= Quasi-long-range order
{wiki}
Quasi-long-range order means that correlations decay algebraically rather than approaching a nonzero constant. It occurs in the low-temperature phase of the two-dimensional <XY model>, where <spin waves> destroy true long-range order but bound vortex pairs preserve power-law correlations.
For a dimensionless <Gaussian> action $F/(k_BT)=(K_R/2)\int d^2r\,|\nabla\theta|^2$, the <phase-difference variance> at $a\ll r\ll L$ is $\log(r/a)/(\pi K_R)$ up to bounded cutoff terms. <Gaussian phase averaging> then gives
$$
g(r)=\left\langle e^{i[\theta(\mathbf r)-\theta(0)]}\right\rangle
\propto(r/a)^{-\eta},\qquad \eta=\frac{1}{2\pi K_R}.
$$
Every positive $\eta$ makes $g(r)\to0$, so there is no conventional <long-range order>, even when the <phase stiffness> is finite. This is how an accumulation of gentle, cheap twists destroys global alignment without making every local twist free.
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