  Lars Onsager and Richard Feynman envisioned that the three-dimensional (3D)
superfluid-to-normal $\lambda$ transition in $^{4}$He occurs through the
proliferation of vortices. This process should hold for every phase transition
in the same universality class. The role of topological defects in
symmetry-breaking phase transitions has become a prime topic in cosmology and
high-temperature superconductivity, even though direct imaging of these defects
is challenging. Here we show that the U(1) continuous symmetry that emerges at
the ferroelectric critical point of multiferroic hexagonal manganites leads to
a similar proliferation of vortices. Moreover, the disorder field (vortices) is
coupled to an emergent U(1) gauge field, which becomes massive by means of the
Higgs mechanism when vortices condense (span the whole system) upon heating
above the ferroelectric transition temperature. Direct imaging of the vortex
network in hexagonal manganites offers unique experimental access to this dual
description of the ferroelectric transition, while enabling tests of the
Kibble-Zurek mechanism.
