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

TL;DR

This paper introduces a stochastic gauge fixing method to define gauge-dependent order parameters in gauge theories, revealing that gauge-fixed correlations exhibit the same critical behavior as standard models, unifying their universality classes.

Contribution

The paper proposes a novel stochastic gauge fixing procedure that enables the definition of gauge-dependent order parameters, clarifying the nature of symmetry breaking in gauge theories.

Findings

Gauge-fixed correlations match those of standard models.

Gauge-fixed order parameters confirm universality class equivalence.

Numerical simulations validate the proposed gauge fixing approach.

Abstract

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