Three-dimensional $Z_2$-gauge $N$-vector models

We study the phase diagram and critical behaviors of three-dimensional lattice $Z_2$-gauge $N$-vector models, in which an $N$-component real field is minimally coupled with $Z_2$-gauge link variables. These models are invariant under global $O(N$) and local $Z_2$ transformations. They present three phases characterized by the spontaneous breaking of the global $O(N$) symmetry and by the different topological properties of the $Z_2$-gauge correlations. We address the nature of the three transition lines separating the three phases. The theoretical predictions are supported by numerical finite-size scaling analyses of Monte Carlo data for the $N=2$ model. In this case, continuous transitions can be observed along both transition lines where the $N$-component spins order, in the regimes of small and large inverse gauge coupling $K$. Even though these continuous transitions belong to the same $XY$ universality class, their critical modes turn out to be different. When the gauge variables are disordered (small $K$), the relevant order-parameter field is a gauge-invariant bilinear combination of the vector field. On the other hand, when the gauge variables are ordered (large $K$), the order-parameter field is the gauge-dependent $N$-vector field, whose critical behavior can only be probed by using a stochastic gauge fixing that reduces the gauge freedom.