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This lecture is the fourth part of a series exploring the Berezinskii-Kosterlitz-Thouless (BKT) phase transition in statistical physics, presented by Christophe Garban at the Hausdorff Center for Mathematics. Delve into the fascinating world of spin systems on lattices and how they fluctuate with temperature changes. Understand the fundamental differences between discrete spin systems (like the Ising model) and continuous spin systems (such as the XY model with unit circle S¹ spins or the classical Heisenberg model with unit sphere S² spins). Discover how continuous symmetry systems exhibit drastically different phase transition behaviors, particularly the unique BKT phase transition in two dimensions predicted in the 1970s. Learn about the critical role of "vortices" (special monodromies) in triggering this transition, explore recent mathematical proofs confirming its existence, and examine cutting-edge results in this field during this one-hour advanced physics lecture.
