Scaling properties of three-dimensional Bose-Einstein condensed gases

The behavior of three-dimensional bosonic gases at low temperatures is characterized by the formation of a Bose-Einstein condensate (BEC). The Bose-Einstein condensation is a phase transition of a bosonic gas, occurring usually when the temperature is lowered below a certain critical value. The gas shifts from the normal phase to a new one where the lowest-energy state becomes macroscopically occupied. The wavefunction associated to such state is called condensate wavefunction. In three-dimensional systems the phase of the condensate wavefunction is usually coherent all over the condensate size. Both the phase-coherence properties of the BEC phase and the critical behavior at the BEC transition turn out to be particularly sensitive to the inhomogeneous conditions arising from spatially-dependent confining potentials and to the geometry of the atomic-gas system. In this work we show how coherence properties vary when the system shape is changed resulting in a change of the effective dimensionality of the system, for example when it shifts from a cube to a very long parallelepiped thus becoming an effective one-dimensional system or to a very thin slab which is instead an effective two-dimensional system. Atomic gases in elongated homogeneous boxes and harmonic traps show a dimensional crossover from a 3D behavior to a quasi-1D behavior, which gives rise to a substantial phase decoherence along the longer axial direction. This crossover is characterized by a peculiar anisotropic finite-size scaling and can be described by an effective 3D spin-wave theory. The behavior of homogeneous gases in slab geometries is more complex. Indeed in two dimensions we do not have condensates but bosonic gases undergo another kind of transition to a quasi-long range ordered phase, where correlations decrease with the distance with a power which depends on the temperature. This transition is known as the Berezinskii-Kosterlitz-Thouless (BKT) transition. The interplay of the BEC and BKT critical modes gives rise to a quite complex behavior. The dimensional-crossover scenario can be described by a transverse finite-size scaling limit for systems in slab geometries. Moreover in inhomogeneous conditions if the trapping potential is smooth enough we can assist to the peculiar presence of different phases in different spatial regions, separated by a spatial critical surface. We give a description of this peculiar critical behavior around the spatial critical surface. All our theoretical predictions are supported by numerical analyses of the 3D hard-core Bose-Hubbard model, which models gases of bosonic atoms in optical lattices. We perform quantum Monte Carlo simulations of the Bose-Hubbard model at several values of the temperature and with different geometries and external potentials, measuring in particular its phase-coherence properties