Correlated Noise and Critical Dimensions

In equilibrium, the Mermin-Wagner theorem prohibits the continuous symmetry breaking for all dimensions $d\leq 2$. In this work, we discuss that this limitation can be circumvented in non-equilibrium systems driven by the spatio-temporally long-range anticorrelated noise. We first compute the lower and upper critical dimensions of the $O(n)$ model driven by the spatio-temporally correlated noise by means of the dimensional analysis. Next, we consider the spherical model, which corresponds to the large $n$ limit of the $O(n)$ model and allows us to compute the critical dimensions and critical exponents, analytically. Both results suggest that the critical dimensions increase when the noise is positively correlated in space and time, and decrease when anticorrelated. We also report that the spherical model with the correlated noise shows the hyperuniformity and giant number fluctuation even well above the critical point.