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

arXiv:cond-mat/9601130 · Physical Review Letters

Finite-Size Scaling and Universality above the Upper Critical Dimensionality

Erik Luijten, Henk W. J. Blöte

Published Mar 4, 1996• 98 citations
arXivHTMLPDF

Abstract

According to renormalization theory, Ising systems above their upper critical dimensionality ${d}_{u}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4$ have classical critical behavior and the ratio of magnetization moments $Q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}〈{m}^{2}{〉}^{2}/〈{m}^{4}〉$ has the universal value $0.456947\dots{}$. However, Monte Carlo simulations of $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5$ Ising models have been reported which yield strikingly different results, suggesting that the renormalization scenario is incorrect. We investigate this issue by simulation of a more general model in which ${d}_{u}<4$, and a careful analysis of the corrections to scaling. Our results are in perfect agreement with the renormalization theory and provide an explanation of the discrepancy mentioned.

DOI
10.1103/physrevlett.76.1557

Related papers

Universal finite-size scaling analysis of Ising models with long-range interactions at the upper critical dimensionality: Isotropic case

Daniel Grüneberg, Alfred Hucht

We investigate a two-dimensional Ising model with long-range interactions that emerge from a generalization of the magnetic dipolar interaction in spin systems with in-plane spin orientation. This interaction is, in general, anisotropic whereby in the present work we focus on the isotropic case for which the model is found to be at its upper critical dimensionality. To investigate the critical behavior the temperature and field dependence of several quantities are studied by means of Monte Carlo simulations. On the basis of the Privman-Fisher hypothesis and results of the renormalization group the numerical data are analyzed in the framework of a finite-size scaling analysis and compared to finite-size scaling functions derived from a Ginzburg-Landau-Wilson model in zero mode (mean-field) approximation. The obtained excellent agreement suggests that at least in the present case the concept of universal finite-size scaling functions can be extended to the upper critical dimensionality.

arXiv:cond-mat/0310252•Physical Review E•Mar 9, 2004•16 citations

Finite-size scaling above the upper critical dimension revisited: The case of the five-dimensional Ising model

Erik Luijten, Kurt Binder, Henk W. J. Blöte

. Monte Carlo results for the moments #M k # of the magnetization distribution of the nearestneighbor Ising ferromagnet in a L d geometry, where L (4 # L # 22) is the linear dimension of a hypercubic lattice with periodic boundary conditions in d = 5 dimensions, are analyzed in the critical region and compared to a recent theory of Chen and Dohm (CD) [X.S. Chen and V. Dohm, Int. J. Mod. Phys. C (1998)]. We show that this finite-size scaling theory (formulated in terms of two scaling variables) can account for the longstanding discrepancies between Monte Carlo results and the so-called "lowest-mode" theory, which uses a single scaling variable tL d/2 where t = T/T c - 1 is the temperature distance from the critical temperature, only to a very limited extent. While the CD theory gives a somewhat improved description of corrections to the "lowest-mode" results (to which the CD theory can easily be reduced in the limit t # 0, L ##, tL d/2 fixed) for the fourth-order cum...

•Jan 1, 1998•42 citations

Universality and the five-dimensional Ising model

Henk W. J. Blöte, Erik Luijten

We solve the long-standing discrepancy between Monte Carlo results and the renormalization prediction for the Binder cumulant of the five-dimensional Ising model. Our conclusions are based on accurate Monte Carlo data for systems with linear sizes up to L=22. A detailed analysis of the corrections to scaling allows the extrapolation of these results to L=\infinity. Our determination of the critical point, K_c=0.1139150 (4), is more than an order of magnitude more accurate than previous estimates.

arXiv:cond-mat/9702172•Europhysics Letters (EPL)•Jun 10, 1997•41 citations

Universality and the five-dimensional Ising model

Henk W. J. Blöte, Erik Luijten

. -- We solve the long-standing discrepancy between Monte Carlo results and the renormalization prediction for the Binder cumulant of the five-dimensional Ising model. Our conclusions are based on accurate Monte Carlo data for systems with linear sizes up to L = 22. A detailed analysis of the corrections to scaling allows the extrapolation of these results to L = #. Our determination of the critical point, Kc =0.1139150 (4), is more than an order of magnitude more accurate than previous estimates. Introduction. -- A much-debated issue in recent years is the question of universality of the five-dimensional Ising model [1]-[8]. This question focuses on the value of the renormalized coupling constant g at criticality, which is related to the Binder cumulant B [9]. For d- dimensional systems with periodic boundary conditions, hypercubic geometry and d # d u , where d u denotes the upper critical dimension, renormalization theory [10] predicts that this cumulant has a universal value. Howe...

