Replicating Higgs fields in Ising gauge theory: the registry order

We consider $Z_2$ gauge field theory coupled to "Higgs" matter fields invoking several copies of such matter, interacting entirely through the gauge fields, the $Z_2 \times Z_2 \times Z_2\cdots / Z_2$ and the $O(N) \times O(N) \times O(N) \cdots / Z_2$ families of theories. We discover that the Higgs phase of such theories is characterized by a hitherto unidentified "registry" order parameter. This is characterized by a gauge invariant $p = 2^{N_{\text{rep}}}/2$ Potts type symmetry where $N_{\text{rep}}$ is the number of matter copies. The meaning of this registry is that the different matter copies align their vectors locally in strictly parallel or anti-parallel fashion, even dealing with the continuous $O(2)$ symmetry. Supported by Monte-Carlo simulations, we identify the origin of this registry order in terms of the gauge interactions mediated by the fluxes ("visons") associated with the $Z_2$ gauge fields, indirectly imposing the discrete symmetry in the gauge invariant global symmetry controlled effective order parameter theory. In addition, it appears that our simulations reveal a hitherto unidentified "pseudo-universality" associated with the very similar form of the overall phase diagrams of the various theories suggesting a remarkable "governance" by the gauge field part of the dynamics.