Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Journal of Modern Physics C

Is Finite-Size Scaling Universal for Ising Models?

B. Müller

Published Jun 1, 1998• 5 citations
Publisher

Abstract

The sensitivity on shape and boundaries at the Curie point of the universal critical ratio LdM2/χ for the Ising model is confirmed by a comparison of square and rectangular lattices. Tests with three different algorithms give the same results for the universal critical ratio on various lattice types and always the same dependence on the system's aspect ratio.

DOI
10.1142/s0129183198000480

Related papers

Test of universal finite-size scaling in two-dimensional site percolation

Amnon Aharony A, Dietrich Stauffer

Traditional two-scale-factor universality, concerning the number of clusters within a correlation volume near the percolation threshold, is reconfirmed by a comparison of site percolation on square and triangular lattices. We also test universality at the threshold, e.g. of the ratio , where is the correlation length and L the lattice size. At the threshold, the result is sensitive to the system's shape and aspect ratio, boundary conditions, the algorithm, the detailed units measuring and L, and possibly other factors as yet unexplored.

Journal of Physics A Mathematical and General•May 21, 1997•20 citations

Universal ratio of magnetization moments in two-dimensional Ising models

G. Kamieniarz, Henk W. J. Blöte

The authors calculate the universal critical-point ratios of the square of the second and the fourth moment of the magnetization for ferromagnetic Ising models on the square and on the triangular lattices. Periodic boundary conditions are used in accordance with the four-fold and six-fold rotational symmetries of the respective lattices. These results, which are obtained by means of an analysis of finite-size data computed with a transfer-matrix technique, have an accuracy of the order of one millionth. This analysis is also applied to rectangular systems with arbitrary aspect ratios.

Journal of Physics A Mathematical and General•Jan 21, 1993•198 citations

The square lattice Ising model on the rectangle II: finite-size scaling limit

Alfred Hucht

Abstract Based on the results published recently (Hucht 2017 J. Phys. A: Math. Theor . 50 065201), the universal finite-size contributions to the free energy of the square lattice Ising model on the L × M rectangle, with open boundary conditions in both directions, are calculated exactly in the finite-size scaling limit L , M → ∞ , T → T c , with fixed temperature scaling variable x ∝ ( T / T c − 1 ) M and fixed aspect ratio ρ ∝ L / M . We derive exponentially fast converging series for the related Casimir potential and Casimir force scaling functions. At the critical point T = T c we confirm predictions from conformal field theory (Cardy and Peschel 1988 Nucl. Phys . B 300 377, Kleban and Vassileva 1991 J. Phys. A: Math. Gen . 24 3407). The presence of corners and the related corner free energy has dramatic impact on the Casimir scaling functions and leads to a logarithmic divergence of the Casimir potential scaling function at criticality.

arXiv:1701.08722•Journal of Physics A Mathematical and Theoretical•Apr 5, 2017•21 citations

Ising universality in three dimensions: a Monte Carlo study

Henk W. J. Blöte, Erik Luijten, Jouke R. Heringa

We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin-1/2 Ising model with nearest-neighbour interactions, a spin-1/2 model with nearest-neighbour and third-neighbour interactions, and a spin-1 model with nearest-neighbour interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbour interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y t =1.587 (2), y h =2.4815 (15) and y i =-0.82 (6). The universal ratio Q=(m 2 ) 2 /(m 4 ) is equal to 0.6233 (4) for periodic systems with cubic symmetry. The critical point of the nearest-neighbour spin-1/2 model is K c =0.2216546 (10).

arXiv:cond-mat/9509016•Journal of Physics A Mathematical and General•Nov 21, 1995•308 citations

Universality in dynamic critical phenomena

Fu-Gao Wang, Chin‐Kun Hu

We use heat bath dynamics to evaluate the dynamic critical exponent $z$ and the dynamic finite-size scaling function of an Ising model on square, planar triangular, and honeycomb lattices. We find convincing evidence that $z$ is universal and, by choosing an aspect ratio and a nonuniversal metric factor for the scaled time of each lattice, we can obtain a universal dynamic finite-size scaling function for the Ising model on the planar lattices. Our results suggest many interesting problems for further research.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 1, 1997•73 citations

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.

•Jan 1, 2000•2 citations
The sensitivity on shape and boundaries at the Curie point of the universal critical ratio LdM2/χ for the Ising model is confirmed by a comparison of square and rectangular lattices. Tests with three different algorithms give the same results for the universal critical ratio on various lattice types and always the same dependence on the system's aspect ratio.
Traditional two-scale-factor universality, concerning the number of clusters within a correlation volume near the percolation threshold, is reconfirmed by a comparison of site percolation on square and triangular lattices. We also test universality at the threshold, e.g. of the ratio , where is the correlation length and L the lattice size. At the threshold, the result is sensitive to the system's shape and aspect ratio, boundary conditions, the algorithm, the detailed units measuring and L, and possibly other factors as yet unexplored.
The authors calculate the universal critical-point ratios of the square of the second and the fourth moment of the magnetization for ferromagnetic Ising models on the square and on the triangular lattices. Periodic boundary conditions are used in accordance with the four-fold and six-fold rotational symmetries of the respective lattices. These results, which are obtained by means of an analysis of finite-size data computed with a transfer-matrix technique, have an accuracy of the order of one millionth. This analysis is also applied to rectangular systems with arbitrary aspect ratios.
Abstract Based on the results published recently (Hucht 2017 J. Phys. A: Math. Theor . 50 065201), the universal finite-size contributions to the free energy of the square lattice Ising model on the L × M rectangle, with open boundary conditions in both directions, are calculated exactly in the finite-size scaling limit L , M → ∞ , T → T c , with fixed temperature scaling variable x ∝ ( T / T c − 1 ) M and fixed aspect ratio ρ ∝ L / M . We derive exponentially fast converging series for the related Casimir potential and Casimir force scaling functions. At the critical point T = T c we confirm predictions from conformal field theory (Cardy and Peschel 1988 Nucl. Phys . B 300 377, Kleban and Vassileva 1991 J. Phys. A: Math. Gen . 24 3407). The presence of corners and the related corner free energy has dramatic impact on the Casimir scaling functions and leads to a logarithmic divergence of the Casimir potential scaling function at criticality.
We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin-1/2 Ising model with nearest-neighbour interactions, a spin-1/2 model with nearest-neighbour and third-neighbour interactions, and a spin-1 model with nearest-neighbour interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbour interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y t =1.587 (2), y h =2.4815 (15) and y i =-0.82 (6). The universal ratio Q=(m 2 ) 2 /(m 4 ) is equal to 0.6233 (4) for periodic systems with cubic symmetry. The critical point of the nearest-neighbour spin-1/2 model is K c =0.2216546 (10).
We use heat bath dynamics to evaluate the dynamic critical exponent zzz and the dynamic finite-size scaling function of an Ising model on square, planar triangular, and honeycomb lattices. We find convincing evidence that zzz is universal and, by choosing an aspect ratio and a nonuniversal metric factor for the scaled time of each lattice, we can obtain a universal dynamic finite-size scaling function for the Ising model on the planar lattices. Our results suggest many interesting problems for further research.
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.