Magnetic critical exponent of the two-dimensional Potts model with long-range correlated bond disorder beyond leading perturbative order

The two-dimensional $q$-state Potts model with quenched bond disorder whose correlations decay as a power law $r^{-a}$ flows, below the marginal value $a \approx 2$ and above an instability value $a^{*}(q)$ at which Monte-Carlo studies locate the takeover by the infinite-disorder fixed point ($a^{*} \approx 1$ at $q = 2$), to a long-range (LR) disorder fixed point distinct from both the pure and the short-range random-bond universality classes. The magnetic exponent $(\beta/\nu)^{LR}$ — the scaling dimension of the Potts spin field at this fixed point — is currently known only to leading order $O(\epsilon^3)$ in a double expansion in $\epsilon_{SR} \simeq 4(q-2)/(3\pi)$ and $\epsilon_{LR} = 1 - a/2 + \epsilon_{SR}/2$ [Lecce, Picco, Santachiara, Phys. Rev. E 110, 064154 (2024)]. The open problem is a controlled determination of $(\beta/\nu)^{LR}(a,q)$ beyond this perturbative window: its value in the strongly long-range regime approaching $a^{*}(q)$, where the leading-order prediction already deviates from Monte-Carlo data; the two-loop and higher magnetic-sector corrections and their dependence on higher cumulants of the disorder distribution; and the fixed-point disorder amplitude $\mu_{LR}^2$ together with the correction-to-scaling exponents at the LR fixed point ($\omega$ and the approach-to-fixed-point eigenvalue), whose present indeterminacy blocks unambiguous numerical extrapolation.

Background

The $q$-state Potts model assigns to each site $i$ of a square lattice a spin taking $q$ values and couples nearest neighbours through bonds $J_{ij}$; in two dimensions the pure model is critical for $q \le 4$ and its continuum limit is a conformal field theory (CFT). Quenched (frozen-in) bond disorder replaces the uniform couplings by random variables; averaging over the disorder via the replica trick yields, in the continuum limit, $n$ replicas of the Potts CFT perturbed by products of energy-density (thermal) operators $\varepsilon$, followed by the limit $n \to 0$.

For uncorrelated — short-range (SR) — disorder this replica perturbed-CFT renormalization group (RG) was developed by Dotsenko, Picco and Pujol: the SR disorder coupling has RG eigenvalue $\epsilon_{SR} = 4 - 6\pi/(2\pi - \arccos((q-2)/2)) \simeq 4(q-2)/(3\pi)$, which is positive (relevant disorder) for $q > 2$ and vanishes at $q = 2$, and the magnetic exponent at the SR fixed point is known as a series in $\epsilon_{SR}$ starting at order $\epsilon_{SR}^3$.

Long-range (LR) correlated disorder, in which the random couplings have algebraically decaying correlations $\overline{(J_{ij}-\bar J)(J_{kl}-\bar J)} \propto r^{-a}$, is far less understood. The extended Harris criterion of Weinrib and Halperin states that such disorder is relevant when $a\,\nu < 2$, with $\nu$ the correlation-length exponent of the pure model (at the Ising point $q=2$, $\nu = 1$, so relevance requires $a < 2$); their perturbative RG also led to the conjecture $\nu^{LR} = 2/a$ at the disorder-induced fixed point. On the lattice, such disorder is realized by taking the random variable on each site to be the product of the spins of $m$ independent critical two-dimensional Ising models, which produces correlations decaying as $r^{-m/4}$ (i.e. $a = m/4$); with bimodal bonds $J_1, J_2$ the critical manifold is exactly known from self-duality, $(e^{J_1}-1)(e^{J_2}-1) = q$, and $\mu^2 = (J_1-J_2)^2/4$ measures the disorder strength. Monte-Carlo studies established the fixed-point structure: the pure point ($\mu^2 = 0$), an SR fixed point, a finite-disorder LR fixed point, and an infinite-disorder point whose universality class is Bernoulli percolation (independent bond occupation) for $a > 3/2$ and long-range percolation for $a < 3/2$. The same simulations established that, for each $q \in [1,4]$, there is a value $a^{*}(q)$ — with $a^{*} \approx 1$ at $q = 2$ — at which the LR fixed point becomes unstable: for $a < a^{*}(q)$ the model is described by the infinite-disorder point.

