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This lecture is the first in a four-part series by Christophe Garban from Université Lyon I, introducing the Berezinskii-Kosterlitz-Thouless (BKT) phase transition in statistical physics. Explore how spins displayed along the lattice Z^d interact and fluctuate with temperature changes, with particular focus on continuous symmetry systems like the XY model. Learn about the fundamental differences between discrete and continuous spin systems, and discover why the BKT phase transition in two dimensions represents a unique phenomenon caused by the behavior of vortices. The lecture covers main examples of systems exhibiting BKT transitions (XY and Villain models, Coulomb gas, clock models, integer-valued height functions) and provides physics explanations distinguishing between S¹ and S² behaviors. This two-hour session lays the groundwork for subsequent lectures that will address mathematical approaches to BKT, domain attraction, and non-linear sigma models.
