From the $SU(2)$ Quantum Link Model on the Honeycomb Lattice to the Quantum Dimer Model on the Kagome Lattice: Phase Transition and Fractionalized Flux Strings

We consider the $(2+1)$-d $SU(2)$ quantum link model on the honeycomb lattice and show that it is equivalent to a quantum dimer model on the Kagome lattice. The model has crystalline confined phases with spontaneously broken translation invariance associated with pinwheel order, which is investigated with either a Metropolis or an efficient cluster algorithm. External half-integer non-Abelian charges (which transform non-trivially under the $mathbb{Z}(2)$ center of the $SU(2)$ gauge group) are confined to each other by fractionalized strings with a delocalized $mathbb{Z}(2)$ flux. The strands of the fractionalized flux strings are domain walls that separate distinct pinwheel phases. A second-order phase transition in the 3-d Ising universality class separates two confining phases; one with correlated pinwheel orientations, and the other with uncorrelated pinwheel orientations.