On the slow decay ofO(2) correlations in the absence of topological excitations: Remark on the Patrascioiu-Seiler model
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For spin models withO(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, includingT=∞. 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.
Key WordsContinuous symmetry Kosterlitz-Thouless transition decay of correlations Fortuin-Kasteleyn representation topological charges
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