arXiv:cond-mat/9907076 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Cluster analysis and finite-size scaling for Ising spin systems

Yusuke Tomita, Yutaka Okabe, Chin‐Kun Hu

Published Sep 1, 1999• 39 citations

Abstract

Based on the connection between the Ising model and a correlated percolation model, we calculate the distribution function for the fraction (c) of lattice sites in percolating clusters in subgraphs with n percolating clusters, f(n)(c), and the distribution function for magnetization (m) in subgraphs with n percolating clusters, p(n)(m). We find that f(n)(c) and p(n)(m) have very good finite-size scaling behavior and that they have universal finite-size scaling functions for the model on square, plane triangular, and honeycomb lattices when aspect ratios of these lattices have the proportions 1:square root[3]/2:square root[3]. The complex structure of the magnetization distribution function p(m) for the system with large aspect ratio could be understood from the independent orientations of two or more percolation clusters in such a system.

DOI: 10.1103/physreve.60.2716

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arXiv:cond-mat/9907076 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Cluster analysis and finite-size scaling for Ising spin systems

Yusuke Tomita, Yutaka Okabe, Chin‐Kun Hu

Published Sep 1, 1999• 39 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physreve.60.2716

Related papers

Cluster Analysis of the Ising Model and Universal Finite-Size Scaling

Yutaka Okabe, Kazuhisa Kaneda, Yusuke Tomita +2 more

The recent progress in the study of finite-size scaling (FSS) properties of the Ising model is briefly reviewed. We calculate the universal FSS functions for the Binder parameter g and the magnetization distribution function p(m) for the Ising model on L1×L2 two-dimensional lattices with tilted boundary conditions. We show that the FSS functions are universal for fixed sets of the aspect ratio L1/L2 and the tilt parameter. We also study the percolating properties of the Ising model, giving attention to the effects of the aspect ratio of finite systems. We elucidate the origin of the complex structure of p(m) for the system with large aspect ratio by the multiple-percolating-cluster argument.The recent progress in the study of finite-size scaling (FSS) properties of the Ising model is briefly reviewed. We calculate the universal FSS functions for the Binder parameter g and the magnetization distribution function p(m) for the Ising model on L1×L2 two-dimensional lattices with tilted boundary conditions. We show that the FSS functions are universal for fixed sets of the aspect ratio L1/L2 and the tilt parameter. We also study the percolating properties of the Ising model, giving attention to the effects of the aspect ratio of finite systems. We elucidate the origin of the complex structure of p(m) for the system with large aspect ratio by the multiple-percolating-cluster argument.

Jan 1, 20002 citations

Chin‐Kun Hu, Chai‐Yu Lin, Jau-Ann Chen

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Geometrical clusters in two-dimensional random-field Ising models

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We consider geometrical or Ising clusters (i.e., domains of parallel spins) in the square lattice random-field Ising model by varying the strength of the Gaussian random field $\ensuremath{\Delta}$. In agreement with the conclusion of a previous investigation [Phys. Rev. E 63, 066109 (2001)], the geometrical correlation length, i.e., the average size of the clusters $\ensuremath{\xi}$ is finite for $\ensuremath{\Delta}>{\ensuremath{\Delta}}_{c}\ensuremath{\approx}1.65$ and divergent for $\ensuremath{\Delta}\ensuremath{\leqslant}{\ensuremath{\Delta}}_{c}$. The scaling function of the distribution of the mass of the clusters as well as the geometrical correlation function are found to involve the scaling exponents of critical percolation. On the other hand, the divergence of the correlation length, $\ensuremath{\xi}(\ensuremath{\Delta})\ensuremath{\sim}{(\ensuremath{\Delta}\ensuremath{-}{\ensuremath{\Delta}}_{c})}^{\ensuremath{-}\ensuremath{\nu}}$, with $\ensuremath{\nu}\ensuremath{\approx}2$, is related to that of tricritical percolation. It is verified numerically that critical geometrical correlations transform conformally.We consider geometrical or Ising clusters (i.e., domains of parallel spins) in the square lattice random-field Ising model by varying the strength of the Gaussian random field $\ensuremath{\Delta}$. In agreement with the conclusion of a previous investigation [Phys. Rev. E 63, 066109 (2001)], the geometrical correlation length, i.e., the average size of the clusters $\ensuremath{\xi}$ is finite for $\ensuremath{\Delta}>{\ensuremath{\Delta}}_{c}\ensuremath{\approx}1.65$ and divergent for $\ensuremath{\Delta}\ensuremath{\leqslant}{\ensuremath{\Delta}}_{c}$. The scaling function of the distribution of the mass of the clusters as well as the geometrical correlation function are found to involve the scaling exponents of critical percolation. On the other hand, the divergence of the correlation length, $\ensuremath{\xi}(\ensuremath{\Delta})\ensuremath{\sim}{(\ensuremath{\Delta}\ensuremath{-}{\ensuremath{\Delta}}_{c})}^{\ensuremath{-}\ensuremath{\nu}}$, with $\ensuremath{\nu}\ensuremath{\approx}2$, is related to that of tricritical percolation. It is verified numerically that critical geometrical correlations transform conformally.

