Uncovering gauge-dependent critical order-parameter correlations by a stochastic gauge fixing at O($N$)${}^{\ast }$ and Ising${}^{\ast }$ continuous transitions

Abstract

We study the O($N$)${}^{\ast }$ transitions that occur in the 3D ${\mathbb{Z}}_2$-gauge $N$-vector model, and the analogous Ising${}^{\ast }$ transitions occurring in the 3D ${\mathbb{Z}}_2$-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($N$)${}^{\ast }$/Ising${}^{\ast }$ and O($N$)/Ising universality classes.

15 pages, 16 pdf figures, minor changes