State Diagram of the Non-Reciprocal Cahn-Hilliard Model and the Effects of Symmetry

TL;DR

This work interrogates how non-reciprocal interactions in conserved, phase-separating systems alter standard Cahn–Hilliard behavior by comparing two symmetry-reduced NRCH models: a continuous SO(2) variant and a discrete C4 variant. By constructing three-dimensional linear stability diagrams and analyzing eigenvalues, exceptional planes, and Takens–Bogdanov–like CEPs, the authors classify instabilities and connect NRCH to equilibrium limits, revealing non-equilibrium steady states with nonzero currents. They derive exact traveling-wave and domain-wall solutions, quantify entropy production, and show how symmetry governs phase separation—continuous symmetry yields traveling waves without sharp walls, while discrete symmetry preserves finite walls and moving domain structures under nonreciprocity. The paper further relates the NRCH dynamics to the Complex Ginzburg–Landau framework, clarifying when conserved NRCH behaves similarly to or differently from CGLE, and outlines future directions to explore transitions to complex, possibly chaotic patterns near the static-traveling boundary in active matter. Overall, the results establish a symmetry-based categorization of NRCH phenomenology and provide analytical benchmarks for studying current-carrying steady states in non-equilibrium, conserved systems.

Abstract

Interactions between active particles may be non-reciprocal, breaking action-reaction symmetry and leading to novel physics not observed in equilibrium systems. The non-reciprocalCahn-Hilliard (NRCH) model is a phenomenological model that captures the large-scale effects of non-reciprocity in conserved, phase-separating systems. In this work, we explore the consequences of different variations of this model corresponding to different symmetries, inspired by the importance of symmetry in equilibrium universality classes. In particular, we contrast two models, one with a continuous SO(2) symmetry and one with a discrete C_4 symmetry. We analyze the corresponding models by constructing three-dimensional linear stability diagrams. With this, we connect the models with their equilibrium limits, highlight the role of mean composition, and classify qualitatively different instabilities. We further demonstrate how non-reciprocity gives rise to out-of-equilibrium steady states with non-zero currents and present representative closed-form solutions that help us understand characteristic features of the models in different parts of the parameter space.

Figures (8)

• Figure 1: (a) The Mexican hat potential of the model with continuous $\mathrm{SO}(2)$ symmetry. (b) The potential of the model with discrete $\mathrm{C}_4$ symmetry.
• Figure 2: The state diagram of the model with continuous symmetry. The exceptional plane is highlighted in purple, and the planes between the stable, type-A unstable, and type-B unstable are highlighted in black, blue, and green, respectively. The red line is the intersection of all these surfaces.
• Figure 3: The four non-trivial slices of the full three-dimensional state diagram shown in \ref{['fig: phase surface']}. Panel (a) corresponds to the equilibrium case $\alpha = 0$, while panels (b)-(d) illustrate three orthogonal slices for finite $\alpha$, at constant $\alpha$, $\bar{\varphi}$, and $r$, respectively. Light blue and hashed areas mark the type-A and type-B unstable regions. Black, blue dash-dotted, and green lines separate stable and type-A, type-A and type-B, and stable and type-B, respectively. The purple dashed line marks the exceptional points, the yellow dot the critical point, and the red dots the critical exceptional points where all the lines intersect.
• Figure 4: The state diagram of the model with discrete symmetry, scaled by $|r|$. Light blue and hashed areas mark the type-A and type-B unstable regions. Black, blue, and green planes separate stable and type-A, type-A and type-B, and stable and type-B regions, respectively. The exceptional plane is highlighted in purple, and the red line marks the critical exceptional points where all the planes intersect.
• Figure 5: Constant $\alpha$ slices of the linear stability diagram illustrated in \ref{['fig: phase 3d assym']}. The gray dashed lines represent the line connecting the minima of the potential illustrated in \ref{['fig: potentials']}(a), and form a square centered at the origin and with side lengths $\sqrt{u/|r|} \bar{\varphi} = \sqrt{2}$. Light blue and hashed areas mark the type-A and type-B unstable regions. Black, blue dash-dotted, and green lines separate stable and type-A, type-A and type-B, and stable and type-B, respectively. The purple lines mark the exceptional points, the yellow dots the critical point, and the reds dot the critical exceptional points where all the lines intersect.