Scaling Behaviors in Active Model B+ via the Functional Renormalization Group

TL;DR

This work applies a nonperturbative functional renormalization group (FRG) framework to the active model B+ (AMB+), formulated via the nonequilibrium Martin–Siggia–Rose action. By computing the scale-dependent β functions for all couplings in generic dimension $d$ and enforcing regulator-compatible projections, the authors identify RG-closed submodels and map the fixed-point structure, including a bicritical fixed point $F_4$ conjectured to control the bulk-to-microphase transition. The FRG analysis reveals substantial differences from perturbative RG: global flows do not connect $F_4$ to Wilson–Fisher fixed points as $d\to 2^+$, and large-activity flows tend to strong coupling, suggesting a first-order microphase transition; FRG also uncovers RG-invariant subcases not visible in perturbative treatments. These results establish a robust nonperturbative picture of universality and scaling in active matter and point to new directions for exploring nonequilibrium critical phenomena beyond perturbation theory.

Abstract

We study the scaling behaviors of the active model B+ using the functional renormalization group (FRG) approach, based on the nonequilibrium effective action formulated via the Martin-Siggia-Rose path-integral formalism. We derive the $β$ functions for all couplings of the system in generic $d$ dimensions, revealing regulator independence in various contributions to the renormalization group (RG) flow at specific values for $d$. After identifying specific regions of the parameter space that define submodels closed under RG transformations, we determine all fixed points of potential physical relevance. We confirm the existence of a bicritical fixed point, which was conjectured within the perturbative momentum-shell RG method for being responsible for the transition from bulk phase separation to microphase separation in active systems. We argue that, within the FRG approach, global flows significantly differ from those obtained in its perturbative counterpart.

Figures (4)

• Figure 1: Two-dimensional phase portrait of the equilibrium model obtained using the perturbative RG. Each fixed point has a relevant direction perpendicular to the plane, that is along the $\bar{a}$ axis. The dashed line in gray indicates a separatrix.
• Figure 2: Three-dimensional phase portrait of the equilibrium model within the FRG. Compared to the perturbative RG, a new fixed point appears, namely R$_1$, shown as a magenta blob. The indices $i$ of the fixed points R$_i$ refers to the number of relevant directions indicated in Table \ref{['tab:FP']}. The flows shown in red go to regions of the parameter space where $\bar{a}>0$.
• Figure 3: The diagrams in the first (second) row are obtained from the first (second) generic triangle diagram of Eq. \ref{['Eq:Z4u_flow']}. The fields $\phi$ and $\pi$ are depicted with solid and dashed lines, respectively. A solid line connecting two vertices corresponds to the propagator $d^{\phi\phi}_{k,R}$, a dashed-solid line to $d^{\pi\phi}_{k,R}$ and a solid-dashed line to $d^{\phi\pi}_{k,R}$.
• Figure 4: Box diagrams obtained from the corresponding generic diagram in Eq. \ref{['Eq:Z4u_flow']}. See Fig. \ref{['Fig:triangles']} for the convention on the lines.