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 $\Theta(\phi)=D(\phi)/M(\phi)$ 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 $y_{\Theta_1}=-\Delta_\phi$, where $\Delta_\phi=(d-2+\eta)/2$. 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 $y_{\rm ch}\simeq-0.89$. A translation Ward identity expresses the current operator as the divergence of the stress tensor. Together with conservation and It\^o causality, this forbids chemical operators from generating the current mode, making the nonequilibrium stability matrix triangular; an independent two-loop calculation in $d=4-\varepsilon$ 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.