Analytically, the thermal sector of the two-dimensional LR-disordered Potts model was computed at two loops by Chippari, Picco and Santachiara: the central charges agree well with numerics, while $\nu^{LR}$ obeys the Weinrib–Halperin value only for Gaussian disorder, the two-loop results being sensitive to the fourth cumulant of the disorder distribution. The magnetic sector is known only at leading order. Working in a double expansion in $\epsilon_{SR}$ and $\epsilon_{LR} = 1 - a/2 + \epsilon_{SR}/2$ (both small when $q \to 2^+$ and $a \to 2^-$), Lecce, Picco and Santachiara obtained for the exponent of the spin pair-correlation function at the LR fixed point — equal to the scaling dimension of the spin field, since $\beta/\nu = x_\sigma$ —

$$\left(\frac{\beta}{\nu}\right)^{LR} = \left(\frac{\beta}{\nu}\right)^{P} - \frac{1}{32}\,\epsilon_{LR}\left(\epsilon_{LR}-\epsilon_{SR}\right)\left(4\epsilon_{LR}+\epsilon_{SR}\,\xi^{2}\right) + O(\epsilon^{4}),$$

where $\xi = 2\,\Gamma(1/6)\Gamma(-2/3)/(\Gamma(-1/6)\Gamma(-1/3))$; at the Ising point this reduces to $(\beta/\nu)^{LR} = \frac{1}{8} - \frac{(2-a)^3}{64} + O((2-a)^4)$, reproducing the Monte-Carlo value $0.123$ at $q=2$, $a=1.5$. The prediction agrees with simulations for $a \lesssim 2$ but deviates for $a \le 1$, where the expansion parameters are no longer small. The numerical determination is itself obstructed: the value $\mu_{LR}^2$ of the disorder strength at the LR fixed point on the lattice critical manifold is unknown, and two competing corrections — corrections to scaling with irrelevant exponent $\omega$ and the approach-to-fixed-point contribution governed by the disorder RG eigenvalue at the LR fixed point (negative, since that fixed point is attractive, and distinct from the disorder-coupling eigenvalue $y_d = 2\epsilon_{LR}$ verified near the pure point) — of unknown relative size prevent an unambiguous finite-size extrapolation (at $q=2$, $a=1.5$ the effective exponent drifts within $0.12$–$0.125$). Simulations indicate that below $a^{*}(q)$ the LR fixed point gives way to the infinite-disorder fixed point, but an analytic description of that changeover is not established; and the LR$\to$SR crossover near the marginal value $a \approx 2$ has been evidenced numerically through the magnetization scaling function $m(\mu^2,L)/m(0,L) = f(\mu^2 L^{y_d})$ with $y_d = 2\epsilon_{LR}$, but lacks an analytic characterization.

Problem Statement

Determine the magnetic critical exponent $(\beta/\nu)^{LR}$ — the scaling dimension of the Potts spin field at the long-range disorder fixed point — of the two-dimensional $q$-state Potts model with quenched bond disorder whose correlations decay as $r^{-a}$, beyond the leading perturbative order, throughout the region of the $(a,q)$ plane in which that fixed point governs the critical behavior (for $q = 2$ this is the interval $a^{*}(q) < a < 2$, between the instability towards the infinite-disorder fixed point at $a^{*}(q) \approx 1$ and the crossover to the short-range class near $a \approx 2$; for $q > 2$ the region is bounded analogously by the crossovers to the short-range and to the infinite-disorder fixed points). The problem comprises two jointly linked components:

(a) A controlled determination of $(\beta/\nu)^{LR}(a,q)$ away from the double-perturbative point $q \to 2$, $a \to 2$ — whether by a higher-order or resummed renormalization-group computation, an exact or nonperturbative result, or a numerical determination with controlled errors — in particular in the strongly long-range regime approaching the instability value $a^{*}(q)$ (for $q = 2$, $a^{*} \approx 1$), where the leading-order prediction already deviates from Monte-Carlo data, together with an analytic account of the destabilization of the LR fixed point at $a^{*}(q)$ and of the changeover to the infinite-disorder class, which is established numerically but not characterized analytically;

(b) the two-loop and higher-order contributions to the magnetic sector and their dependence on higher cumulants of the disorder distribution (the leading magnetic correction depends only on the first two cumulants, while two-loop thermal-sector results already involve the fourth cumulant).

