= Compact phase sectors
{title2=$\pi_1(S^1)=\mathbb Z$}
An <XY model> orientation takes values on a circle, so a circuit around an excluded defect core can carry integer <winding number>. Smooth single-valued <spin waves> cannot change that integer without a core event or boundary crossing. The locally quadratic <order-parameter phase> action therefore describes a sector, rather than automatically summing every compact-field configuration. Vortex-antivortex unbinding changes the large-distance <correlation function> and renormalized <phase stiffness>, even though ordinary <long-range order> is already absent in two dimensions at positive <temperature>.
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