Uncovering gauge-dependent critical order-parameter correlations by a stochastic gauge fixing at O(N)* and Ising* continuous transitions

We study the $\mathrm{O}{\left(N\right)}^{*}$ transitions that occur in the 3D ${\mathbb{Z}}_2$-gauge $N$-vector model and the analogous ${\mathrm{Ising}}^{*}$ transitions occurring in the 3D ${\mathbb{Z}}_2$-gauge Higgs model, corresponding to the ${\mathbb{Z}}_2$-gauge $N$-vector model with $N=1$. At these transitions, gauge-invariant correlations behave as in the usual $N$-vector (Ising for $N=1$) model. Instead, the non-gauge-invariant spin correlations are trivial and therefore the spin order parameter that characterizes the spontaneous breaking of the $\mathrm{O}\left(N\right)$ symmetry in standard $N$-vector (Ising) systems is apparently absent. We define a gauge fixing procedure—we name it stochastic gauge fixing—that allows us to define a gauge-dependent vector field that orders at the transition and is therefore the appropriate order parameter for the $\mathrm{O}\left(N\right)$ symmetry breaking. To substantiate this approach, we perform numerical simulations for $N=3$ and $N=1$. A finite-size scaling analysis of the numerical data allows us to confirm the general scenario: the gauge-fixed spin correlation functions behave as the corresponding functions computed in the usual $N$-vector (Ising) model. The emergence of a critical vector order parameter in the gauge model shows the complete equivalence of the $\mathrm{O}{\left(N\right)}^{*}$ (${\mathrm{Ising}}^{*}$) and $\mathrm{O}\left(N\right)$ (Ising) universality classes.