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arXiv:2604.05772

Percolation in the three-dimensional Ising model

Jinhong Zhu, Tao Chen, Zhiyi Li, Sheng Fang, Youjin Deng

Published Apr 7, 2026• 0 citations
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Abstract

Geometric representations provide a useful perspective on critical phenomena in the Ising model. In a recent study [Phys. Rev. E 112, 034118 (2025)], we found that the two-dimensional critical Ising model exhibits two consecutive percolation transitions for geometric spin clusters as the bond-occupation probability $p$ between parallel spins increases. Here, through extensive Monte Carlo simulations, we show that this phenomenon does not persist in three dimensions, where we observe only a single percolation transition on critical Ising configurations. Further theoretical analysis of the Ising model on the complete graph also yields the same scenario. In addition, we study percolation on a two-dimensional layer embedded in the three-dimensional critical Ising model. For this layer system, we estimate the red-bond exponent $y_p = 0.426(6)$ and the fractal dimensions of the largest cluster, hull, and shortest path as $d_f = 1.8926(20)$, $d_{\rm hull} = 1.663(4)$, and $d_{\rm min} = 1.080(10)$, respectively. These values indicate a distinct universality class induced by coupling to out-of-plane critical correlations.

DOI
10.48550/arxiv.2604.05772

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Geometric representations provide a useful perspective on critical phenomena in the Ising model. In a recent study [Phys. Rev. E 112, 034118 (2025)], we found that the two-dimensional critical Ising model exhibits two consecutive percolation transitions for geometric spin clusters as the bond-occupation probability ppp between parallel spins increases. Here, through extensive Monte Carlo simulations, we show that this phenomenon does not persist in three dimensions, where we observe only a single percolation transition on critical Ising configurations. Further theoretical analysis of the Ising model on the complete graph also yields the same scenario. In addition, we study percolation on a two-dimensional layer embedded in the three-dimensional critical Ising model. For this layer system, we estimate the red-bond exponent yp=0.426(6)y_p = 0.426(6)yp​=0.426(6) and the fractal dimensions of the largest cluster, hull, and shortest path as df=1.8926(20)d_f = 1.8926(20)df​=1.8926(20), dhull=1.663(4)d_{\rm hull} = 1.663(4)dhull​=1.663(4), and dmin=1.080(10)d_{\rm min} = 1.080(10)dmin​=1.080(10), respectively. These values indicate a distinct universality class induced by coupling to out-of-plane critical correlations.
The study of the Ising model from a percolation perspective has played a significant role in the modern theory of critical phenomena. We consider the celebrated square-lattice Ising model and construct percolation clusters by placing bonds, with probability p, between any pair of parallel spins within an extended range beyond nearest neighbors. At the Ising criticality, we observe two percolation transitions as p increases: starting from a disordered phase with only small clusters, the percolation system enters into a stable critical phase that persists over a wide range p_{c_{1}}<p<p_{c_{2}}, and then develops a long-ranged percolation order with giant clusters for both up and down spins. At p_{c1} and for the stable critical phase, the critical behaviors agree well with those for the Fortuin-Kasteleyn random clusters and the spin domains of the Ising model, respectively. At p_{c2}, the fractal dimension of clusters and the scaling exponent along p direction are estimated as y_{h2}=1.9580(6) and y_{p2}=0.552(9), of which the exact values remain unknown. These findings reveal interesting geometric properties of the two-dimensional Ising model that has been studied for more than 100 years.
The fractal dimensions and the percolation exponents of geometrical spin clusters of like sign at criticality are obtained numerically for an Ising model with temperature dependent annealed bond dilution, also known as the thermalized bond Ising model (TBIM), in two dimensions. For this purpose, a modified Wolff single-cluster Monte Carlo simulation is used to generate equilibrium spin configurations on square lattices in the critical region. A tie-breaking rule is employed to identify nonintersecting spin cluster boundaries along the edges of the dual lattice. The values obtained for the fractal dimensions of the spanning geometrical clusters D c , and their interfaces D I , are in perfect agreement with those reported for the standard two-dimensional ferromagnetic Ising model. Furthermore, the variance of the winding angles results in a diffusivity κ = 3 for the two-dimensional thermalized bond Ising model, thus placing it in the universality class of the regular Ising model. A finite-size scaling analysis of the largest geometrical clusters results in a reliable estimation of the critical percolation exponents for the geometrical clusters in the limit of an infinite lattice size. The percolation exponents thus obtained are also found to be consistent with those reported for the regular Ising model. These consistencies are explained in terms of the Fisher renormalization relations, which express the thermodynamic critical exponents of systems with annealed bond dilution in terms of those of the regular model system.
