 Universal stiffness jump

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 ${\mathbf{v}}_s=(\hslash /m)\nabla \theta$ into the kinetic energy gives ${\rho }_s={\hslash }^2{n}_s/m$. Consequently the jump can also be writtenwhere ${\lambda }_T$ 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.^[ref]