Flip Networks & Energies in Active Potts Models

• Flip networks and flip energies are a framework to induce and control nonequilibrium spatiotemporal patterns in active Potts models through cyclic state transitions.
• Modulating the flip energy h drives transitions between homogeneous cycling and multistate spiral-wave regimes, demonstrating tunable dynamical order.
• The topology of the flip network, defined by cycle structures like three-state and four-state cycles, directly dictates the number of coexisting states and emergent patterns.

Flip networks and associated flip energies define a framework for inducing and controlling nonequilibrium spatiotemporal patterns in active Potts models through competing cyclic transitions among discrete states. By tailoring the topology of locally allowed state-flip cycles (the “flip network”) and modulating the energy scale associated with directed flips (the flip energy h), a spectrum of collective behaviors emerges, including multistate spiral waves, stochastic phase switching, and spatially homogeneous cyclic dynamics. This approach enables direct and programmable control over the number of coexisting states and the qualitative nature of dynamical order in lattice systems, as demonstrated through comprehensive numerical studies of active–Potts models with varied underlying cycle graphs (Noguchi, 1 Dec 2025).

1. Architecture of Flip Networks

In active Potts models equipped with flip networks, each site carries a Potts “spin” $\sigma\in\{0,1,\ldots, q-1\}$, which both (i) interacts ferromagnetically with its lattice neighbors and (ii) undergoes locally-driven transitions along closed cyclic loops defined by a prescribed subgraph of the complete state space. The specific flip network determines the type and multiplicity of allowed cycles, which crucially impact collective pattern formation.

Three-State Cycles

• Octahedral Network ($q=6$): States occupy vertices of a regular octahedron, with each of the eight triangular faces corresponding to a directed three-cycle (e.g., $0\rightarrow1\rightarrow2\rightarrow0$). In the cycle-graph representation, this yields eight shared-edge triangles at each site.
• Square-Antiprism Network ($q=8$): States sit on a square antiprism. Eight triangles (3-cycles) and two squares (4-cycles) are present, but pattern dynamics are primarily governed by the three-state cycles in the relevant regime.

At each site, any of these local cycles may be activated. Under high flip energy, all cycles compete, leading to spatial coexistence of all possible three-state spiral waves (e.g., $W_8$ mode in both the octahedral and antiprism networks). Lower energy favors dominance of just one or a subset of cycles, and at very low energies, locally homogeneous cycling (HC) emerges.

Four-State Cycles

• Two-Cycle Six-State Network ($q=6$): States ${0,1,2,3}$ form one 4-cycle, and ${0,4,5,3}$ another, sharing nodes $0$ and $3$.
• Cubic Network ($q=8$): The eight state vertices of a cube provide six square faces, each defining a directed 4-cycle.

With four-state cycles, competition is so strong that a single state dominates globally for all examined flip energies; the formation of persistent multistate spiral waves is inhibited.

2. Model Hamiltonian and Flip Energies

The system is governed by an energy functional:

$$H[\{\sigma\}] = H_{\text{int}}[\{\sigma\}] + \epsilon E_{\text{flip}}[\{\sigma\}]$$

Where:

• $$H_{\text{int}} = -J \sum_{\langle i, j \rangle} \delta_{\sigma_i, \sigma_j}$$ introduces nearest-neighbor ferromagnetic coupling (with $J>0$; $J=2$ in typical simulations).
• $E_{\text{flip}}$ is a non-reciprocal on-site potential driving cyclic transitions, implemented via edge-specific flip energies $h_{s,s'}$ for transitions $s\rightarrow s'$. For each directed edge, $h_{a,b} = +h$, and the reverse $h_{b,a} = -h$, so that each closed $n$-cycle accumulates nonzero $\sum h_{s,s'}$ and globally breaks detailed balance, though edge reversals remain locally balanced.

In simulation protocols, $\epsilon$ is often absorbed into $h$, rendering $h$ the practical “drive amplitude.” The Metropolis update assigns transition probabilities proportional to $\exp(h_{s,s'})$.

