🌀 nonlinear sciences

Criticality without Temperature in an Ising Spin System

This paper demonstrates that critical behavior in an Ising spin system can emerge without temperature or Hamiltonian-driven dynamics, arising instead from an adaptive, memory-dependent connectivity mechanism controlled by local quiet times.

Original authors: Christos Kakalis, Fotios K. Diakonos

Published 2026-09-29

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CC BY 4.0

Original authors: Christos Kakalis, Fotios K. Diakonos

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of physics, there is a long-standing idea that for a group of tiny particles to act in unison, they must be warmed up or cooled down to a specific temperature. Think of a crowd of people in a room; usually, they only start moving together in a coordinated way if the room gets hot enough to make them restless or cold enough to make them huddle. This principle, known as thermal equilibrium, has been the standard explanation for how order emerges from chaos in materials like magnets. For decades, scientists believed that this temperature-driven process was the only way to reach a "critical" state, a delicate point where a system is perfectly balanced between disorder and order, ready to shift dramatically with the slightest nudge.

However, a new study challenges this deep-seated assumption. Researchers have discovered that a system of magnetic spins can reach this critical, collective state without any heat, without any energy exchange, and without a single degree of temperature involved. Instead of relying on thermal energy to drive the system, they found that the simple act of remembering how long it has been since a particle last moved is enough to organize the entire group. This finding suggests that the history of a system's own movements can create complex, large-scale patterns, offering a new way to understand how order might arise in living things or other systems where energy is not the primary driver.

The researchers, working with a model known as the Ising system, set out to see if they could generate this critical behavior using a completely different set of rules. In a traditional setup, the behavior of these tiny magnetic units, or spins, is dictated by a mathematical formula called a Hamiltonian, which acts like an energy map. The spins flip and settle based on how much energy they have, which is controlled by temperature. In this new experiment, the team removed the temperature entirely. They disconnected the spins from any energy-based rules. Instead, they introduced a simple rule based on time and memory. They assigned each spin a "quiet time," which is simply a counter that ticks up every moment the spin sits still and does not change.

The system works by grouping these spins into clusters. The rule for joining a cluster is based on the quiet time of the neighbors. If a spin has been sitting quietly for a long time, it becomes more likely to be picked up and flipped along with its neighbors. The longer a spin waits, the more "eager" it becomes to participate in a group change. Once a group is formed, all the spins inside it flip their direction instantly, regardless of energy or cost. After the flip, their quiet time resets to zero, and they must wait again before they can be part of a new group. The only control the scientists had was a single number that determined how strongly the quiet time influenced the chance of joining a group.

As the researchers adjusted this control number, the system underwent a dramatic transformation. At low settings, the spins flipped randomly and independently, resulting in a chaotic mess with no overall pattern. But as the control number increased, the system began to organize itself. The spins started to form large, connected groups that moved together. When the control number reached a specific threshold, the system hit a critical point. Here, the behavior changed from random noise to a state of intense, collective fluctuation. The spins would suddenly align in one direction, then flip to the opposite direction, and back again, creating a rhythm that spanned the entire grid.

The most striking evidence of this new kind of criticality was found in the size of the groups formed. In the disordered state, the groups were small and their sizes dropped off quickly, like a pile of sand where most grains are tiny and very few are large. But as the system approached the critical point, the distribution of group sizes changed. Instead of dropping off, the sizes followed a broad pattern where groups of all sizes, from very small to massive, appeared with a predictable frequency. This is the signature of criticality, a state where the system has no single characteristic size and is sensitive to changes at every scale. Remarkably, this happened even though the researchers never measured the energy of the system or used it to guide the spins.

The team also looked at the total energy of the system, not to drive it, but to see how it reacted to the changes. Even though energy played no role in the rules of the game, the system's energy responded sharply and strangely as the control number crossed the critical threshold. It showed a sudden, singular change, behaving as if it were undergoing a phase transition, even though no heat was involved. This confirmed that the collective behavior was real and robust, not just a random fluctuation. The system had organized itself into a critical state purely through the feedback loop of its own memory.

This discovery reveals that the ability to remember past activity is a powerful force for creating order. The "quiet time" acts as a memory that tells the system when it is time to act. By waiting longer, the spins build up a kind of potential that eventually forces them to move together. This mechanism allows the system to self-organize without needing an external energy source or a temperature setting. It suggests that in complex systems, such as those found in biology or social networks, critical behavior might emerge simply because the components keep track of their own history. The study proves that temperature is not the only path to collective intelligence; a system can learn to coordinate itself just by knowing how long it has been since it last moved.

Technical Summary: Criticality without Temperature in an Ising Spin System

Problem Statement
Conventional criticality in the Ising model is generated by Hamiltonian dynamics and controlled by temperature, where thermal fluctuations drive spontaneous symmetry breaking. While nonequilibrium systems can exhibit ordering through dynamical rules (e.g., absorbing-state transitions), the specific mechanism by which an Ising system could display critical-like behavior entirely decoupled from its Hamiltonian and devoid of temperature-like parameters remains an open question. This paper addresses whether memory-dependent connectivity alone can generate collective critical behavior in an Ising spin system without underlying energetic dynamics.

