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

Yang-Mills theories at $\theta$ and $\theta +2\pi$ are unitarily equivalent, but their $2\pi$ periodicity has a nontrivial realization. Recent developments in generalized global symmetries show that confinement vacua at $\theta =0$ and $2\pi$ 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 $\theta =2\pi$ 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 then obtain numerical evidence consistent with dyon condensation at $\theta =2\pi$, rather than monopole condensation, as theoretically predicted.