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Universal finite-size scaling functions for critical systems with tilted boundary conditions

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We calculate finite-size scaling functions (FSSF's) of Binder parameter g and magnetization distribution function $p(m)$ for the Ising model on ${L}_{1}\ifmmode\times\else\texttimes\fi{}{L}_{2}$ square lattices with periodic boundary conditions in the horizontal ${L}_{1}$ direction and tilted boundary conditions in the vertical ${L}_{2}$ direction such that the $i\mathrm{th}$ site in the first row is connected with the $\mathrm{mod}{(i+cL}_{1}{,L}_{1})\mathrm{th}$ site in the ${L}_{2}$ row of the lattice, where $1<~i<~{L}_{1}.$ For fixed sets of $(a,c)$ with ${a=L}_{1}{/L}_{2},$ the FSSF's of g and $p(m)$ are universal and in such cases ${a/(c}^{2}{a}^{2}+1)$ is an invariant. For percolation on lattices with fixed a, the FSSF of the existence probability (also called spanning probability) is not affected by c.We calculate finite-size scaling functions (FSSF's) of Binder parameter g and magnetization distribution function $p(m)$ for the Ising model on ${L}_{1}\ifmmode\times\else\texttimes\fi{}{L}_{2}$ square lattices with periodic boundary conditions in the horizontal ${L}_{1}$ direction and tilted boundary conditions in the vertical ${L}_{2}$ direction such that the $i\mathrm{th}$ site in the first row is connected with the $\mathrm{mod}{(i+cL}_{1}{,L}_{1})\mathrm{th}$ site in the ${L}_{2}$ row of the lattice, where $1<~i<~{L}_{1}.$ For fixed sets of $(a,c)$ with ${a=L}_{1}{/L}_{2},$ the FSSF's of g and $p(m)$ are universal and in such cases ${a/(c}^{2}{a}^{2}+1)$ is an invariant. For percolation on lattices with fixed a, the FSSF of the existence probability (also called spanning probability) is not affected by c.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsFeb 1, 199943 citations

Scaling behavior in very small percolation lattices

J. B. Elliott, M. L. Gilkes, J. A. Hauger +7 more

We examine the average cluster distribution as a function of lattice probability for a very small $(L=6)$ lattice and determine the scaling function of three-dimensional percolation. The behavior of the second moment, calculated from the average cluster distribution of $L=6$ and $L=63$ lattices, is compared to power-law behavior predicted by the scaling function. We also examine the finite-size scaling of the critical point and the size of the largest cluster at the critical point. This analysis leads to estimates of the critical exponent $\ensuremath{\nu}$ and the ratio of critical exponents $\ensuremath{\beta}/\ensuremath{\nu}$.We examine the average cluster distribution as a function of lattice probability for a very small $(L=6)$ lattice and determine the scaling function of three-dimensional percolation. The behavior of the second moment, calculated from the average cluster distribution of $L=6$ and $L=63$ lattices, is compared to power-law behavior predicted by the scaling function. We also examine the finite-size scaling of the critical point and the size of the largest cluster at the critical point. This analysis leads to estimates of the critical exponent $\ensuremath{\nu}$ and the ratio of critical exponents $\ensuremath{\beta}/\ensuremath{\nu}$.

