  The three-dimensional integer-valued lattice gauge theory, which is also
known as a "frozen superconductor," can be obtained as a certain limit of the
Ginzburg-Landau theory of superconductivity, and is believed to be in the same
universality class. It is also exactly dual to the three-dimensional XY model.
We use this duality to demonstrate the practicality of recently developed
methods for studying topological defects, and investigate the critical behavior
of the phase transition using numerical Monte Carlo simulations of both
theories. On the gauge theory side, we concentrate on the vortex tension and
the penetration depth, which map onto the correlation lengths of the order
parameter and the Noether current in the XY model, respectively. We show how
these quantities behave near the critical point, and that the penetration depth
exhibits critical scaling only very close to the transition point. This may
explain the failure of superconductor experiments to see the inverted XY model
scaling.
