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An elementary proof of phase transition in the planar XY model
Diederik van Engelenburg, Marcin Lis · 2021-10-19 · via math.PR updates on arXiv.org

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.

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