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

TL;DR

This paper develops a real-space renormalization group approach using percolation probability to analyze confinement in two-dimensional $Z_2$ lattice gauge theories, providing analytical phase diagrams and confirming confinement conditions.

Contribution

It introduces a novel RG formalism based on percolation for $Z_2$ LGTs, enabling analytical exploration of confinement with matter and fluctuations.

Findings

Good agreement with numerical and exact results.

Finite matter density enforces confinement at finite temperature.

Analytical phase diagram for 2D $Z_2$ LGTs with matter.

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.