Fig. 1: \({{\mathbb{Z}}}_{2}\) Loop-chain.
a Hardcore bosons live in the sites of a one-dimensional lattice. Two links depart from each site in a loop geometry and host spin-\(\frac{1}{2}\) gauge-field variables \({\sigma }_{1}^{\alpha }\) and \({\sigma }_{2}^{\alpha }\). Each loop encloses a gauge flux taking values in {0, π} and corresponding to the eigenvalues +1 (0 flux) or -1 (π flux) of \({\sigma }_{1}^{z}{\sigma }_{2}^{z}\). The hopping amplitude between sites connected by a loop in a π-flux state vanishes due to destructive Aharonov-Bohm interference. b Gauge invariant configurations in the spin-1 reduction of the loop-chain model, where \(| {\Phi }^{\pm }\rangle\) and \(| {\Psi }^{+}\rangle\) are common eigenstates of the spin-1 operators \({({S}^{x})}^{2}\) and \({({S}^{z})}^{2}\), defined at the end of the “Additional local symmetry and reduction to spin-1" subsection in the Methods. Occupied sites are filled in dark blue.
