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arXiv:2606.22398

Percolation of Zero-Weight Paths and the Shape of the Phase Boundary in the Two-Dimensional Random-Bond Ising Model

Amirhossein Manouchehri, Kirill Shtengel

Published Jun 21, 2026• 0 citations
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Abstract

We explore the connection between the low-temperature boundary of the ferromagnetic phase in the two-dimensional $\pm J$ random-bond Ising model, where antiferromagnetic bonds occur with probability $p$ and a geometric transition dubbed ``zero-weight percolation''. We argue that the onset of this percolation characterized by the emergence of a percolating path containing an equal number of $+J$ and $-J$ bonds is incompatible with ferromagnetic ordering. Due to its purely geometrical nature, this percolation criterion is a property of a disorder realization and is independent of the temperature, which in turn suggests that the ferromagnetic phase boundary is vertical below the Nishimori point in the $(p,T)$ plane. Using a dynamic-programming algorithm combined with finite-size scaling, we identify the critical disorder at which zero-weight paths first percolate as $p_c = 0.1000(2)$, and we extract the associated critical exponents $ν= 1.26(1)$, $β/ν= 0.85(1)$, $γ/ν= 0.264(5)$, and fractal dimension $d_f \approx 1.11$. The value of $p_c$ is below the previously reported values of the critical disorder strength corresponding to the loss of the ferromagnetic order, both at zero temperature and the Nishimori point. Nevertheless, we argue that the percolation transition studied in this paper is behind the loss of ferromagnetism and thus provides a new, purely geometrical perspective on the stability of ferromagnetic order in disordered spin systems.

