Topological phase transitions in four dimensions

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