Methodology: The Quiet-Time Cluster Model (QTCM)
The authors propose a novel adaptive cluster dynamics for Ising spins ( $s_i = \pm 1$ ) on a square lattice of size $N \times N$ . The core innovation is the decoupling of the dynamical rules from the Ising Hamiltonian ( $H = -\sum_{\langle ij \rangle} s_i s_j$ ). Instead of using a temperature-dependent bond probability, the system evolves based on a "quiet-time" variable, $q(i, t)$ , defined as the time elapsed since spin $i$ was last updated.

The dynamics proceed as follows:

Cluster Construction: Clusters are built similarly to standard cluster algorithms, but connectivity is governed by the quiet times of the spins. A neighbor $s_j$ of a cluster spin $s_i$ is added to the cluster only if $s_j = s_i$ . The probability of adding this spin at time $t$ is:
$P_{con}(j, t) = 1 - \exp\left(-\alpha \frac{q(j, t)}{\langle q(t) \rangle}\right)$
where $\alpha$ is a control parameter regulating connectivity strength, and $\langle q(t) \rangle$ is the global average quiet time.
Memory Update: After a cluster is formed, all spins within it are flipped without any energetic acceptance criterion. Their quiet times are reset to zero ( $q(i, t+1) = 0$ for $i \in C(t)$ ), while the quiet times of all other spins increment by one ( $q(i, t+1) = q(i, t) + 1$ ).
Observables: The Ising Hamiltonian is treated solely as an observable to characterize the resulting configurations. The authors monitor global observables including magnetization ( $m$ ), mean energy ( $E$ ), cluster-size distribution ( $n(s)$ ), and the average connectivity probability ( $P$ ).

Simulations were performed on lattices ranging from $32 \times 32$ to $256 \times 256$ over $10^6$ to $4 \times 10^6$ algorithmic steps, varying $\alpha$ in the range $[0.5, 1.2]$ .

Key Results
The variation of the connectivity parameter $\alpha$ drives the system through distinct dynamical regimes, culminating in a critical-like transition:

Spontaneous Symmetry Breaking and Magnetization Dynamics:
- For low $\alpha$ (e.g., 0.5), magnetization fluctuates with tiny amplitude around zero.
- As $\alpha$ increases, fluctuations grow. Around $\alpha \approx 1.0$ , the system exhibits dynamics similar to a ferromagnetic system at a thermal critical point, with a flattened magnetization distribution.
- At $\alpha \approx 1.12$ , the system undergoes a transition where the magnetization distribution develops two dominant symmetric peaks near $m = \pm 1$ , indicating spontaneous symmetry breaking. However, unlike equilibrium systems, the system does not settle into a single state but continuously jumps between positive and negative values.
- For $\alpha > 1.12$ (e.g., 1.2), the magnetization time series becomes almost periodic, jumping rapidly between $m \approx -1$ and $m \approx 1$ .
Singular Energy Response:
Despite the Hamiltonian not driving the dynamics, the mean energy per spin, $\langle E \rangle$ , exhibits a singular response to $\alpha$ . The derivative $-\partial \langle E \rangle / \partial \alpha$ shows a one-sided power-law divergence near $\alpha_s \approx 1.12$ of the form $(\alpha - \alpha_s)^{-0.93}$ .
Cluster-Size Distribution:
- Small $\alpha$ yields an exponential decay in cluster sizes.
- As $\alpha$ approaches the critical region $[1, 1.12]$ , the distribution develops an extended power-law regime ( $n(s) \sim s^{-\tau}$ ) with an exponent $\tau \approx -1.08$ to $-1.11$.
- This power-law behavior extends over several decades in cluster size and scales with system size ( $N^2$ ), terminating in a small finite-size hump at the largest scales.
- Above $\alpha_c \approx 1.12$ , the power-law regime disappears, and the distribution becomes disconnected with an exponential decay followed by a sharp hump at very large sizes.
Connectivity and Percolation:
The mean connectivity probability $P$ evolves with $\alpha$ . At low $\alpha$ , $P$ is constant. As $\alpha$ increases, the distribution of $P$ broadens. Near the critical point ( $\alpha \approx 1.12$ ), the distribution peaks around the site percolation threshold for a square lattice ( $P \approx 0.59$ ). For $\alpha > 1.12$ , the statistical weight of $P$ values within the critical percolation domain vanishes, correlating with the disappearance of the power-law cluster distribution.

Significance and Claims
The paper claims to demonstrate that critical-like collective behavior can emerge in an Ising spin system without Hamiltonian dynamics or a thermodynamic control parameter. The distinctive mechanism is a memory-dependent connectivity rule, where the temporal persistence of individual spins (quiet time) provides a feedback loop capable of organizing the system on macroscopic scales.

The authors assert that this construction reverses the conventional role of the Ising Hamiltonian: rather than generating dynamics, it serves as an observable characterizing configurations produced by an adaptive process. The results suggest that adaptive temporal memory alone can generate collective criticality, challenging the presumption that such behavior in complex systems must be rooted in an effective energy landscape. The paper notes that while the transition shares features with percolation (specifically the proximity of connectivity probabilities to percolation thresholds), establishing universality with specific percolation classes requires further finite-size scaling analysis, which is left for future work.

Technical Summary: Criticality without Temperature in an Ising Spin System

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