标题： 三维$\mathbb{Z}_2$ 显示英文标题

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

摘要： 通过大规模蒙特卡洛模拟和有限尺度标度分析，我们研究了Wegner的三维$\mathbb{Z}_2$格点规范模型在平衡状态下的渗流特性。 我们证实，穿过多激发面plaquettes的环在热临界点$T_{\rm c}$处发生渗流，并且它们的临界指数与三维Ising模型对偶表示中的环的临界指数一致。 然后，我们使用随机簇表示构建Fortuin-Kasteleyn（FK）簇，并发现它们也在$T_c$处发生渗流，而且可以进一步获得所有热临界指数。 两种环和FK簇的渗流阶参量的Binder累积量展示了伪一阶相变。 这项研究揭示了量子纠错和格点规范理论的关键性质。 显示英文摘要

摘要： With intensive Monte Carlo simulations and finite size scaling we undertake a percolation analysis of Wegner's three-dimensional $\mathbb{Z}_2$ lattice gauge model in equilibrium. We confirm that the loops threading excited plaquettes percolate at the thermal critical point $T_{\rm c}$ and we show that their critical exponents coincide with the ones of the loop representation of the dual 3D Ising model. We then construct Fortuin-Kasteleyn (FK) clusters using a random-cluster representation and find that they also percolate at $T_c$ and moreover give access to all thermal critical exponents. The Binder cumulants of the percolation order parameter of both loops and FK clusters demonstrate a pseudo first order transition. This study sheds light on the critical properties of quantum error correction and lattice gauge theories.