= 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 winding
$$
\boxed{q=\frac1{2\pi}\oint\nabla\theta\cdot d\boldsymbol\ell\in\mathbb Z.}
$$
Smooth single-valued small fluctuations remain in the zero-winding sector. Vortices with $q\ne0$ 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> $a$,
$$
\beta E_q=\frac{\bar K}{2}\int_a^L2\pi r\,dr\,\frac{q^2}{r^2}+\beta E_c
=\pi\bar Kq^2\log(L/a)+\beta E_c.
$$
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 is
$$
\beta F_1\simeq(\pi\bar K-2)\log(L/a)+\beta E_c.
$$
Actual 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>.
\Image[/past-exam-of-the-phase-transitions-and-collective-phenomena-course-of-the-university-of-cambridge/2012/paper-1-ptcp2012-vortex-pair.png]
{title=Compact phase with opposite winding}
{description=Original constructed XY orientation field theta=arg(z+1)-arg(z-1), with unit positive and negative winding about the two excluded cores. Arrows show magnetization orientation, not fluid velocity; a circuit around both cores has zero net winding.}
The sharp criterion uses the long-distance renormalized stiffness $K_R$, not a bare coefficient: $K_R(T_{\rm BKT}^-)=2/\pi$, 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.<Statistical Field Theory by David Tong>[§4.4 and §4.4.1: compact angle, vortex energy/entropy and the two-dimensional transition]{ref}
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