🔬 condensed matter

Phase Diagram and Critical Behaviour of the Two-Dimensional Potts Model with Long-Range Quenched Disorder

This paper refines the phase diagram of the two-dimensional $q=3$ and $q=8$ Potts models with long-range quenched disorder by utilizing Fortuin-Kasteleyn cluster wrapping probabilities to map the transition between finite- and infinite-disorder fixed points, while simultaneously clarifying the origin of double-peak magnetic susceptibility structures and confirming the validity of hyper-scaling relations across all investigated regimes.

Original authors: Rodolfo Rocha, Leticia F. Cugliandolo, Marco Picco

Published 2026-09-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rodolfo Rocha, Leticia F. Cugliandolo, Marco Picco

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

Matter often exists in a state of delicate balance, where tiny changes in temperature or pressure can trigger a dramatic shift in its fundamental nature. This is the realm of phase transitions, the moment water turns to ice or a magnet loses its pull. For decades, physicists have studied how these transitions behave in perfectly ordered materials, but the real world is rarely perfect. It is filled with impurities, defects, and random variations that act as a form of "disorder." When this disorder is random and short-ranged, its effects are well understood. However, when the disorder stretches out over long distances, following a specific pattern where distant parts of the material still influence one another, the rules change. This is the territory of long-range correlated disorder, a complex landscape where the usual laws of physics struggle to predict how a material will behave as it approaches a critical point. Understanding these systems is not just an academic exercise; it helps scientists grasp how complexity and randomness shape the physical world, from the behavior of magnetic materials to the structure of the universe itself.

In a recent study, researchers set out to map this complex landscape using a mathematical model known as the Potts model, a versatile tool that generalizes the behavior of magnetic spins to include multiple possible states. They focused on two specific versions of this model, one with three possible states and another with eight, and introduced a special kind of disorder that decays slowly over distance. By running massive computer simulations on a grid that mimics a two-dimensional surface, they watched how the system reacted as they tweaked the strength of the disorder and the range of its correlations. Their goal was to determine whether the material would settle into a predictable, finite state of disorder or be driven toward a chaotic, infinite state where standard rules break down. The team used a clever method involving the probability of clusters of connected points wrapping around the grid to pinpoint exactly where these transitions occurred, refining a previous map that had left some regions unclear.

The results painted a clear picture of two distinct regimes. When the disorder was correlated over longer distances (corresponding to smaller decay exponents), the system was pushed toward an "infinite-disorder" state. In this extreme regime, the material effectively loses its thermal character and behaves more like a random network of connections, similar to how water seeps through a porous rock. However, when the disorder was correlated over shorter distances, the system settled into a stable, finite state, behaving like a typical material undergoing a smooth phase transition. This distinction is crucial because it tells us that the nature of the randomness itself dictates the fundamental behavior of the material, not just the amount of disorder present.

One of the most intriguing findings concerned the magnetic susceptibility, a measure of how easily the material can be magnetized. In some of their simulations, the researchers observed a strange double-peak structure in the data, where the material seemed to react strongly at two different temperatures. For a long time, such a pattern was thought to signal a "Griffiths phase," a mysterious extended region where the material hovers between order and chaos. However, the new study argues against this interpretation. By carefully analyzing how these peaks shifted as the size of the simulated system grew, the team showed that for certain types of disorder, the double peak is actually an illusion created by the finite size of the simulation. As the system becomes larger, these two peaks merge into a single, sharp transition, revealing a standard critical point. This suggests that the earlier interpretation of a Griffiths phase was a misreading of the data, and that the system is actually behaving in a more conventional way than previously suspected.

There is, however, one case where the double-peak structure is real and persists even in an infinitely large system. This occurs when the disorder is correlated over very long distances, specifically at the decay exponent a = 0.5, driving the system toward that infinite-disorder state. In this specific scenario, the two peaks represent two distinct physical processes happening at different scales. The first peak corresponds to the activation of strong bonds between atoms, which form large clusters quickly. The second peak, appearing at a much lower temperature, marks the activation of the remaining weak bonds. The system essentially operates on two separate clocks: one for the strong connections and one for the weak ones. This separation of scales means the material does not behave like a standard magnet but rather like a percolation network, where connectivity is the dominant feature. The researchers confirmed that even in this chaotic regime, the fundamental laws of scaling still hold true, provided one accounts for the strong corrections caused by the system's size.

By combining these observations, the study provides a comprehensive map of how long-range disorder reshapes the critical behavior of magnetic materials. It clarifies that the transition between finite and infinite disorder is not a vague boundary but a sharp divide determined by the range of the correlations. It also corrects a long-standing misunderstanding about the nature of double peaks in susceptibility, showing that they are often just artifacts of limited system size, except in the most extreme cases of disorder. The work demonstrates that even in systems driven by randomness and complexity, there is an underlying order and predictability, provided one looks closely enough at how the pieces fit together. This deeper understanding of how disorder influences phase transitions brings us closer to a unified theory of critical behavior in the messy, imperfect materials that make up our world.

