Emergent Photons and Confinement: A Numerical Study on Z N Lattice Gauge Theory

We numerically study <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:msub> <a:mrow> <a:mi mathvariant="double-struck">Z</a:mi> </a:mrow> <a:mrow> <a:mi>N</a:mi> </a:mrow> </a:msub> </a:mrow> </a:math> lattice gauge theories in 4D as prototypical models of systems with <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:msub> <d:mi mathvariant="double-struck">Z</d:mi> <d:mi>N</d:mi> </d:msub> </d:math> 1-. For <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mrow> <g:mi>N</g:mi> <g:mo>≥</g:mo> <g:mn>3</g:mn> </g:mrow> </g:math> , we provide evidence that such systems exhibit not only the expected phases with spontaneously broken/restored symmetry but also a third photon phase. When present, the 1-form symmetry provides a precise notion of confinement, and it is commonly believed that confinement ensues due to the proliferation of extended, stringlike objects known as , which carry a <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi mathvariant="double-struck">Z</i:mi> <i:mi>N</i:mi> </i:msub> </i:math> flux. However, this picture is challenged by the three-phase scenario investigated here. We show that both the confined and the photon phases are associated with the proliferation of center vortices and that the key difference between them lies in whether or not vortex junctions—the proliferate.