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

We study the O(N)^* transitions that occur in the 3D Z₂-gauge N-vector model, and the analogous Ising^* transitions occurring in the 3D Z₂-gauge Higgs model, corresponding to an N-vector model with $N=1$. At these transitions, gauge-invariant correlations behave as in the usual N-vector/Ising model. Instead, the nongauge invariant spin correlations are trivial and therefore the spin order parameter that characterizes the spontaneous breaking of the O(N) symmetry in standard N-vector/Ising systems is apparently absent. We define a novel 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 O(N) 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 O(N)^*/Ising^* and O(N)/Ising universality classes.