Exponential decay of transverse correlations for O(N) spin systems and related models

TL;DR

This paper proves exponential decay of transverse correlations in the Spin O(N) model for any non-zero external magnetic field and any N > 1, extending understanding of correlation decay in multi-component spin systems.

Contribution

It establishes exponential decay of correlations for N > 3 without relying on the Lee-Yang theorem, using a novel path representation and sampling method.

Findings

Exponential decay proven for N > 3 without Lee-Yang theorem.

Applicable to a wide class of multi-component spin systems.

Provides bounds on the typical length of open paths.

Abstract

We prove exponential decay of transverse correlations in the Spin O(N) model for arbitrary (non-zero) values of the external magnetic field and arbitrary spin dimension N > 1. Our result is new when N > 3, in which case no Lee-Yang theorem is available, it is an alternative to Lee-Yang when N = 2, 3, and also holds for a wide class of multi-component spin systems with continuous symmetry. The key ingredients are a representation of the model as a system of coloured random paths, a `colour-switch' lemma, and a sampling procedure which allows us to bound from above the `typical' length of the open paths.