= Phase stiffness
{title2=$\rho_s$}
= Helicity modulus
{synonym}
The phase stiffness is the coefficient of $\tfrac12\int d^dx\,(\nabla\theta)^2$ in the long-distance <free energy>. Its renormalized value determines the <thermal phase fluctuations> and algebraic <correlation function> of a two-dimensional <O(N) model> with $N=2$.
Operationally impose a small phase difference $\phi$ between opposite sides of a square sample of side $L$. The smooth minimum has $\theta=\phi x/L$, so the two-dimensional excess <free energy> is
$$
F(\phi)-F(0)=\tfrac12\rho_s\phi^2+o(\phi^2),
\qquad \rho_s=\left.\frac{\partial^2F}{\partial\phi^2}\right|_{\phi=0}.
$$
The derivative concerns a prescribed twist at the stated observation scale; in $d$ dimensions its conventional size factor is $L^{d-2}$. This response definition includes thermal fluctuations and <vortices> after equilibration, whereas a microscopic gradient coefficient need not. A finite <phase stiffness> requires no nonzero spatially uniform <order parameter>; the <Berezinskii–Kosterlitz–Thouless transition> is the instructive example.
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