= Universal stiffness jump
{title2=$\rho_s(T_{\mathrm{BKT}}^-)=2k_BT_{\mathrm{BKT}}/\pi$}
The renormalized <phase stiffness> has this discontinuity at the <Berezinskii–Kosterlitz–Thouless transition>. The algebraic <correlation function> has exponent $1/4$ at the transition from the ordered side.
The stiffness here is an energy in two dimensions, not an areal mass density. For a neutral <superfluid> of particles of mass $m$, let $n_s$ be the <superfluid number density> per unit area. Substituting the <superfluid velocity> $\mathbf v_s=(\hbar/m)\nabla\theta$ into the kinetic energy gives $\rho_s=\hbar^2n_s/m$. Consequently the jump can also be written
$$
n_s(T_{\rm BKT}^-)\lambda_T^2=4,
\qquad \lambda_T^2=\frac{2\pi\hbar^2}{mk_BT},
$$
where $\lambda_T$ is the <thermal wavelength>. It is the large-distance <superfluid number density>, rather than the total density or a locally inferred condensate density, that enters this relation.
Within the <vortex-pair renormalization flow>, $y$ changes from irrelevant to relevant at $\pi K_R=2$, with $K_R$ the large-distance dimensionless <phase stiffness>. Combining this with the <spin wave> result $\eta=1/(2\pi K_R)$ gives $\eta=1/4$. Above the transition the homogeneous equilibrium <phase stiffness> vanishes at infinite observation scale. A discontinuous response coefficient is compatible with a diverging <correlation length>; it does not imply a <latent heat>.<Topological Defects and Phase Transitions by John Michael Kosterlitz>[Section 3, pp. 92–94: fugacity flow and renormalized superfluid-density jump]{ref}
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