Physical Review CMar 1, 199727 citations

Percolation, clusters, and phase transitions in spin models

Chin‐Kun Hu

The phase transition in the Ising model and the percolation transition in the lattice percolation model have many common characteristics which have motivated researchers to explore whether the former is a percolation transition of a correlated percolation model. Previous attempts to draw such a connection have been either unsuccessful or unsatisfactory. Considering each lattice site with an Ising spin occupied and the nearest-neighbor (NN) coupling between occupied sites as a bond with a bond probability $p$ depending on the NN coupling constant $J$ and the temperature $T$ , we formally show that the partition function of the Ising model is the generating function of bond-correlated percolation model (BCPM) with a bond probability $p=1\ensuremath{-}\mathrm{exp}(\frac{\ensuremath{-}2J}{\mathrm{kT}})$. The BCPM has the Ising critical temperature and exponents, including $\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}},\ensuremath{\eta},\ensuremath{\beta},\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}}$, and $\ensuremath{\gamma}$ (perhaps also ${\ensuremath{\gamma}}^{\ensuremath{'}}$). From the connection between the Ising model and the BCPM, we also derive and hence give a geometrical meaning of the finite-size scaling and broadening at first-order phase transitions of the Ising model. Our approach may easily be extended to many spin models and give geometrical meaning to other properties of spin models.The phase transition in the Ising model and the percolation transition in the lattice percolation model have many common characteristics which have motivated researchers to explore whether the former is a percolation transition of a correlated percolation model. Previous attempts to draw such a connection have been either unsuccessful or unsatisfactory. Considering each lattice site with an Ising spin occupied and the nearest-neighbor (NN) coupling between occupied sites as a bond with a bond probability $p$ depending on the NN coupling constant $J$ and the temperature $T$, we formally show that the partition function of the Ising model is the generating function of bond-correlated percolation model (BCPM) with a bond probability $p=1\ensuremath{-}\mathrm{exp}(\frac{\ensuremath{-}2J}{\mathrm{kT}})$. The BCPM has the Ising critical temperature and exponents, including $\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}},\ensuremath{\eta},\ensuremath{\beta},\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}}$, and $\ensuremath{\gamma}$ (perhaps also ${\ensuremath{\gamma}}^{\ensuremath{'}}$). From the connection between the Ising model and the BCPM, we also derive and hence give a geometrical meaning of the finite-size scaling and broadening at first-order phase transitions of the Ising model. Our approach may easily be extended to many spin models and give geometrical meaning to other properties of spin models.

Physical review. B, Condensed matterMay 1, 1984146 citations

We calculate finite-size scaling functions (FSSF's) of Binder parameter g and magnetization distribution function

p(m)

for the Ising model on ${L}_{1}\ifmmode\times\else\texttimes\fi{}{L}_{2}$ square lattices with periodic boundary conditions in the horizontal

{L}_{1}

direction and tilted boundary conditions in the vertical

{L}_{2}

direction such that the

i\mathrm{th}

site in the first row is connected with the

\mathrm{mod}{(i+cL}_{1}{,L}_{1})\mathrm{th}

site in the

{L}_{2}

row of the lattice, where

1<~i<~{L}_{1}.

For fixed sets of

(a,c)

with

{a=L}_{1}{/L}_{2},

the FSSF's of g and

p(m)

are universal and in such cases

{a/(c}^{2}{a}^{2}+1)

is an invariant. For percolation on lattices with fixed a, the FSSF of the existence probability (also called spanning probability) is not affected by c.

We examine the average cluster distribution as a function of lattice probability for a very small

(L=6)

lattice and determine the scaling function of three-dimensional percolation. The behavior of the second moment, calculated from the average cluster distribution of

L=6

and

L=63

lattices, is compared to power-law behavior predicted by the scaling function. We also examine the finite-size scaling of the critical point and the size of the largest cluster at the critical point. This analysis leads to estimates of the critical exponent $\ensuremath{\nu}$ and the ratio of critical exponents $\ensuremath{\beta}/\ensuremath{\nu}$.

p

depending on the NN coupling constant

J

and the temperature

T

, we formally show that the partition function of the Ising model is the generating function of bond-correlated percolation model (BCPM) with a bond probability $p=1\ensuremath{-}\mathrm{exp}(\frac{\ensuremath{-}2J}{\mathrm{kT}})$. The BCPM has the Ising critical temperature and exponents, including $\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}},\ensuremath{\eta},\ensuremath{\beta},\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}}$, and $\ensuremath{\gamma}$ (perhaps also ${\ensuremath{\gamma}}^{\ensuremath{'}}$). From the connection between the Ising model and the BCPM, we also derive and hence give a geometrical meaning of the finite-size scaling and broadening at first-order phase transitions of the Ising model. Our approach may easily be extended to many spin models and give geometrical meaning to other properties of spin models.