•Jan 1, 1997•23 citations

Comment on “Finite-size scaling of the 5D Ising model"

Erik Luijten

In a recent letter [1], Mon addresses the disagreement between a renormalization prediction [2] and Monte Carlo (MC) analyses of the critical value of the Binder cumulant in the five-dimensional Ising model. Whereas there is firm evidence supporting his overall conclusion, namely that the discrepancy can be explained by strong finite-size corrections [3], there is no such evidence for Mon's identification of the nature of this correction. This is highly relevant for the discussion of this problem, because neglecting the dominant correction to scaling can evidently explain the incorrect conclusions drawn from previous simulations.

Europhysics Letters (EPL)•Mar 1, 1997•9 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
According to renormalization theory, Ising systems above their upper critical dimensionality du=4{d}_{u}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4du​=4 have classical critical behavior and the ratio of magnetization moments Q=〈m2〉2/〈m4〉Q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}〈{m}^{2}{〉}^{2}/〈{m}^{4}〉Q=〈m2〉2/〈m4〉 has the universal value 0.456947…0.456947\dots{}0.456947…. However, Monte Carlo simulations of d=5d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5d=5 Ising models have been reported which yield strikingly different results, suggesting that the renormalization scenario is incorrect. We investigate this issue by simulation of a more general model in which du<4{d}_{u}<4du​<4, and a careful analysis of the corrections to scaling. Our results are in perfect agreement with the renormalization theory and provide an explanation of the discrepancy mentioned.
We investigate a two-dimensional Ising model with long-range interactions that emerge from a generalization of the magnetic dipolar interaction in spin systems with in-plane spin orientation. This interaction is, in general, anisotropic whereby in the present work we focus on the isotropic case for which the model is found to be at its upper critical dimensionality. To investigate the critical behavior the temperature and field dependence of several quantities are studied by means of Monte Carlo simulations. On the basis of the Privman-Fisher hypothesis and results of the renormalization group the numerical data are analyzed in the framework of a finite-size scaling analysis and compared to finite-size scaling functions derived from a Ginzburg-Landau-Wilson model in zero mode (mean-field) approximation. The obtained excellent agreement suggests that at least in the present case the concept of universal finite-size scaling functions can be extended to the upper critical dimensionality.
. Monte Carlo results for the moments #M k # of the magnetization distribution of the nearestneighbor Ising ferromagnet in a L d geometry, where L (4 # L # 22) is the linear dimension of a hypercubic lattice with periodic boundary conditions in d = 5 dimensions, are analyzed in the critical region and compared to a recent theory of Chen and Dohm (CD) [X.S. Chen and V. Dohm, Int. J. Mod. Phys. C (1998)]. We show that this finite-size scaling theory (formulated in terms of two scaling variables) can account for the longstanding discrepancies between Monte Carlo results and the so-called "lowest-mode" theory, which uses a single scaling variable tL d/2 where t = T/T c - 1 is the temperature distance from the critical temperature, only to a very limited extent. While the CD theory gives a somewhat improved description of corrections to the "lowest-mode" results (to which the CD theory can easily be reduced in the limit t # 0, L ##, tL d/2 fixed) for the fourth-order cum...
We solve the long-standing discrepancy between Monte Carlo results and the renormalization prediction for the Binder cumulant of the five-dimensional Ising model. Our conclusions are based on accurate Monte Carlo data for systems with linear sizes up to L=22. A detailed analysis of the corrections to scaling allows the extrapolation of these results to L=\infinity. Our determination of the critical point, K_c=0.1139150 (4), is more than an order of magnitude more accurate than previous estimates.
. -- We solve the long-standing discrepancy between Monte Carlo results and the renormalization prediction for the Binder cumulant of the five-dimensional Ising model. Our conclusions are based on accurate Monte Carlo data for systems with linear sizes up to L = 22. A detailed analysis of the corrections to scaling allows the extrapolation of these results to L = #. Our determination of the critical point, Kc =0.1139150 (4), is more than an order of magnitude more accurate than previous estimates. Introduction. -- A much-debated issue in recent years is the question of universality of the five-dimensional Ising model [1]-[8]. This question focuses on the value of the renormalized coupling constant g at criticality, which is related to the Binder cumulant B [9]. For d- dimensional systems with periodic boundary conditions, hypercubic geometry and d # d u , where d u denotes the upper critical dimension, renormalization theory [10] predicts that this cumulant has a universal value. Howe...
In a recent letter [1], Mon addresses the disagreement between a renormalization prediction [2] and Monte Carlo (MC) analyses of the critical value of the Binder cumulant in the five-dimensional Ising model. Whereas there is firm evidence supporting his overall conclusion, namely that the discrepancy can be explained by strong finite-size corrections [3], there is no such evidence for Mon's identification of the nature of this correction. This is highly relevant for the discussion of this problem, because neglecting the dominant correction to scaling can evidently explain the incorrect conclusions drawn from previous simulations.
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).