Geometric phase transition of the three-dimensional Z2 lattice gauge model

After fifty years of lattice gauge theories (LGTs), the nature of the transition between their topological phases (confinement/deconfinement) remains elusive due to the absence of a local order parameter. In this work, we conduct a percolation analysis of Wegner’s three-dimensional Z2 lattice gauge model using intensive Monte Carlo simulations and finite-size scaling, offering fresh insights into the topological phase transitions of gauge-invariant systems. We demonstrate that loops threading excited plaquettes, regardless of the connection rules, percolate precisely at the thermal critical point Tc, with critical exponents coinciding with those of the loop representation of the dual 3D Ising model. Further, we construct Fortuin-Kasteleyn (FK) clusters in a random-cluster representation, showing that they also percolate at Tc, enabling access to all thermal critical exponents. Strikingly, the Binder cumulants of the percolation order parameters for both loops and FK clusters reveal a pseudo-first-order transition. This work sheds new light on the critical behavior of pure LGTs, with potential implications for condensed matter systems and quantum error correction.