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 a 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 spins order, in the regime 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.