 Phase-difference variance

For the physical free energythe Boltzmann factor is $e^{-F_{\theta }/(k_{B}T)}$. At $T>0$, the Gaussian thermal phase fluctuations therefore obeyThe temperature factor is necessary when ${\rho }_s$ is the energy stiffness defined by the parent. Defining a dimensionless stiffness $K={\rho }_s/(k_{B}T)$ instead replaces $k_{B}T/{\rho }_s$ by $1/K$.

The difference cancels the spatially uniform phase mode. At any fixed separation, $1-\cos (\mathbf{k}\cdot \mathbf{x})$ removes the small-$k$ singularity. What grows in low dimensions is the variance as the separation tends to infinity.

In one dimension, the integral equals $|x|/2$, so the variance is $(k_{B}T/{\rho }_s)|x|$. In two dimensions, with microscopic cutoff $a$ and $r\gg a$, it grows asFor the Gaussian field, the order parameter correlation isIt decays exponentially in one dimension and algebraically in two. Thus nonzero-temperature Gaussian fluctuations do not leave conventional long-range order in these dimensions. This is the long-wavelength mechanism related to the Mermin-Wagner theorem; vortices and interactions require the fuller theory of the Berezinskii–Kosterlitz–Thouless transition in the two-dimensional case.