Classical XY Model in 1.99 Dimensions

TL;DR

This paper investigates the classical XY model on fractal-like lattices with dimensions between 1 and 2, providing rigorous bounds that challenge previous predictions of phase transitions in such systems.

Contribution

It offers a mathematical proof that rules out phase transitions in the XY model on fractal dimensions 1<D<2, contradicting earlier harmonic approximation predictions.

Findings

Exponential decay of correlations at non-zero temperatures

No phase transition occurs in the studied fractal dimensions

Rigorous bounds contradict previous harmonic approximation results

Abstract

We consider the classical XY model (O(2) nonlinear sigma-model) on a class of lattices with the (fractal) dimensions 1<D<2. The Berezinskii's harmonic approximation suggests that the model undergoes a phase transition in which the low temperature phase is characterized by stretched exponential decay of correlations. We prove an exponentially decaying upper bound for the two-point correlation functions at non-zero temperatures, thus excluding the possibility of such a phase transition.