Abstract
The fifth lecture was devoted entirely to the Berezinskii-Kosterlitz-Thouless transition. Starting from the famous example of the one-dimensional Ising chain with interactions in 1/r2, which was introduced by Anderson, Yval and Hamann in 1971 in connection with the Kondo problem, renormalization equations can be written by combining, at each step, the elimination of the smallest domains with a change of scale. The elimination of small domains leads to a renormalization of the interaction strength in 1/r2 and, depending on the initial temperature, either a recrudescence of domain walls (disordered phase) or a dilute gas of domain wall pairs (ferromagnetic phase) is observed on a large scale. The same type of approach applies to the XY model in dimension 2, which was introduced to study the superfluid transition of helium films : at each renormalization step, the closest pairs of vortices are eliminated, thus renormalizing the interactions. The lecture ended by showing the equivalence of the XY model in dimension 2 with surface models, such as the Villain model, which feature a rough transition, or two-dimensional Coulomb gas models.
