Hyperuniformity at the Absorbing State Transition: Perturbative RG for Random Organization

TL;DR

This paper uses perturbative renormalization group analysis to determine the hyperuniformity exponent at the absorbing state transition in the random organization universality class, revealing new insights into critical behavior and universality distinctions.

Contribution

It provides the first analytic calculation of the hyperuniformity exponent in the random organization class using Doi-Peliti field theory and perturbative RG, clarifying universality class distinctions.

Findings

Found ^+ and = 2/9 + O(^2) in dimensions above and near 4.

Demonstrated hyperuniformity arises from anticorrelation of active and passive densities.

Established differences in hyperuniformity exponents between RO and q-EW models.

Abstract

Hyperuniformity, where the static structure factor obeys $S(q)\sim q^{\varsigma}$ with $\varsigma> 0$, emerges at criticality in systems having multiple, symmetry-unrelated, absorbing states. Important examples arise in periodically sheared suspensions and amorphous solids; these lie in the random organisation (RO) universality class, for which analytic results for $\varsigma$ are lacking. Here, using Doi-Peliti field theory and perturbative RG about a Gaussian model, we find $\varsigma = 0^+$ and $\varsigma= 2\epsilon/9 + O(\epsilon^2)$ in dimension $d>d_c=4$ and $d=4-\epsilon$ respectively. Our calculations assume that renormalizability is sustained via a certain pattern of cancellation of strongly divergent terms. These cancellations allow the upper critical dimension to remain $d_c = 4$, as is known for RO, while generic perturbations (e.g., those violating particle conservation) would typically flow to a fixed point with $d_c=6$. The assumed cancellation pattern is closely reminiscent of a long-established one near the tricritical Ising fixed point. (This has $d_c=3$, although generic perturbations flow towards the Wilson-Fisher fixed point with $d_c = 4$.) We show how hyperuniformity in RO emerges from anticorrelation of strongly fluctuating active and passive densities. Our calculations also yield the remaining exponents to order $\epsilon$, surprisingly without recourse to functional RG. These exponents coincide as expected with the Conserved Directed Percolation (C-DP) class which also contains the Manna Model and the quenched Edwards-Wilkinson (q-EW) model. Importantly however, our $\varsigma$ differs from one found via a mapping to q-EW. That mapping neglects a conserved noise in the RO action, which we argue to be dangerously irrelevant. Thus, although other exponents are common to both, the RO and C-DP universality classes have different exponents for hyperuniformity.