 Solution

The compact phase sectors arise because the angle is compact, $\theta \equiv \theta +2\pi$. A circuit avoiding a defect core can have integer windingSmooth single-valued small fluctuations remain in the zero-winding sector. Vortices with $q\ne 0$ cannot be unwound without a core or a boundary crossing; the smooth Gaussian theory misses those sectors. For $\theta =q\arg (x+iy)$ outside a core of radius $a$,The positional entropy of a freely placed defect is approximately $S/k_B=2\log (L/a)$. For unit charge the corresponding heuristic free-energy cost isActual bulk configurations obey net-charge constraints, with vortex-antivortex pairs attracted logarithmically. At low temperature they remain bound; they renormalize the phase stiffness but do not screen away the algebraic correlations. Heating can unbind the pairs into a screening vortex plasma with short-range correlations: this is the Berezinskii–Kosterlitz–Thouless transition.

The sharp criterion uses the long-distance renormalized stiffness $K_R$, not a bare coefficient: $K_R(T_{\mathrm{BKT}}^{-})=2/\pi$, giving algebraic exponent ${\eta }_c=1/4$. Both phases have zero conventional spontaneous magnetization at positive temperature; they differ in vortex binding, correlations and stiffness.^[ref]