An elementary proof of phase transition in the planar XY model

TL;DR

This paper provides an elementary proof of the phase transition in the planar XY model by establishing a power-law lower bound on the two-point function at low temperatures, confirming the Berezinskii-Kosterlitz-Thouless transition.

Contribution

It introduces a new loop representation of spin correlations and combines recent results and classical inequalities for a simplified proof of the phase transition.

Findings

Power-law lower bound on two-point function at low temperatures

Confirmation of Berezinskii-Kosterlitz-Thouless phase transition

New loop representation of spin correlations

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 Fr\"{o}hlich 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.