These components jointly serve one objective: a quantitatively reliable theory of the spin-sector critical behavior at the long-range disorder fixed point, with its domain of validity specified. A numerical determination of $(\beta/\nu)^{LR}$ additionally requires the fixed-point disorder amplitude $\mu_{LR}^2$ on the lattice critical manifold and the leading correction-to-scaling exponents at the LR fixed point, which are presently unknown and obstruct unambiguous finite-size extrapolation.

A solution counts as a resolution only when both components (a) and (b) are supplied jointly, each in full as specified above — component (a) including its analytic account of the destabilization of the LR fixed point at $a^{*}(q)$ and of the changeover to the infinite-disorder class, and component (b) requiring the higher-order magnetic-sector contributions and their dependence on the higher cumulants of the disorder distribution to be supplied — with the domain of validity of the resulting expression or numerical estimate stated. A partial result — for example, magnetic-sector corrections established only within the double-perturbative window $q \to 2$, $a \to 2$, or a single numerical value of $(\beta/\nu)^{LR}$ at one point of the $(a,q)$ plane — does not by itself resolve the task. The fixed-point disorder amplitude $\mu_{LR}^2$ and the correction-to-scaling exponents at the LR fixed point are required only insofar as the determination of $(\beta/\nu)^{LR}$ is numerical, where they are needed to make the finite-size extrapolation unambiguous; for an analytic or nonperturbative determination they are supporting results rather than mandatory deliverables.

References

1. Ivan Lecce, Marco Picco, Raoul Santachiara, 'Magnetic exponent for the long-range bond disordered Potts model', Phys. Rev. E 110, 064154 (2024). DOI: 10.1103/PhysRevE.110.064154; arXiv:2407.13456. https://doi.org/10.1103/PhysRevE.110.064154
2. Francesco Chippari, Marco Picco, Raoul Santachiara, 'Long-range quenched bond disorder in the bidimensional Potts model', J. Stat. Mech. (2023) 043301. DOI: 10.1088/1742-5468/acc72a; arXiv:2301.13727. https://doi.org/10.1088/1742-5468/acc72a
3. Francesco Chippari, Marco Picco, Raoul Santachiara, 'Two-dimensional Ising and Potts model with long-range bond disorder: A renormalization group approach', SciPost Phys. 15, 135 (2023). DOI: 10.21468/SciPostPhys.15.4.135; arXiv:2306.01887. https://doi.org/10.21468/SciPostPhys.15.4.135
4. Connor Behan, Dario Benedetti, Fanny Eustachon, Edoardo Lauria, 'Long-range minimal models', J. High Energy Phys. 02 (2026) 001. DOI: 10.1007/jhep02(2026)001; arXiv:2509.26372. https://doi.org/10.1007/jhep02(2026)001
5. Abel Weinrib, B. I. Halperin, 'Critical phenomena in systems with long-range-correlated quenched disorder', Phys. Rev. B 27, 413 (1983). DOI: 10.1103/PhysRevB.27.413. https://doi.org/10.1103/PhysRevB.27.413
6. Vladimir Dotsenko, Marco Picco, Pierre Pujol, 'Renormalisation-group calculation of correlation functions for the 2D random bond Ising and Potts models', Nucl. Phys. B 455, 701 (1995). DOI: 10.1016/0550-3213(95)00534-Y. https://doi.org/10.1016/0550-3213(95)00534-Y
7. M. Dudka, A. A. Fedorenko, V. Blavatska, Yu. Holovatch, 'Critical behavior of the two-dimensional Ising model with long-range correlated disorder', Phys. Rev. B 93, 224422 (2016). DOI: 10.1103/PhysRevB.93.224422. https://doi.org/10.1103/PhysRevB.93.224422
8. Christophe Chatelain, 'Griffiths phase and critical behavior of the two-dimensional Potts models with long-range correlated disorder', Phys. Rev. E 89, 032105 (2014). DOI: 10.1103/PhysRevE.89.032105. https://doi.org/10.1103/PhysRevE.89.032105
9. Zhijin Li, 'Conformality loss and short-range crossover in long-range conformal field theories', arXiv:2409.19392 (2024). https://arxiv.org/abs/2409.19392