Percolation renormalization group analysis of confinement in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi mathvariant="double-struck">Z</a:mi><a:mn>2</a:mn></a:msub></a:math> 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi mathvariant="double-struck">Z</c:mi><c:mn>2</c:mn></c:msub></c:math> 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi>T</e:mi><e:mo>&lt;</e:mo><e:mi>∞</e:mi></e:mrow></e:math> in the model we consider. Our RG scheme enables future analytical studies of <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:msub><f:mi mathvariant="double-struck">Z</f:mi><f:mn>2</f:mn></f:msub></f:math> LGTs with matter and quantum fluctuations and beyond.