Quasi-long-range order

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 $F/(k_{B}T)=(K_R/2)\int d^{2}r|\nabla \theta {|}^2$, 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 givesEvery positive $\eta$ makes $g(r)\to 0$, 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.