  Phase transitions give crucial insight into many-body systems, as crossovers
between different regimes of order are determined by the underlying dynamics.
These dynamics, in turn, are often constrained by dimensionality and geometry.
For example, in one- and two-dimensional systems with continuous symmetry,
thermal fluctuations prevent the formation of long-range order[1,2].
Two-dimensional systems are particularly significant, as vortices can form in
the plane but cannot tilt out of it. At high temperatures, random motion of
these vortices destroys large-scale coherence. At low temperatures, vortices
with opposite spin can pair together, cancelling their circulation and allowing
quasi-long-range order to appear. This Berezenskii-Kosterlitz-Thouless (BKT)
transition[3,4] is essentially classical, arising for example in the
traditional XY model for spins, but to date experimental evidence has been
obtained only in cold quantum systems. Measurements of superfluid sound
speed[5] and critical velocity[6] have been consistent with scaling
predictions, and vortices have been observed directly in cold atom
experiments[7,8]. However, the presence of trapping potentials restricts
measurement to vortex density, rather than number, and obscures the process of
vortex unbinding. Further, atom and fluid experiments suffer from parasitic
heating and difficulties in phase recording, leading to results that differ
from theory in many quantitative aspects. Here, we use a nonlinear optical
system to directly observe the ideal BKT transition, including vortex pair
dynamics and the correlation properties of the wavefunction, for both repulsive
and attractiveinteractions (the photonic equivalent of ferromagnetic and
antiferromagnetic conditions[9]). The results confirm the thermodynamics of the
BKT transition and expose outstanding issues in the crossovers to superfluidity
and Bose-Einstein condensation.
