An elementary proof of phase transition in the planar XY model

Abstract

Using elementary methods we obtain a power-law lower bound on the two-point function of the planar XY spin model at low temperatures. This was famously first rigorously obtained by Fröhlich and Spencer and establishes a Berezinskii-Kosterlitz-Thouless phase transition in the model. Our argument relies on a new loop representation of spin correlations, a recent result of Lammers on delocalisation of integer-valued height functions, and classical correlation inequalities.

arXiv comment25 pages, statement of main result slightly improved

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• Delocalisation and absolute-value-FKG in the solid-on-solid model
• The Discrete Gaussian model, I. Renormalisation group flow at high temperature
• Random partitions under the Plancherel-Hurwitz measure, high genus Hurwitz numbers and maps
• The Discrete Gaussian model, II. Infinite-volume scaling limit at high temperature
• Geometric analysis of Ising models, Part III
• Coexistence, enhancements and short loops in random walk loop soups
• On the duality between height functions and continuous spin models
• Phase transitions for the $XY$ model in non-uniformly elliptic and Poisson-Voronoi environments
• Vector-valued Gaussian free field conditioned to avoid a ball: Entropic repulsion of the norm and Freezing of spins