The geometric phase transition of the three-dimensional $\mathbb{Z}_2$ lattice gauge model

TL;DR

This study uses percolation analysis and Monte Carlo simulations to reveal the geometric nature of topological phase transitions in a 3D $ ext{Z}_2$ lattice gauge model, linking percolation phenomena to critical behavior.

Contribution

It introduces a novel percolation-based approach to analyze topological phase transitions in lattice gauge theories, connecting geometrical loops and FK clusters to critical phenomena.

Findings

Percolation of geometrical loops occurs at the critical temperature $T_c$.

FK clusters also percolate at $T_c$, allowing extraction of critical exponents.

Binder cumulants indicate a pseudo-first-order transition.

Abstract

After fifty years of lattice gauge theories (LGTs), the nature of the transition between their topological phases (confinement/deconfinement) remains challenging due to the absence of a local order parameter. In this work, we conduct a percolation analysis of Wegner's three-dimensional $\mathbb{Z}_2$ 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, regardless of the connection rules, geometrical loops, constructed by piercing excited plaquettes percolate precisely at the thermal critical point $T_{\rm c}$, 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 $T_{\rm c}$, 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.