Numerical Hints for Dyon Condensation at $θ=2π$ via Wilson-'t Hooft Loops in $SU(2)$ Yang-Mills Theory

Abstract

Yang-Mills theories at $θ$ and $θ+2π$ are unitarily equivalent, but their $2π$ periodicity has a nontrivial realization. Recent developments in generalized symmetries rigorously prove that confinement vacua at $θ=0$ and $2π$ should belong to different symmetry-protected topological (SPT) states with the $1$-form center symmetry. For its examination, we measure the Wilson-'t Hooft loop operators at $θ=2π$ for the $SU(2)$ Wilson lattice gauge action and discuss their long-distance behaviors. This requires us to identify the gauge topological charge in the presence of defects, and we employ the $1$-form covariant DBW2 gradient flow to smear lattice gauge fields. We find a clear perimeter-law signal for the dyonic Wilson-'t Hooft loop at $θ=2π$, providing numerical evidence in the pure $SU(2)$ Yang-Mills theory for the theoretically expected dyon condensation at $θ=2π$.