An Elementary Proof of Phase Transition in the Planar XY Model
Diederik van Engelenburg, Marcin Lis
Abstract
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 (Commun Math Phys 81(4):527–602, 1981) 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 (Probab Relat Fields, 2021) on delocalisation of general integer-valued height functions, and classical correlation inequalities.
Topics & Concepts
Classical XY modelElementary proofPlanarPhase transitionMathematicsInteger (computer science)Mathematical physicsKosterlitz–Thouless transitionSpin (aerodynamics)Transition pointCombinatoricsPhysicsQuantum mechanicsPure mathematicsComputer scienceMechanicsThermodynamicsProgramming languageComputer graphics (images)Theoretical and Computational PhysicsStochastic processes and statistical mechanicsMarkov Chains and Monte Carlo Methods