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

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.

Automated summary

The study demonstrates the exponential decay of transverse correlations in the Spin O(N) model for all N > 1, with a novel result when N > 3. It serves as an alternative to the Lee-Yang theorem for N = 2, 3 and extends to a wide class of multi-component spin systems with continuous symmetry.