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

We study the O(N)* N ) * transitions that occur in the 3D Z2 2-gauge N-vector model and the analogous Ising* * transitions occurring in the 3D Z2 2-gauge Higgs model, corresponding to the Z2 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 O(N) N ) 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 O(N) 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)* N ) * (Ising*) * ) and O(N) N ) (Ising) universality classes.