3. Dynamical Regimes and Critical Energy Thresholds

Numerical studies on lattices ($L=128,256,512$) reveal three principal regimes as $h$ is varied:

• Homogeneous Cycling (HC) ($h < h_c^1$): The system settles into global cycling between spatially homogeneous single-state phases.
• Spiral-Wave Regime ($h > h_c^2$): For sufficiently large $h$, persistent spiral waves of multiple (typically three) sequential states emerge, with each spiral type corresponding to a distinct triangle in the flip network. Domains of these types meet at spiral cores, producing vortex–antivortex patterns ($W_n$).
• Intermediate Coexistence ($h_c^1 < h < h_c^2$): Both HC and $W_n$ regimes can persist for long times. In small systems, stochastic switching occurs between regimes; in large systems, strong hysteresis prevents switching.

Typical empirical thresholds:

• Octahedral ($L=512$): $h_c^1 \approx 0.6$, $h_c^2 \approx 0.85$
• Square-Antiprism: $h_c^1 \approx 0.6$, $h_c^2 \approx 0.9$
• Four-State Networks: $h_c^2 \to \infty$ (no stable waves observed)

Time-scale arguments: crossover $h_c^1$ occurs when the nucleation time for HC states matches the spiral-wave traversal time, $\tau_{\text{HC}}(h)\sim\tau_{\text{wave}}(h)$, while $h_c^2$ is set by the drive’s ability to overcome domain coarsening.

4. Dynamical Observables and Scaling Laws

The exploration of pattern dynamics employs several quantitative observables:

• Spatial Correlation Length ($\xi(h)$): Extracted from $C(r) = \langle \delta_{\sigma_0, \sigma_r} \rangle$, decays exponentially for $h < h_c^1$ with $\xi \sim |h-h_c^1|^{-1/2}$, peaking near regime coexistence and saturating at spiral-arm spacing at high $h$.
• Spiral-Wave Speed ($v(h)$): Grows from zero at $h_c^2$ following $v(h) \approx A(h-h_c^2)^{1/2}$, saturating to $v_0 \approx 0.3$ lattice-sites/MC-step at large $h$.
• Domain-Area Distribution ($P(A)$): In the HC phase, $P(A)\sim A^{-\tau}$ with $\tau \approx 1.8$ up to $A_{\text{max}} \sim \xi^2$; in the $W_n$ regime, $P(A)$ becomes bimodal with distinct spiral sector and transient bubble populations.
• Residence-Time Statistics: In the intermediate regime, state-residence times follow an exponential distribution, $$\langle \tau_{\text{switch}}\rangle \sim \exp(\alpha L^2 (h-h_c^1))$$.

Simulation methodology employs $10^7$–$10^8$ MC steps per $h$ value, with time- and ensemble-averages taken over several independent runs.

5. Control of Multistate Coexistence

The flip-network topology dictates the maximal number of coexisting spiral-wave types ($n_{\text{max}}$) achievable at high drive:

Network Type	$q$	$n_{\text{max}}$	Regime at Large $h$
Two-triangle (triangular)	4	2	$W_4$: two competing 3-cycles
Octahedral	6	8	$W_6$ (practically $W_8$)
Square-Antiprism	8	8	$W_8$ or incomplete $W_I$
Cubic (4-state cycles)	8	0	Single-state dominance
Two 4-cycles (six-state)	6	0	Single-state dominance

As $h$ is decreased, the number of coexisting spiral types reduces stepwise: from full $n_{\text{max}}$ at high $h$, through subsets corresponding to cycles sharing high-connectivity states, down to a single dominant cycle and eventually to HC. In four-state cycle networks, no persistent $W_4$ regime is found; only dominance by a single state and transient, shrinking domains of others are observed.

6. Significance and Implications

The architecture of the flip network and the scale of the flip energy $h$ provide a means for programmable control over the global dynamical order in active Potts lattices. By tuning these parameters, it is possible to select among globally homogeneous oscillations (HC), robust multistate spiral-wave patterns with prescribed cycle-length and multiplicity, or complex coexistence/hysteretic regimes. This framework therefore enables the design of spatiotemporal materials in which the local “chemistry” of allowed state-flip cycles dictates emergent patterning at the macroscale (Noguchi, 1 Dec 2025).

A plausible implication is that the principles outlined for competitive flip-cycle-driven active Potts models may be readily generalized to other driven, discrete-state lattice systems. The direct link between cycle-graph topology, flip-energy scale, and emergent dynamical order highlights the potential for flip networks as fundamental pattern-control elements in nonequilibrium statistical mechanics.