arXiv:hep-lat/0007035 · Journal of Physics A Mathematical and General

Cluster percolation inO(n) spin models

Ph Blanchard, S Digal, Santo Fortunato, D Gandolfo, T Mendes, H Satz

Published Nov 24, 2000• 22 citations

Abstract

The spontaneous symmetry breaking in the Ising model can be equivalently described in terms of percolation of Wolff clusters. In O ( n ) spin models similar clusters can be built in a general way, and they are currently used to update these systems in Monte Carlo simulations. We show that for three-dimensional O (2), O (3) and O (4) such clusters are indeed the physical `islands' of the systems, i.e. they percolate at the physical threshold and the percolation exponents are in the universality class of the corresponding model. For O (2) and O (3) the result is proven analytically, for O (4) we derived it by numerical simulations.

DOI: 10.1088/0305-4470/33/48/301

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arXiv:hep-lat/0007035 · Journal of Physics A Mathematical and General

Cluster percolation inO(n) spin models

Ph Blanchard, S Digal, Santo Fortunato, D Gandolfo, T Mendes, H Satz

Published Nov 24, 2000• 22 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/0305-4470/33/48/301

Related papers

Percolation in the three-dimensional Ising model

Jinhong Zhu, Tao Chen, Zhiyi Li +2 more

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.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.

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The magnetization of the 3D Ising model

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We present highly accurate Monte Carlo results for simple cubic Ising lattices containing up to spins. These results were obtained by means of the Cluster Processor, a newly built special-purpose computer for the Wolff cluster simulation of the 3D Ising model. We find that the spontaneous magnetization M(t) is accurately described by , where , in a wide temperature range 0.0005 < t < 0.26. Any corrections to scaling with higher powers of t could not be resolved from our data, which implies that they are very small. The magnetization exponent is determined as . An analysis of the magnetization distribution near criticality yields a new determination of the critical point: , with a standard deviation of .We present highly accurate Monte Carlo results for simple cubic Ising lattices containing up to spins. These results were obtained by means of the Cluster Processor, a newly built special-purpose computer for the Wolff cluster simulation of the 3D Ising model. We find that the spontaneous magnetization M(t) is accurately described by , where , in a wide temperature range 0.0005 < t < 0.26. Any corrections to scaling with higher powers of t could not be resolved from our data, which implies that they are very small. The magnetization exponent is determined as . An analysis of the magnetization distribution near criticality yields a new determination of the critical point: , with a standard deviation of .

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In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point ${T}_{c}$ of the thermal transition and the percolation exponents belong to a special universality class. By introducing a bond probability ${p}_{B}<1,$ the corresponding site-bond clusters keep on percolating at ${T}_{c}$ and the exponents do not change, until ${p}_{B}{=p}_{\mathrm{CK}}=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}2J/kT):$ for this special expression of the bond weight the critical percolation exponents switch to the two-dimensional Ising universality class. We show here that the result is valid for a wide class of bidimensional models with a continuous magnetization transition: there is a critical bond probability ${p}_{c}$ such that, for any ${p}_{B}>~{p}_{c},$ the onset of percolation of the site-bond clusters coincides with the critical point of the thermal transition. The percolation exponents are the same for ${p}_{c}<{p}_{B}<~1$ but, for ${p}_{B}{=p}_{c},$ they suddenly change to the thermal exponents, so that the corresponding clusters are critical droplets of the phase transition. Our result is based on Monte Carlo simulations of various systems near criticality.In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point ${T}_{c}$ of the thermal transition and the percolation exponents belong to a special universality class. By introducing a bond probability ${p}_{B}<1,$ the corresponding site-bond clusters keep on percolating at ${T}_{c}$ and the exponents do not change, until ${p}_{B}{=p}_{\mathrm{CK}}=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}2J/kT):$ for this special expression of the bond weight the critical percolation exponents switch to the two-dimensional Ising universality class. We show here that the result is valid for a wide class of bidimensional models with a continuous magnetization transition: there is a critical bond probability ${p}_{c}$ such that, for any ${p}_{B}>~{p}_{c},$ the onset of percolation of the site-bond clusters coincides with the critical point of the thermal transition. The percolation exponents are the same for ${p}_{c}<{p}_{B}<~1$ but, for ${p}_{B}{=p}_{c},$ they suddenly change to the thermal exponents, so that the corresponding clusters are critical droplets of the phase transition. Our result is based on Monte Carlo simulations of various systems near criticality.

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Critical Ising Spin Dynamics near the Percolation Threshold

R. Rammal, A. D. Benoit

A low-temperature approach for the critical Ising spin dynamics on percolation clusters is proposed. The average energy barrier to overturning a cluster of $s$ spins is shown to scale as $\mathrm{ln}s$ . This basic result is supported by exact numerical calculations of the energy-barrier distribution for two-and three-dimensional percolation clusters, and is argued to be a consequence of scale invariance. The resulting temperature as well as size dependence of the relaxation-time scales agrees very well with the results of our Monte Carlo simulations.A low-temperature approach for the critical Ising spin dynamics on percolation clusters is proposed. The average energy barrier to overturning a cluster of $s$ spins is shown to scale as $\mathrm{ln}s$. This basic result is supported by exact numerical calculations of the energy-barrier distribution for two-and three-dimensional percolation clusters, and is argued to be a consequence of scale invariance. The resulting temperature as well as size dependence of the relaxation-time scales agrees very well with the results of our Monte Carlo simulations.

Physical Review LettersAug 12, 198549 citations

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.

In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point

{T}_{c}

of the thermal transition and the percolation exponents belong to a special universality class. By introducing a bond probability

{p}_{B}<1,

the corresponding site-bond clusters keep on percolating at

{T}_{c}

and the exponents do not change, until ${p}_{B}{=p}_{\mathrm{CK}}=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}2J/kT):$ for this special expression of the bond weight the critical percolation exponents switch to the two-dimensional Ising universality class. We show here that the result is valid for a wide class of bidimensional models with a continuous magnetization transition: there is a critical bond probability

{p}_{c}

such that, for any

{p}_{B}>~{p}_{c},

the onset of percolation of the site-bond clusters coincides with the critical point of the thermal transition. The percolation exponents are the same for

{p}_{c}<{p}_{B}<~1

but, for

{p}_{B}{=p}_{c},

they suddenly change to the thermal exponents, so that the corresponding clusters are critical droplets of the phase transition. Our result is based on Monte Carlo simulations of various systems near criticality.

A low-temperature approach for the critical Ising spin dynamics on percolation clusters is proposed. The average energy barrier to overturning a cluster of

s

spins is shown to scale as

\mathrm{ln}s

. This basic result is supported by exact numerical calculations of the energy-barrier distribution for two-and three-dimensional percolation clusters, and is argued to be a consequence of scale invariance. The resulting temperature as well as size dependence of the relaxation-time scales agrees very well with the results of our Monte Carlo simulations.