finally, we consider phase diffusion of the Bose-Einstein condensate in the superfluid phase and find that phase diffusion indeed occurs in a finite-sized system.