Percolation as a confinement order parameter in $\mathbb{Z}_2$ lattice gauge theories

Lattice gauge theories (LGTs) were introduced in 1974 by Wilson to study quark confinement. These models have been shown to exhibit (de-)confined phases, yet it remains challenging to define experimentally accessible order parameters. Here we propose percolation-inspired order parameters (POPs) to probe confinement of dynamical matter in $\mathbb{Z}_2$ LGTs using electric field basis snapshots accessible to quantum simulators. We apply the POPs to study a classical $\mathbb{Z}_2$ LGT and find a confining phase up to temperature $T=\infty$ in 2D (critical $T_c$, i.e. finite-$T$ phase transition, in 3D) for any non-zero density of $\mathbb{Z}_2$ charges. Further, using quantum Monte Carlo we demonstrate that the POPs reproduce the square lattice Fradkin-Shenker phase diagram at $T=0$ and explore the phase diagram at $T>0$. The correlation length exponent coincides with the one of the 3D Ising universality class and we determine the POP critical exponent characterizing percolation. Our proposed POPs provide a geometric perspective of confinement and are directly accessible to snapshots obtained in quantum simulators, making them suitable as a probe for quantum spin liquids.