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

Abstract

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 symmetries rigorously prove 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 find a clear perimeter-law signal for the dyonic Wilson-'t Hooft loop at $\theta =2\pi$, providing numerical evidence in the pure $SU(2)$ Yang-Mills theory for the theoretically expected dyon condensation at $\theta =2\pi$.

Comments: 1+28 pages, 15 figures, 3 tables, v2:reference added. v3:discussion and Table 3 added, abstract modified