We present a short review of how the effective action formalism, well known in relativistic quantum field theory, can be used to discuss Bose-Einstein condensation of non-relativistic gases. This method lends itself very naturally to an interpretation of Bose-Einstein condensation in terms of symmetry breaking. It also allows for the definition of a very elegant regularization technique involvi...

This article surveys a number of theoretical problems and open questions in the field of two-dimensional dilute Bose gases with weak repulsive interactions. In contrast to three dimensions, in two dimensions the formation of long-range order is prohibited by the Bogoliubov-Hohenberg theorem, and Bose-Einstein condensation is not expected to be realized. Nevertheless, the first experimental indi...

The Bose gas in an external potential is studied by means of the local density approximation. An analytical result is derived for the dependence of the critical temperature of Bose-Einstein condensation on the mutual interaction in a generic power-law potential. PACS numbers: 03.75.Fi, 32.80.Pj In an ideal Bose-Einstein gas, the zero momentum state can become macroscopically occupied, and the s...

The model of spatial permutations is related to the Feynman-Kac representation of the Bose gas. The transition to infinite cycles corresponds to Bose-Einstein condensation. We review the general setting and some results, and we derive a multi-body interaction between permutation jumps, that is due to the original interactions between quantum particles.

The momentumand frequency-dependent one-body correlation function of the one-dimensional interacting Bose gas (Lieb-Liniger model) in the repulsive regime is studied using the Algebraic Bethe Ansatz and numerics. We first provide a determinant representation for the field form factor which is welladapted to numerical evaluation. The correlation function is then reconstructed to high accuracy fo...

Coherent resonance is the effect of resonant excitation of nonlinear coherent modes in trapped Bose condensates. This novel effect is shown to be feasible for Bose-condensed trapped gases. Conditions for realizing this effect are derived. A method of stabilizing Bose condensates with attractive interactions is advanced. The origin of dynamic critical phenomena is elucidated. Interference effect...

The Bose-Hubbard hamiltonian will be introduced and justified as an effective description of some relevant physical systems (dilute alkali gases in optical lattices, arrays of Josephson junctions). I will also show a mean field theory solution and the resulting phase diagram. Finally, I will describe in detail one of the more important experiments with an optical lattice of Rb atoms. 1. The Bos...

We show that a many-atom Bose system at zero temperature has, in general, a liquid phase in addition to its well-known gaseous phase. A universal phase diagram is presented that is applicable to all Bose systems with a -C6/r6 type of interaction at large interparticle separations. We show that the predicted phase structure has implications on the stability of a gaseous Bose-Einstein condensate ...

We investigate the particle-number fluctuations of condensate due to collective excitations for trapped two-dimensional (2D) and quasi-2D Bose gases at low temperature. We show that the fluctuations of the condensate are proportional to the square of the total particle number of the system. For a quasi-2D Bose gas the fluctuations are also proportional to the square root of the trapping frequen...

Phase transitions, as the condensation of a gas to a liquid, are often revealed by a discontinuous behaviour of thermodynamic quantities. For liquid helium, for example, a divergence of the specific heat signals the transition from the normal fluid to the superfluid state. Apart from liquid helium, determining the specific heat of a Bose gas has proven to be a challenging task, for example, for...