Stability of discrete-symmetry flocks: sandwich state, traveling domains and motility-induced pinning

TL;DR

Problem: assess the stability of polar order in discrete flocking models. Approach: analyze the 4-state active Potts model (APM) using a coarse-grained hydrodynamic description and Monte Carlo simulations with artificial droplets and low-diffusion spontaneous nucleation. Findings: the globally ordered phase is metastable across a broad parameter range; droplets induce a sandwich state or complete reversal, and low diffusion with high propulsion promotes spontaneous transverse droplets leading to SRO and motility-induced pinning (MIP); comprehensive phase diagrams reveal distinct states including gas, long-range ordered liquid, stripe, lane, SRO, and MIP as functions of $D$, $β$, $ρ_0$, and $ε$. Significance: provides a unified picture of how thermal fluctuations, self-propulsion, diffusion, and finite-size effects shape flock stability in discrete active matter and offers experimentally testable predictions for dry active-matter platforms.

Abstract

Polar flocks in discrete active systems are often assumed to be robust, yet recent studies reveal their fragility under both imposed and spontaneous fluctuations. Here, we revisit the four-state active Potts model (APM) and show that its globally ordered phase is metastable across a broad swath of parameter space. Small counter-propagating droplets disrupt the flocking phase by inducing a persistent sandwich state, where the droplet-induced opposite-polarity lane remains embedded within the original flock, particularly pronounced at low noise, influenced by spatial anisotropy. In contrast, small transversely propagating droplets, when introduced into the flock, can trigger complete phase reversal due to their alignment orthogonal to the dominant flow and their enhanced persistence. At low diffusion and strong self-propulsion, such transverse droplets also emerge spontaneously, fragmenting the flock into multiple traveling domains and giving rise to a short-range order (SRO) regime. We further identify a motility-induced pinning (MIP) transition in small diffusion and low-temperature regimes when particles of opposite polarity interact, flip their state, hop, and pin an interface. Our comprehensive phase diagrams, encompassing full reversal, sandwich coexistence, stripe bands, SRO, and MIP, delineate how thermal fluctuations, self-propulsion strength, and diffusion govern flock stability in discrete active matter systems.

Figures (19)

