Nonequilibrium corrections to conserved Ising criticality in scalar active matter: Ward identities, spectrum, and long crossovers

Abstract

We identify the slowest-decaying nonequilibrium perturbations near the three-dimensional conserved Ising critical point and determine their impact on finite-size observables. We study two classes of perturbations: a field-dependent noise-to-mobility ratio $\Theta (\varphi )=D(\varphi )/M(\varphi )$ and the gradient activity of Active Model B+. Starting from the Martin-Siggia-Rose-Janssen-De Dominicis action, we compute the linearized flow using the functional renormalization group. The transport sector is block triangular, with leading odd eigenvalue $y_{{\Theta }_1}=-{\Delta }_{\varphi }$, where ${\Delta }_{\varphi }=(d-2+\eta )/2$. In the gradient sector, removing the detailed-balance direction leaves two genuinely nonequilibrium modes, chemical and current-like. Two smooth regulators give $y_{{\Theta }_1}\simeq -0.52$, $y_J\simeq -0.56$, and $y_{\mathrm{ch}}\simeq -0.89$. A translation Ward identity expresses the current operator as the divergence of the stress tensor. Together with conservation and Itô causality, this forbids chemical operators from generating the current mode, making the nonequilibrium stability matrix triangular; an independent two-loop calculation in $d=4-\epsilon$ finds no additional current contact counterterm. Within the FRG truncation, $y_J-y_{{\Theta }_1}=-\eta$; beyond it, this relation requires the absence of an additional contact anomaly. Using the 3D Ising value $\eta =0.0362978(20)$ [Kos et al., 2016] gives $y_{{\Theta }_1}\simeq -0.5181$ and $y_J\simeq -0.5544$. Because these exponents nearly coincide, single-power fits yield amplitude-dependent apparent exponents and crossover lengths may exceed accessible system sizes. We derive the resulting finite-size scaling rules: odd block observables respond linearly to activity, while even observables receive only quadratic corrections.

29 pages, 7 figures