DOI
10.48550/arxiv.2606.22398

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We explore the connection between the low-temperature boundary of the ferromagnetic phase in the two-dimensional ±J\pm J±J random-bond Ising model, where antiferromagnetic bonds occur with probability ppp and a geometric transition dubbed ``zero-weight percolation''. We argue that the onset of this percolation characterized by the emergence of a percolating path containing an equal number of +J+J+J and −J-J−J bonds is incompatible with ferromagnetic ordering. Due to its purely geometrical nature, this percolation criterion is a property of a disorder realization and is independent of the temperature, which in turn suggests that the ferromagnetic phase boundary is vertical below the Nishimori point in the (p,T)(p,T)(p,T) plane. Using a dynamic-programming algorithm combined with finite-size scaling, we identify the critical disorder at which zero-weight paths first percolate as pc=0.1000(2)p_c = 0.1000(2)pc​=0.1000(2), and we extract the associated critical exponents ν=1.26(1)ν= 1.26(1)ν=1.26(1), β/ν=0.85(1)β/ν= 0.85(1)β/ν=0.85(1), γ/ν=0.264(5)γ/ν= 0.264(5)γ/ν=0.264(5), and fractal dimension df≈1.11d_f \approx 1.11df​≈1.11. The value of pcp_cpc​ is below the previously reported values of the critical disorder strength corresponding to the loss of the ferromagnetic order, both at zero temperature and the Nishimori point. Nevertheless, we argue that the percolation transition studied in this paper is behind the loss of ferromagnetism and thus provides a new, purely geometrical perspective on the stability of ferromagnetic order in disordered spin systems.
The random-bond Ising model on the square lattice has several disordered critical points, depending on the probability distribution of the bonds. There are a finite-temperature multicritical point, called the Nishimori point, and a zero-temperature fixed point, for both a binary distribution where the coupling constants take the values ± J and a Gaussian disorder distribution. Inclusion of dilution in the ± J distribution ( J = 0 for some bonds) gives rise to another zero-temperature fixed point which can be identified with percolation in the non-frustrated case ( J ≥ 0). We study these fixed points using numerical (transfer matrix) methods. We determine the location, critical exponents, and central charge of the different fixed points and study the spin–spin correlation functions. Our main findings are the following. (1) We confirm that the Nishimori point is universal with respect to the type of disorder, i.e. we obtain the same central charge and critical exponents for the ± J and Gaussian distributions of disorder. (2) The Nishimori point, the zero-temperature fixed point for the ± J and Gaussian distributions of disorder, and the percolation point in the diluted case all belong to mutually distinct universality classes. (3) The paramagnetic phase is re-entrant below the Nishimori point, i.e. the zero-temperature fixed points are not located exactly below the Nishimori point, either for the ± J distribution, or for the Gaussian distribution.
Renyi mutual information, computed from second Renyi entropies, can identify classical phase transitions from their finite-size scaling at critical points. We apply this technique to examine the presence or absence of finite temperature phase transitions in various two-dimensional models on a square lattice, which are extensions of the conventional Ising model by adding a quenched disorder. When the quenched disorder causes the nearest neighbor bonds to be both ferromagnetic and antiferromagnetic, (a) a spin glass phase exists only at zero temperature, and (b) a ferromagnetic phase exists at a finite temperature when the antiferromagnetic bond distributions are sufficiently dilute. Furthermore, finite temperature paramagnetic-ferromagnetic transitions can also occur when the disordered bonds involve only ferromagnetic couplings of random strengths. In our numerical simulations, the 'zero temperature only' phase transitions are identified when there is no consistent finite-size scaling of the Renyi mutual information curves, while for finite temperature critical points, the curves can identify the critical temperature T-c by their crossings at T-c and 2 T-c.
We show that an out-of-equilibrium percolation transition occurs after quenching ferromagnetic Ising-like systems across their magnetic first-order transitions. As a paradigmatic example, we consider a two-dimensional Ising system driven across its low-temperature first-order transition line by a quench of the magnetic field hhh from hi<0h_i<0hi​<0 to h>0h>0h>0. In the thermodynamic limit and for finite values of hhh, the post-quench evolution under a purely relaxational dynamics is characterized by a dynamic transition at a finite critical time tc(h)t_c(h)tc​(h) from the metastable negatively magnetized phase to the positive one, marked by the percolation of the largest clusters of positive and negative spins. This out-of-equilibrium percolation transition displays a finite-size scaling behavior as in the standard random-percolation case. However, while the fractal dimension of the percolating clusters is consistent with the random-percolation value, the exponent controlling the approach to criticality differs and depends on hhh. We also show that the percolation critical behavior is related to the spinodal-like behavior of the magnetization in the small-hhh limit, which implies that the percolation time tc(h)t_c(h)tc​(h) exhibits a spinodal-like exponential dependence on hhh.
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.
Abstract Rényi mutual information, computed from second Rényi entropies, can identify classical phase transitions from their finite-size scaling at critical points. We apply this technique to examine the presence or absence of finite temperature phase transitions in various two-dimensional models on a square lattice, which are extensions of the conventional Ising model by adding a quenched disorder. When the quenched disorder causes the nearest neighbor bonds to be both ferromagnetic and antiferromagnetic, (a) a spin glass phase exists only at zero temperature, and (b) a ferromagnetic phase exists at a finite temperature when the antiferromagnetic bond distributions are sufficiently dilute. Furthermore, finite temperature paramagnetic-ferromagnetic transitions can also occur when the disordered bonds involve only ferromagnetic couplings of random strengths. In our numerical simulations, the ‘zero temperature only’ phase transitions are identified when there is no consistent finite-size scaling of the Rényi mutual information curves, while for finite temperature critical points, the curves can identify the critical temperature T c by their crossings at T c and 2 T c .
The four-dimensional $\ifmmode\pm\else\textpm\fi{}J$ random-bond Ising model is studied using ground-state calculations. System sizes up to N=64{N=6}^{4}N=64 spins are considered. Here it is found that the ferromagnetic--spin-glass transition occurs at a critical concentration pc=0.28(1){p}_{c}=0.28(1)pc​=0.28(1) of the antiferromagnetic bonds, which is comparable to values found previously by high-temperature series expansions. The transition is characterized by a correlation-length exponent $\ensuremath{\nu}=1.0(1)$ and an order-parameter exponent $\ensuremath{\beta}=0.4(1).$ Thus, this transition is in a different universality class from four-dimensional bond percolation, where $\ensuremath{\nu}=0.678(50)$ and $\ensuremath{\beta}=0.639(20).$