Technical Summary: Phase Diagram and Critical Behaviour of the Two-Dimensional Potts Model with Long-Range Quenched Disorder

Problem Statement
This work investigates the critical properties and phase diagrams of the two-dimensional $q$ -state Potts model ( $q=3$ and $q=8$ ) subject to spatially correlated quenched disorder. The disorder correlations decay algebraically with an exponent $a$ . Previous theoretical work by Weinrib and Halperin established an extended Harris criterion for Gaussian disorder, predicting a new fixed point with correlation length exponent $\nu_{LR} = 2/a$ when disorder is relevant ( $\nu < 2/a$ ). However, recent studies utilizing non-Gaussian disorder constructions (coupling Potts bonds to products of auxiliary Ising spins) suggested a more complex phase diagram involving a transition between finite-disorder and infinite-disorder fixed points as a function of $a$ . Specifically, for $q=2$ and $q=3$ , a finite-disorder fixed point was identified for $a \geq 1.0$ , while $a \leq 0.5$ drives the system to an infinite-disorder fixed point. The regime $0.5 < a < 1.0$ remained ambiguous. Furthermore, the introduction of disorder in the $q=8$ case (which is first-order in the pure limit) softens the transition, but the nature of the critical points and the validity of hyperscaling relations in the presence of long-range correlations required further clarification.

Methodology
The authors employ a numerical approach based on Monte Carlo simulations of the Potts model on a two-dimensional square lattice with periodic boundary conditions (torus). The disorder is generated using a specific construction where $n$ independent copies of the critical Ising model are simulated on the same lattice. The product of spins from these copies determines the bond configuration of the Potts model, resulting in a bimodal disorder distribution with correlation exponent $a = n/4$ .

Key observables and techniques include:

Fortuin-Kasteleyn (FK) Wrapping Probabilities: The probability $W(r, L)$ that an FK cluster wraps around the torus is computed for various system sizes $L$ and disorder strengths $r = J_1/J_2$ . This quantity serves as a probe for the underlying fixed points, interpolating between the pure Potts value and the correlated percolation value.
Finite-Size Scaling (FSS): The authors analyze the scaling collapse of rescaled wrapping probabilities and magnetic observables to determine the nature of fixed points (attractive vs. repulsive) and extract critical exponents.
Magnetic Susceptibility and Magnetisation: The temperature dependence of the magnetic susceptibility $\chi$ and magnetisation density $m$ is studied to investigate the double-peak structure observed in previous works and to test hyperscaling relations.
Hyperscaling Analysis: The ratio of critical exponents $\gamma/\nu$ is compared against the hyperscaling prediction $2 - 2\beta_m/\nu$ to verify the validity of standard scaling relations in the presence of long-range disorder.

Key Contributions and Results

Refined Phase Diagram: By analyzing wrapping probabilities, the authors refine the phase diagram proposed by Chippari et al. [3].
- For $a = 1.0$ ( $q=3$ and $q=8$ ): The data confirms the existence of an attractive finite-disorder fixed point. The wrapping probabilities exhibit a clear crossing, and scaling collapses are achieved using the predicted disorder scaling dimension $y_d = 2 - a + 0.4$ .
- For $a = 0.5$ ( $q=3$ and $q=8$ ): The wrapping probabilities do not cross, and the data collapses onto a curve consistent with an infinite-disorder fixed point. The scaling variable involves a negative exponent, indicating the flow is towards infinite disorder. In this regime, the system behaves like correlated percolation.
- For $a = 0.75$ : A crossover regime is identified. For $q=3$ , a crossing occurs at large disorder ( $r \approx 20$ ), suggesting a finite-disorder fixed point, though with a broad range of effective critical disorder. For $q=8$ , a crossing is observed at $r \approx 25$ , also indicating a finite-disorder fixed point.
Origin of the Double-Peak Structure in Magnetic Susceptibility:
- The study addresses the double-peak structure in the magnetic susceptibility previously interpreted as a Griffiths phase.
- For $a \geq 0.75$ : The double-peak structure is identified as a finite-size effect. As system size $L$ increases, the two peaks merge into a single peak at the critical temperature $\beta = 1$ .
- For $a = 0.5$ : The double-peak structure persists in the thermodynamic limit. The authors attribute this to a two-scale activation process: at low temperatures (relative to the disorder scale), strong bonds activate first (forming a percolating cluster), and at much higher temperatures, weak bonds activate. This separation of energy scales leads to two distinct peaks in susceptibility, corresponding to the activation of strong and weak bonds, rather than an extended Griffiths phase.
Validation of Hyperscaling:
- The authors resolve previous claims of hyperscaling violation in Potts models with long-range disorder.
- They demonstrate that while the effective exponent $\gamma/\nu$ exhibits strong finite-size corrections (deviating significantly from the hyperscaling prediction for small $L$ ), it converges to the value $2 - 2\beta_m/\nu$ in the large-system-size limit.
- This confirms that the hyperscaling relation $\gamma/\nu = 2 - 2\beta_m/\nu$ holds across all investigated regimes ( $a=0.5, 0.75, 1.0$ ) and for both $q=3$ and $q=8$ .

Significance
The paper provides a comprehensive numerical characterization of the interplay between long-range correlated disorder and critical fluctuations in low-dimensional Potts models. By utilizing wrapping probabilities, the authors definitively distinguish between finite-disorder and infinite-disorder fixed points, clarifying the phase diagram for $q=3$ and extending the analysis to the first-order $q=8$ case. The work offers a physical interpretation of the double-peak susceptibility structure, distinguishing between finite-size artifacts and a genuine two-scale activation mechanism in the infinite-disorder limit. Finally, by accounting for strong finite-size corrections, the study reaffirms the validity of hyperscaling in these systems, correcting earlier interpretations of hyperscaling violation. These results support and extend the predictions of perturbative renormalization-group calculations based on conformal perturbation theory for non-Gaussian disorder.

Technical Summary: Phase Diagram and Critical Behaviour of the Two-Dimensional Potts Model with Long-Range Quenched Disorder

More like this