Percolation renormalization group analysis of confinement in mathbb{Z}₂ lattice gauge theories

The analytical study of confinement in lattice gauge theories (LGTs) remains a difficult task to this day. Taking a geometric perspective on confinement, we develop a real-space renormalization group (RG) formalism for mathbb{Z}₂ LGTs using percolation probability as a confinement order parameter. The RG flow we analyze is constituted by both the percolation probability and the coupling parameters. We consider a classical mathbb{Z}₂ LGT in two dimensions, with matter and thermal fluctuations, and analytically derive the confinement phase diagram. We find good agreement with numerical and exact benchmark results and confirm that a finite matter density enforces confinement at T<infty in the model we consider. Our RG scheme enables future analytical studies of mathbb{Z}₂ LGTs with matter and quantum fluctuations and beyond.