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

Yutaka Okabe, Kazuhisa Kaneda, Yusuke Tomita, Macoto Kikuchi, Chin‐Kun Hu

Published Jan 1, 2000• 2 citations
OpenAlex

Abstract

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.

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arXiv:cond-mat/9905202•Physical review. B, Condensed matter•Jan 1, 2000•17 citations
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.
We calculate finite-size scaling functions (FSSF's) of Binder parameter g and magnetization distribution function p(m)p(m)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 L1{L}_{1}L1​ direction and tilted boundary conditions in the vertical L2{L}_{2}L2​ direction such that the ithi\mathrm{th}ith site in the first row is connected with the mod(i+cL1,L1)th\mathrm{mod}{(i+cL}_{1}{,L}_{1})\mathrm{th}mod(i+cL1​,L1​)th site in the L2{L}_{2}L2​ row of the lattice, where 1< i< L1.1<~i<~{L}_{1}.1< i< L1​. For fixed sets of (a,c)(a,c)(a,c) with a=L1/L2,{a=L}_{1}{/L}_{2},a=L1​/L2​, the FSSF's of g and p(m)p(m)p(m) are universal and in such cases a/(c2a2+1){a/(c}^{2}{a}^{2}+1)a/(c2a2+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.
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.
We study the finite-size scaling properties of the Ising model on the Möbius strip and the Klein bottle. The results are compared with those of the Ising model under different boundary conditions, that is, the free, cylindrical, and toroidal boundary conditions. The difference in the magnetization distribution function p(m) for various boundary conditions is discussed in terms of the number of the percolating clusters and the cluster size. We also find interesting aspect-ratio dependence of the value of the Binder parameter at T = T(c) for various boundary conditions. We discuss the relation to the finite-size correction calculations for the dimer statistics.
The finite-size scaling (FSS) functions for anisotropic three-dimensional (3D) Ising models of size L 1 × L 1 × aL 1 ( a : anisotropy parameter) are studied by Monte Carlo simulations. We study the a dependence of FSS functions of the Binder parameter g and the magnetization distribution function p ( m ). We have shown that the FSS functions for p ( m ) at the critical temperature change from a two-peak structure to a single-peak one by increasing or decreasing a from 1. We also study the FSS near the critical temperature of the layered square-lattice Ising model, when the systems have a large two-dimensional (2D) anisotropy. We have found the 3D and 2D FSS behaviour depending on the parameter which is fixed; a unified view of 3D and 2D FSS behaviour has been obtained for the anisotropic 3D Ising models.
We present detailed calculations for the partition function and the free energy of the finite two-dimensional square lattice Ising model with periodic and antiperiodic boundary conditions, variable aspect ratio, and anisotropic couplings, as well as for the corresponding universal free energy finite-size scaling functions. Therefore, we review the dimer mapping, as well as the interplay between its topology and the different types of boundary conditions. As a central result, we show how both the finite system as well as the scaling form decay into contributions for the bulk, a characteristic finite-size part, and – if present – the surface tension, which emerges due to at least one antiperiodic boundary in the system. For the scaling limit we extend the proper finite-size scaling theory to the anisotropic case and show how this anisotropy can be absorbed into suitable scaling variables.
It is shown by the Monte Carlo method that finite-size scaling (FSS) holds in the two-dimensional randomly coupled Ising ferromagnet. It is also demonstrated that the form of the universal FSS function constructed via a FSS scheme depends on the strength of the random coupling in the case of strongly disordered systems. Monte Carlo measurements of thermodynamic (infinite-volume-limit) data of the correlation length $(\ensuremath{\xi})$ up to $\ensuremath{\xi}\ensuremath{\simeq}200$ along with measurements of the fourth-order cumulant ratio (Binder cumulant ratio) at criticality are analyzed in light of two competing scenarios of weak universality and the multiplicative logarithmic correction. It is demonstrated that the data are likely to be more consistent with the former scenario than the latter which is the conventional scenario.