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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 Z₂ 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 Z₂ 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< in the model we consider. Our RG scheme enables future analytical studies of Z₂ LGTs with matter and quantum fluctuations and beyond.
Dünnweber et al. (Tue,) studied this question.