  We show that four-dimensional systems may exhibit a topological phase
transition analogous to the well-known Berezinskii-Kosterlitz-Thouless vortex
unbinding transition in two-dimensional systems. The realisation of an
engineered quantum system, where the predicted phase transition shall occur, is
also presented. We study a suitable generalization of the sine-Gordon model in
four dimensions and the renormalization group flow equation of its couplings,
showing that the critical value of the frequency is the square of the
corresponding value in $2D$. The value of the anomalous dimension at the
critical point is determined ($\eta=1/32$) and a conjecture for the universal
jump of the superfluid stiffness ($4/\pi^2$) presented.
