Emergence of Chimeras States in One-dimensional Ising model with Long-Range Diffusion

TL;DR

This work addresses the emergence of chimera-like states in a symmetric, one-dimensional Ising chain with non-local Kawasaki diffusion up to range $R$ at zero or low temperature. By combining analytical constructions (defining $\Sigma^{n^+}$ and $B_{n^+}$) with extensive Monte Carlo simulations, the authors identify three regimes—pure chimera states, pure attractors, and chimera–attractor coexistence—organized in a phase diagram on the $(r=R/N,|m_0|)$ plane, with a critical boundary near $|m_0|=2r-1$. They show that chimera states persist for $n^+\le R+1$, while $n^+>R+1$ leads to boundary-spin shifts and attractor formation with $L_{\text{attractor}}\sim e^{cR}$, and they provide a random-walk interpretation for metastable chimeras and their collisions. The results demonstrate that partial synchronization patterns can arise in fully symmetric binary-state systems and offer a framework for understanding complex spatiotemporal organization in networks, with potential implications for neural and other diffusive systems.

Abstract

In this work, we examine the conditions for the emergence of chimera-like states in Ising systems. We study an Ising chain with periodic boundaries in contact with a thermal bath at temperature T, that induces stochastic changes in spin variables. To capture the non-locality needed for chimera formation, we introduce a model setup with non-local diffusion of spin values through the whole system. More precisely, diffusion is modeled through spin-exchange interactions between units up to a distance R, using Kawasaki dynamics. This setup mimics, e.g., neural media, as the brain, in the presence of electrical (diffusive) interactions. We explored the influence of such non-local dynamics on the emergence of complex spatiotemporal synchronization patterns of activity. Depending on system parameters we report here for the first time chimera-like states in the Ising model, characterized by relatively stable moving domains of spins with different local magnetization. We analyzed the system at T=0, both analytically and via simulations and computed the system's phase diagram, revealing rich behavior: regions with only chimeras, coexistence of chimeras and stable domains, and metastable chimeras that decay into uniform stable domains. This study offers fundamental insights into how coherent and incoherent synchronization patterns can arise in complex networked systems as it is, e.g., the brain.

Figures (8)

• Figure 1: System connectivity scheme used in the present work: Schematic of a one-dimensional ring with $N=30$ spins showing long-range Kawasaki interactions. Each spin (blue circle) connects to $R=10$ neighbors on each side (green lines), enabling diffusion through spin-exchange dynamics. The periodic boundary conditions create a closed ring topology.
• Figure 2: Four observed emergent behaviors in the system for a $N=512$ spins ring: (A) chimera states, (B) attractor states, (C) coexistence of chimeras and attractors, and (D) merging of chimeras into attractors. From top to bottom and from left to right model parameters were $r=0.264, 0.205, 0.088, 0.059$ and $|m_0|=0.65,0.55,0.65,0.65,$ respectively.
• Figure 3: Normalized Mean domain Length $\ell_{dom}$ versus interaction range $r=R/N$, demonstrating exponential growth of attractor size with $r.$ Blue circles correspond to all measures of $l_{dom}$ performed in the analysis, while green diamonds correspond to values of $d_{dom}$ used for the fitting. Solid lines correspond to the fit for largest attractor size for a given $r$. In particular the red curve is fitted to $l^c_{dom}=a\cdot a e^{br}$ and the green curve corresponds to $l^c_{dom}=x^a\cdot e^{br}.$ The inset shows the corresponding values of $\tau_{ss}$ for every point in the main plot. Simulations have been performed for $N=512$ and $5\times10^4$ MCS.
• Figure 4: Dependence of chimera features on system size $N$ for $m_0\approx-0.8.$ Top panels (A): For a given $r=R/N=0.2$ the figure shows that chimera state remains as a stable attractor when $N$ increases so under this condition ($r=0.2$ fixed) one expects the chimera to remain in the thermodynamic limit. From left to right $N=500, \,1000,\, 2000$ and $4000$ respectively. It is also remarkable that the fluctuations of the borders decrease as $N$ increases. Bottom panels (B): These figures show the appearance of multichimeras for fixed $R=200$ and increasing values of $N.$ Note that the number of chimeras increases with $N$ but their width decreases with $N.$ Then, one would expect to have infinite number of chimeras with negligible width in the thermodynamic limit. From left to right $N=1000, 2000, 3000$ and $4000.$
• Figure 5: Main plot: Dependence of the mean number of chimeras $N_c$ surviving at observation time $t_{obs}$ as a function of the system size $N$. Each data point has been obtained after averaging the measured $N_c$ over $10$ different simulations. Error bars have been computed using the standard deviation of these data. The figure shows a clear scaling as $N_c\approx \frac{q_0}{\gamma(t_{obs})(R+1)} N$ confirming the theoretical prediction in Eq. \ref{['teornc']} assuming $\gamma(t_{obs})=0.5$. Parameters $t_{obs}=1000$ MCS, $q_0=0.1,$$R=100$ and system sizes $N=500, 1000, 2000, 5000, 10000, 20000$ and $40000$ spins. Inset: Slow decrease of $N_c$ when the observation time $t_{obs}$ increases from $1$ to $2\times 10^4$ MCS for a system size of $N=10^4$. We observe a good agreement between simulation data and theoretical prediction in Eqs. \ref{['solfinal']} for $\tau_c=100$ and $\tau_a=200$ MCS. Each data point has been obtained after averaging over $10$ simulations.

Theorems & Definitions (2)

• Proposition 3.1.1: Energetic asymmetry of boundary spins
• proof