Dynamics due to competitive flip cycles in active Potts models

TL;DR

The paper tackles how competition among multiple identical cyclic loops at each site shapes nonequilibrium spatiotemporal patterns in active Potts models on 2D lattices. Using extensive Monte Carlo simulations across a variety of flip networks (tetrahedral/octahedral/square-antiprism/cubic), it maps homogeneous cycling, spiral-wave, and mixed modes as functions of flip energy $h$ and the presence of diagonal flips, revealing size-dependent coexistence and hysteresis. Key findings show three-state cycles robustly generate spiral waves (W3/W6/W8 depending on topology), whereas four-state cycles tend to single-state dominance; network topology and flip energies can be tuned to control the number of coexisting spatial states, offering design principles for steering nonequilibrium pattern formation. These insights have potential implications for engineered pattern formation in driven many-body systems and biological-like active matter where competitive cyclic loops are present.

Abstract

Nonequilibrium spatiotemporal patterns have been extensively studied. However, a single oscillator or cyclic loop of states is typically employed at each site in theories and simulations. Here, we investigated how competition among multiple identical cyclic loops at each site alters patterns. We simulated active Potts models with standard Potts interactions between neighboring sites in two-dimensional square lattices. When multiple three-state cycles exist in state flips, such as in octahedral and square-antiprism networks, all types of spiral waves comprising the three states are formed simultaneously at high flip energies. However, at lower energies, only one or a few types emerge and switch stochastically into different types. At even lower energies, cyclic changes in single-state dominant homogeneous phases emerge [homogeneous cycling (HC) mode]. At intermediate flip energies, the spiral wave and HC modes temporally coexist in small systems but do not switch between each other in large systems. Conversely, when multiple four-state cycles exist in six-state and cubic networks, one state remains dominant for the entire range of flip energies, whereas the other states occasionally form domains at intermediate flip energies. Therefore, the number of spatially coexisting states can be controlled using flip networks and energies.

Figures (14)

• Figure 1: Flip networks comprising multiple three-state cycles in active Potts models. (a) Four-state model with two three-state cycles. The broad and narrow arrows represent the forward and backward flips with flip energies $h$ and $-h$, respectively. (b) Four-state model with the tetrahedral network. The flips between $s=1$ and $s=3$ are added in the model shown in (a) with flip energies $h_{\mathrm {a}}$ and $-h_{\mathrm {a}}$, respectively. (c) Six-state model with the octahedral network. (d) Eight-state model with the square-antiprism network. The arrows for backward flips are omitted in (b)--(d). The unfolded graphs are also shown in the bottom panels of (c) and (d).
• Figure 2: Flip energy $h$ dependence of the state densities $N_s/N$ for the tetrahedron network [Fig. \ref{['fig:cart0']}(b)]. The gray, green, blue, and red lines represent the data at $s=0$, $1$, $2$, and $3$, respectively. The solid and dashed lines for $s=1$ and $3$ represent the data at $h_{\mathrm{a}}=1$ and $0.2$, respectively.
• Figure 3: Active Potts model with two three-state cycles [Fig. \ref{['fig:cart0']}(a)] at $L=256$. (a) Snapshots at $h=0.7$, $1.5$, and $2$ (from left to right). The light yellow, green, blue, and red sites (light to dark in grayscale) represent $s=0$, $1$, $2$, and $3$, respectively. (b)--(e) Time development of the number of each state at (b) $h=2$, (c) $h=1.5$, and (d),(e) $h=0.7$. Two three-state spiral waves ($s=0\to 1 \to 2 \to 0$, and $s=3$ instead of $s=1$) spatially coexist (W4) at $h\gtrsim 0.9$, and the $s=0$ state becomes dominant at high $h$. The three-state spiral waves (W3) and homogeneous cycling (HC) are obtained at $h=0.7$, depending on the initial state. In W3, the two types of waves temporally coexist.
• Figure 4: Dependence on the flip energy $h$ in the active Potts model with two three-state cycles [Fig. \ref{['fig:cart0']}(a)]. (a)--(b) Mean number density $\langle N_s\rangle/N$ at (a) $L=256$ and (b) $L=128$. The blue down-pointing triangles in (a) represent $s=2$ in the HC mode. (c) Time fractions $p_{\mathrm{phase}}$ of phases at $L=256$ (solid lines) and $L=128$ (dashed lines). The red down-pointing triangles represent the ratio of the single-state dominant phase in the HC mode at $L=256$, whereas the ratios of multiple-state coexisting phases in the HC mode are omitted for clarity. The bidirectional arrows at the top of (a) represent the ranges of three modes at $L=256$.
• Figure 5: Active Potts model with two three-state cycles and diagonal flips [Fig. \ref{['fig:cart0']}(b)] at $L=128$. (a) Snapshots for $h_{\mathrm {a}}/h=0.01$, $0.2$, and $2$ at $h=2$ (from left to right). (b) Mean number density $\langle N_s\rangle/N$ as a function of $h_{\mathrm{a}}/h$. The solid and dashed lines represent the data at $h=2$ and $1$, respectively.