On the slow decay of O(2) correlations in the absence of topological excitations: Remark on the Patrascioiu-Seiler model

For spin models with O(2)-invariant ferromagnetic interactions, the Patrascioiu-Seiler constraint is |arg(S(x))-arg(S(y))|≤θ₀ for all |x-y|=1. It is shown that in two-dimensional systems of two-component spins the imposition of such contraints with θ₀ small enough indeed results in the suppression of exponential clustering. More explicitly, it is shown that in such systems on every scale the spin-spin correlation function obeys 〈S(x)·S(y)〉≥1/(2|x-y|²) at any temperature, including T=∞. The derivation is along the lines proposed by A. Patrascioiu and E. Seiler, with the yet unproven conjectures invoked there replaced by another geometric argument.