Abstract
This year's lecture was devoted to the study of quantum gases in reduced dimension, essentially in dimension two. Because of this reduced dimension, the condensation phenomenon associated with Bose-Einstein statistics is absent, at least in the thermodynamic limit, i.e. for an infinite system. In other words, the long-range order that characterizes a Bose-Einstein condensate, i.e. the in-phase correlation of the wave function that describes the gas, disappears at two or one dimension. Thermal fluctuations play a more important role than in three dimensions, destroying the order that statistics and interactions tend to create. It was German-born physicist Rudolf Peierls, working in England from 1933 onwards, who first formalized this possible loss of long-range order in reduced-dimensional systems. He was not interested in Bose-Einstein condensates, but in the order of a solid, and showed that a crystal could not exist in a one- or two-dimensional world. In this lecture, we first examined Peierls' arguments in one and then in two dimensions. We then described some recent experiments carried out on colloids, which tested Peierls' prediction. We also showed that the absence of crystalline order does not lead to complete disorder: these colloidal assemblies can be "almost" crystalline at low temperatures, before becoming completely liquid at higher temperatures. This was our first encounter with a phase transition induced by topological defects, a concept formalized by Berezinskii on the one hand, and by Kosterlitz and Thouless on the other (BKT).