The Ashkin-Teller model is a two-layer lattice model where spins in each layer interact ferromagnetically with strength JJJ, and the spin-dipoles (product of spins) interact with neighbors with strength $\ensuremath{\lambda}.$ The model exhibits simultaneous magnetic and electric transitions along a self-dual line on the $\ensuremath{\lambda}\text{\ensuremath{-}}J$ plane with continuously varying critical exponents. In this article, we investigate the percolation of geometric clusters of spins and spin-dipoles denoted, respectively, as magnetic and electric clusters. We find that the largest cluster in both cases becomes macroscopic in size and spans the lattice when interaction exceeds a critical threshold given by the same self-dual line where magnetic and electric transitions occur. The fractal dimension of the critical spanning clusters is related to order parameter exponent ${\ensuremath{\beta}}_{m,e}$ as ${D}_{m,e}=d\ensuremath{-}\frac{5}{12}\frac{{\ensuremath{\beta}}_{m,e}}{\ensuremath{\nu}},$ where d=2d=2d=2 is the spatial dimension and $\ensuremath{\nu}$ is the correlation length exponent. This relation determines all other percolation exponents and their variation with respect to $\ensuremath{\lambda}.$ We show that for magnetic percolation, the binder cumulant, as a function of ${\ensuremath{\xi}}_{2}/L$ with ${\ensuremath{\xi}}_{2}$ being the second-moment correlation length, remains invariant all along the critical line and matches with that of the spin-percolation in the usual Ising model. The function also remains invariant for the electric percolation, forming a new superuniversality class of percolation transition.
We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to L=512L=512L=512, and estimate the percolation thresholds to be pc(bond)=0.24881182(10){p}_{c}(\mathrm{bond})=0.248\phantom{\rule{0.16em}{0ex}}811\phantom{\rule{0.16em}{0ex}}82(10)pc​(bond)=0.24881182(10) and pc(site)=0.3116077(2){p}_{c}(\mathrm{site})=0.311\phantom{\rule{0.16em}{0ex}}607\phantom{\rule{0.16em}{0ex}}7(2)pc​(site)=0.3116077(2). By performing extensive simulations at these estimated critical points, we then estimate the critical exponents $1/\ensuremath{\nu}=1.1410(15)$, $\ensuremath{\beta}/\ensuremath{\nu}=0.477\phantom{\rule{0.16em}{0ex}}05(15)$, the leading correction exponent ${y}_{i}=\ensuremath{-}1.2(2)$, and the shortest-path exponent dmin=1.3756(3){d}_{\mathrm{min}}=1.3756(3)dmin​=1.3756(3). Various universal amplitudes are also obtained, including wrapping probabilities, ratios associated with the cluster-size distribution, and the excess cluster number. We observe that the leading finite-size corrections in certain wrapping probabilities are governed by an exponent $\ensuremath{\approx}$$\ensuremath{-}2$, rather than ${y}_{i}\ensuremath{\approx}\ensuremath{-}1.2$.
We numerically investigate the heterogeneity in cluster sizes in the two-dimensional Ising model and verify its scaling form recently proposed in the context of percolation problems [Phys. Rev. E 84, 010101(R) (2011)]. The scaling exponents obtained via the finite-size scaling analysis are shown to be consistent with theoretical values of the fractal dimension d(f) and the Fisher exponent τ for the cluster distribution. We also point out that strong finite-size effects exist due to the geometric nature of the cluster-size heterogeneity.
We present a Monte Carlo study of the geometric properties of Fortuin-Kasteleyn (FK) clusters of the Ising model on square [two-dimensional (2D)] and simple-cubic [three-dimensional (3D)] lattices. The wrapping probability, a dimensionless quantity characterizing the topology of the FK clusters on a torus, is found to suffer from smaller finite-size corrections than the well-known Binder ratio and yields a high-precision critical coupling as Kc(3D)=0.221654631(8){K}_{c}(3\mathrm{D})=0.221\phantom{\rule{0.16em}{0ex}}654\phantom{\rule{0.16em}{0ex}}631(8)Kc​(3D)=0.221654631(8). We then study other geometric properties of FK clusters at criticality. It is demonstrated that the distribution of the critical largest-cluster size C1{C}_{1}C1​ follows a single-variable function as $P({C}_{1},L)\phantom{\rule{0.16em}{0ex}}d{C}_{1}=\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}(x)\phantom{\rule{0.16em}{0ex}}dx$ with $x\ensuremath{\equiv}{C}_{1}/{L}^{{d}_{\mathrm{F}}}$ (LLL is the linear size), where the fractal dimension dF{d}_{\mathrm{F}}dF​ is identical to the magnetic exponent. An interesting bimodal feature is observed in distribution $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}(x)$ in three dimensions, and attributed to the different approaching behaviors for $K\ensuremath{\rightarrow}{K}_{c}+{0}^{\ifmmode\pm\else\textpm\fi{}}$. To characterize the compactness of the FK clusters, we measure their graph distances and determine the shortest-path exponents as dmin(3D)=1.25936(12){d}_{\mathrm{min}}(3\mathrm{D})=1.259\phantom{\rule{0.16em}{0ex}}36(12)dmin​(3D)=1.25936(12) and dmin(2D)=1.0940(2){d}_{\mathrm{min}}(2\mathrm{D})=1.094\phantom{\rule{0.16em}{0ex}}0(2)dmin​(2D)=1.0940(2). Further, by excluding all the bridges from the occupied bonds, we obtain bridge-free configurations and determine the backbone exponents as dB(3D)=2.1673(15){d}_{\mathrm{B}}(3\mathrm{D})=2.167\phantom{\rule{0.16em}{0ex}}3(15)dB​(3D)=2.1673(15) and dB(2D)=1.7321(4){d}_{\mathrm{B}}(2\mathrm{D})=1.732\phantom{\rule{0.16em}{0ex}}1(4)dB​(2D)=1.7321(4). The estimates of the universal wrapping probabilities for the 3D Ising model and of the geometric critical exponents dmin{d}_{\mathrm{min}}dmin​ and dB{d}_{\mathrm{B}}dB​ either improve over the existing results or have not been reported yet. We believe that these numerical results would provide a testing ground in the development of further theoretical treatments of the 3D Ising model.