Percolation renormalization group analysis of confinement in ${\mathbb{Z}}_2$ lattice gauge theories

Abstract

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.

Comments: 7+6 pages, 8+1 figures