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arXiv:2608.29710

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

Rodolfo Rocha, Leticia F. Cugliandolo, Marco Picco

Published Aug 30, 2026• 0 citations
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Abstract

We study the phase diagram and critical properties of the $q = 3$ and $q = 8$ Potts models with spatially correlated disorder governed by a power-law decay with exponent $a$. Building on the phase diagram proposed by Chippari et al. [3], where a transition from finite-disorder to infinite-disorder fixed points was identified as a function of a, we refine this picture by using wrapping probabilities of Fortuin-Kasteleyn clusters. Furthermore, using magnetic observables, we clarify the physical origin of the double-peak structure in the magnetic susceptibility and establish the validity of the hyper-scaling relation across all investigated regimes.

DOI
10.48550/arxiv.2608.29710

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We study the phase diagram and critical properties of the q=3q = 3q=3 and q=8q = 8q=8 Potts models with spatially correlated disorder governed by a power-law decay with exponent aaa. Building on the phase diagram proposed by Chippari et al. [3], where a transition from finite-disorder to infinite-disorder fixed points was identified as a function of a, we refine this picture by using wrapping probabilities of Fortuin-Kasteleyn clusters. Furthermore, using magnetic observables, we clarify the physical origin of the double-peak structure in the magnetic susceptibility and establish the validity of the hyper-scaling relation across all investigated regimes.
Abstract We study the bidimensional q-Potts model with long-range bond correlated disorder. Similarly to Chatelain (2014 Phys. Rev. E 89 032105), we implement a disorder bimodal distribution by coupling the Potts model to auxiliary spin-variables, which are correlated with a power-law decaying function. The universal behaviour of different observables, especially the thermal and the order-parameter critical exponents, are computed by Monte-Carlo techniques for q = 1 , 2 , 3 -Potts models for different values of the power-law decaying exponent a. On the basis of our conclusions, which are in agreement with previous theoretical and numerical results for q = 1 and q = 2, we can conjecture the phase diagram for q ∈ [ 1 , 4 ] . In particular, we establish that the system is driven to a fixed point at finite or infinite long-range disorder depending on the values of q and a. Finally, we discuss the role of the higher cumulants of the disorder distribution. This is done by drawing the auxiliary spin-variables from different statistical models. While the main features of the phase diagram depend only on the first and second cumulant, we argue, for the infinite disorder fixed point, that certain universal effects are affected by the higher cumulants of the disorder distribution.
The phase transition in the q -state Potts model with homogeneous ferromagnetic couplings is strongly first order for large q, while it is rounded in the presence of quenched disorder. Here we study this phenomenon on different two-dimensional lattices by using the fact that the partition function of the model is dominated by a single diagram of the high-temperature expansion, which is calculated by an efficient combinatorial optimization algorithm. For a given finite sample with discrete randomness the free energy is a piecewise linear function of the temperature, which is rounded after averaging, however, the discontinuity of the internal energy at the transition point (i.e., the latent heat) stays finite even in the thermodynamic limit. For a continuous disorder, instead, the latent heat vanishes. At the phase transition point the dominant diagram percolates and the total magnetic moment is related to the size of the percolating cluster. Its fractal dimension is found d(f) = ( 5 + square root of 5)/4 and it is independent of the type of the lattice and the form of disorder. We argue that the critical behavior is exclusively determined by disorder and the corresponding fixed point is the isotropic version of the so-called infinite randomness fixed point, which is realized in random quantum spin chains. From this mapping we conjecture the values of the critical exponents as beta=2- d(f), beta(s) =1/2, and nu=1.
The q-state Potts model with long-range correlated disorder is studied by means of large-scale Monte Carlo simulations for q=2, 4, 8, and 16. Evidence is given of the existence of a Griffiths phase, where the thermodynamic quantities display an algebraic finite-size scaling, in a finite range of temperatures. The critical exponents are shown to depend on both the temperature and the exponent of the algebraic decay of disorder correlations, but not on the number of states of the Potts model. The mechanism leading to the violation of hyperscaling relations is observed in the entire Griffiths phase.
Phase transitions in a two-dimensional q-state Potts model with long-range disorder in which the correlation decays as a power law approximately r -a for separation r are studied analytically within a Migdal-Kadanoff renormalization group. Correlation terms are introduced into the parameter space of the renormalization group and the recursion relations are derived. We expand q near q 0 , where the specific-heat exponent of the pure system alpha p vanishes, and a near d=2. For small alpha p and delta =2-a<<1, we find three fixed points: 'pure','short-range disorder' and 'long-range disorder'. In our calculation the disorder is relevant if av-2<0 for a<d, where the v are the correlation-length exponents of the fixed points. The correlation-length exponent for the 'long-range disorder' fixed point is v long =2/a. These are consistent with field-theoretic renormalization group results.
We consider the critical behavior of two-dimensional Potts models in the presence of a bond disorder in which the correlation decays as a power law. In some recent work the thermal sector of this theory was investigated by a renormalization group computation based on perturbed conformal field theory. Here we apply the same approach to study instead the magnetic sector. In particular, we compute the leading corrections to the Potts spin scaling dimension. Our results include as a special case the long-range disorder Ising model. We compare our prediction to Monte Carlo simulations. Finally, by studying the magnetization scaling function, we show a clear numerical evidence of a cross-over between the long-range and the short-range class of universality.
The two-dimensional q-state Potts model is subjected to a Z(q) symmetric disorder that allows for the existence of a Nishimori line. At q=2, this model coincides with the +/- J random-bond Ising model. For q>2, apart from the usual pure- and zero-temperature fixed points, the ferro/paramagnetic phase boundary is controlled by two critical fixed points: a weak disorder point, whose universality class is that of the ferromagnetic bond-disordered Potts model, and a strong disorder point which generalizes the usual Nishimori point. We numerically study the case q=3, tracing out the phase diagram and precisely determining the critical exponents. The universality class of the Nishimori point is inconsistent with percolation on Potts clusters.