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Progress of Theoretical Physics Supplement

Effects of Shape and Boundary Conditions on Finite-Size Scaling Functions for Anisotropic Three-Dimensional Ising Systems

Kazuhisa Kaneda, Yutaka Okabe, Macoto Kikuchi

Published Jan 1, 2000• 1 citation
Publisher

Abstract

By use of Monte Carlo simulations, we study the finite-size scaling (FSS) functions for anisotropic three-dimensional (3D) Ising models of size L1 ×L1 ×aL1 (a: anisotropy parameter). Paying attention to the a dependence, we have shown that the FSS functions for the magnetization distribution function 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 effects of boundary conditions on the FSS functions for the anisotropic 3D Ising models.

DOI
10.1143/ptps.138.458

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By use of Monte Carlo simulations, we study the finite-size scaling (FSS) functions for anisotropic three-dimensional (3D) Ising models of size L1 ×L1 ×aL1 (a: anisotropy parameter). Paying attention to the a dependence, we have shown that the FSS functions for the magnetization distribution function 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 effects of boundary conditions on the FSS functions for the anisotropic 3D Ising models.
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.
The magnetization probability density in d = 2 and 3 dimensional Ising models in slab geometry of volume [Formula: see text] is computed through Monte-Carlo simulation at the critical temperature and zero magnetic field. The finite-size scaling of this distribution and its dependence on the system aspect-ratio [Formula: see text]and boundary conditions are discussed. In the limiting case [Formula: see text] of a macroscopically large slab ([Formula: see text]) the distribution is found to scale as a Gaussian function for all tested system sizes and boundary conditions.
The Ising model (and also the Ising model in a transverse field) has been used extensively to investigate ferroelectric phase transitions. Numerical Monte Carlo simulations of phase transitions in three-dimensional Ising systems with variable size (2 ≤ L ≤ 100) have been performed. The susceptibility maximum and peak temperature Tc 3D(L,Lc) for free and periodic boundary conditions have been determinated and have been observed to be well fitted by simple expressions as a function of L. For L → ∞, as expected [Tc 3D]fbc = [Tc 3D]pbc being fbc = free boundary conditions and pbc = periodic boundary conditions. Scaling plots MLβ/v vs |ϵ|Ll/v for larger L with both boundary conditions were obteined and compared. The scaling function for our data up to L ≈ 100 was examined closely and an excellent fit at T → Tc was obtenied.
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.
Finite-size scaling functions are investigated both for the mean-square magnetization fluctuations and for the probability distribution of the magnetization in the one-dimensional Ising model. The scaling functions are evaluated in the limit of the temperature going to zero (T → 0), the size of the system going to infinity (N → ∞) while N[1 − tanh(J/kBT)] is kept finite (J being the nearest neighbour coupling). Exact calculations using various boundary conditions (periodic, antiperiodic, free, block) demonstrate explicitly how the scaling functions depend on the boundary conditions. We also show that the block (small part of a large system) magnetization distribution results are identical to those obtained for free boundary conditions.
Finite-size scaling functions are investigated both for the mean-square magnetization fluctuations and for the probability distribution of the magnetization in the one-dimensional Ising model. The scaling functions are evaluated in the limit of the temperature going to zero (T -> 0), the size of the system going to infinity (N -> oo) while N[1-tanh(J/k_BT)] is kept finite (J being the nearest neighbor coupling). Exact calculations using various boundary conditions (periodic, antiperiodic, free, block) demonstrate explicitly how the scaling functions depend on the boundary conditions. We also show that the block (small part of a large system) magnetization distribution results are identical to those obtained for free boundary conditions.