• Figure 1: (color online) Hydrodynamic theory of the APM with droplet excitations. (a--b) Time-evolution snapshots showing (a) the formation of a sandwich state ($\beta=0.9$, $\epsilon=2.4$), and (b) the complete reversal ($\beta=0.75$, $\epsilon=1.5$) for a counter-propagating droplet ($\sigma=1$). (c--d) Time-evolution snapshots showing (c) the complete reversal ($\beta=0.9$, $\epsilon=2.4$), and (d) the formation of a persistent motion with two orthogonal states ($\beta=0.9$, $\epsilon=1.5$) caused by a transversely-propagating droplet ($\sigma=2$). The initial ordered state is of state $\sigma=3$. Parameters of the droplet: $r_d = 10$ and $\rho_0^d = 5\rho_0$. Arrows indicate the direction of motion, and the colorbar represents the corresponding droplet magnetization. A movie (movie01) of the same can be found at Ref. zenodo. (e) Magnetization profiles of the sandwich state formed by the counter-propagating droplet ($r_d = 10$ and $\rho_0^d = 5\rho_0$), $\beta=0.9$ and different values of $\epsilon$. (f) Corresponding sandwich width $w_{\rm sw}$ as a function of $\epsilon$. The red, green, and cyan stars represent the corresponding widths of the profiles shown in (c). (g--h) $(\rho_0^d,r_d)$ stability diagrams for (g) counter- and (h) transversely-propagating droplets for $\epsilon = 1.2$. Parameters: $D=1$, $\rho_0 = 3$, and $L = 200$.
• Figure 2: (color online) Time evolution and steady-states with droplet excitations. (i--ii) Schematic illustrating the movement of droplets: (i) a counter-propagating droplet and (ii) a transversely-propagating droplet, relative to the motion of the homogeneous liquid phase. (a, c, e, g, i) Early-time snapshots and (b, d, f, h, j) steady-state snapshots for (a--f) counter-propagating and (g--j) transversely propagating droplets in the 4-state APM. A counter-propagating droplet with a very small self-propulsion velocity of $\epsilon=0.3$ is unable to reverse the initial liquid phase (a--b) but reverses it for intermediate velocity $\epsilon=0.9$ (c--d). (e--f) Counter-propagating droplet at large self-propulsion velocity $\epsilon=2.7$ creates a sandwich state of $\sigma=3$ (blue) and $\sigma=1$ (red). (g--h) Transversely propagating droplet at small self-propulsion velocity $\epsilon=0.9$ could not reverse the initial liquid phase, although reverses it for large self-propulsion velocity $\epsilon=2.7$ (i--j). Parameters: $D=1$, $\beta = 1$, $\rho_0 = 10$, $r_d = 10$, $\rho_0^d = 1.2\rho_0$, $L_x = 500$, and $L_y = 50$. Colorbar legend: red $(\sigma=1)$: right; green $(\sigma=2)$: up; blue $(\sigma=3)$: left; black $(\sigma=4)$: down. Arrows indicate the direction of motion. A movie (movie02) of the same can be found at Ref. zenodo.
• Figure 3: (color online) Phase reversal due to counter-propagating droplet. (a) Average relative width ($w_{sw}/L_y$) of the sandwich state versus aspect ratio of the simulation box $(L_x/L_y)$ for fixed $r_d = 10$ and $\epsilon=0.9$. $w_{sw}/L_y=1$ corresponds to a full reversal of the initial ordered state. (b) $w_{sw}/L_y$ as a function of relative droplet size $(2r_d/L_y)$ obtained for fixed $r_d = 6$ and $\epsilon=0.9$. (c) $w_{sw}/L_y$ as a function of $\epsilon$ for counter-propagating droplet with fixed $L_y=50$ and $r_d = 10$. A movie (movie03) showing the sandwich formation for various $\epsilon$ can be found at Ref. zenodo. Parameters: $D=1$, $\beta=1$, $\rho_0=10$, $\rho_0^d = 5\rho_0$ and (a) $L_y=50$, (b--c) $L_x = 200$.
• Figure 4: (color online) State diagrams for counter-propagating droplet excitation. [above] (a) $\beta-\epsilon$ diagram for $\rho_0=10$. (b) $\beta-\rho_0$ diagram for $\epsilon=1.5$. and (c) $\epsilon-\rho_0$ diagram for $\beta=0.75$. Each diagram is partitioned into regions labeled (I)--(VII), corresponding to different steady-states: (I) sandwich state (circle), (II) full reversal of the initial liquid phase by the droplet (square), (III) coexistence band (up triangle), (IV) coexistence sandwich (down triangle), (V) liquid blob of the droplet on a gaseous background (diamond), (VI) gas phase (pentagon), and (VII) longitudinal lane (open square), primarily composed of the initial liquid phase, though it can also form from other states. Parameters: $D=1$, $r_d = 10$, $\rho_0^d = 5\rho_0$, and $L_x = L_y = 200$. [below] Representative snapshots of the seven steady states in the phase diagram for $L_x = L_y = 800$.
• Figure 5: (color online) Liquid stability with a counter-propagating droplet. (a) Sandwich state probability $P_r$ as a function of droplet radius $r_d$ for fixed $\epsilon=0.9$ and $\rho_0^d = 1.2\rho_0$. The dotted line is a hyperbolic tangent fit for extracting the critical radius $r_c$ when $P_r=0.5$. (b) $r_c$ versus $\epsilon$ at a fixed initial droplet density $\rho_0^d = 1.2\rho_0$. (c) The probability of sandwich state, $P_r$, as a function of $\rho_0^d$ for fixed $r_d=5$ and $\epsilon=1.2$. (d) $(\rho_0^d,r_d)$ stability diagram for $\epsilon=0.9$. The color bar denotes the sandwich state probability, $P_r$. Parameters: $D=1$, $\beta=1$, $\rho_0=10$, and $L_x = L_y = 200$.