Exponential decay in $O(n)$-invariant quantum spin systems

TL;DR

This paper proves that in certain symmetric quantum spin systems, correlations decay exponentially when the symmetry parameter n is sufficiently large, extending understanding of phase behavior in these models.

Contribution

It establishes exponential decay of correlations for large n in $O(n)$-invariant quantum spin systems, answering a question about the transition from long-range order.

Findings

Exponential decay of correlations for large n

Long-range order persists for small n in high dimensions

Provides rigorous bounds on correlation decay rates

Abstract

We consider $O(n)$-invariant and reflection-positive quantum spin systems on the integer lattice in any dimension, and prove that spin-spin correlations decay exponentially fast provided n is large enough. This answers a question of Ueltschi, who proved that for small n there is instead long-range order (for d at least 3).