= Phase-difference variance
For the physical <free energy>
$$
F_\theta=\frac{\rho_s}{2}\int d^dx\,|\nabla\theta|^2,
$$
the <Boltzmann factor> is $e^{-F_\theta/(k_BT)}$. At $T>0$, the Gaussian <thermal phase fluctuations> therefore obey
$$
\langle[\theta(\mathbf x)-\theta(0)]^2\rangle
=\frac{2k_BT}{\rho_s}
\int\frac{d^dk}{(2\pi)^d}\frac{1-\cos(\mathbf k\cdot\mathbf x)}{k^2}.
$$
The <temperature> factor is necessary when $\rho_s$ is the <energy> stiffness defined by the parent. Defining a dimensionless stiffness $K=\rho_s/(k_BT)$ instead replaces $k_BT/\rho_s$ 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 $(k_BT/\rho_s)|x|$. In two dimensions, with microscopic cutoff $a$ and $r\gg a$, it grows as
$$
\langle[\theta(\mathbf x)-\theta(0)]^2\rangle
\simeq\frac{k_BT}{\pi\rho_s}\log(r/a).
$$
For the Gaussian field, the <order parameter> correlation is
$$
\langle e^{i[\theta(\mathbf x)-\theta(0)]}\rangle
=\exp\left[-\frac12\langle(\theta(\mathbf x)-\theta(0))^2\rangle\right].
$$
It 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.
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