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arXiv:cond-mat/9702172 · Europhysics Letters (EPL)

Universality and the five-dimensional Ising model

Henk W. J. Blöte, Erik Luijten

Published Jun 10, 1997• 41 citations
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Abstract

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.

DOI
10.1209/epl/i1997-00284-x

Related papers

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

Finite-size scaling of the 5D Ising model

K. K. Mon

We address the long-standing disagreement between renormalization group (RG) and Monte Carlo (MC) study of finite-size scaling for the order parameter distribution function of the 5D Ising model. MC study finds disagreement between the RG and MC on the fourth-order Binder cumulant. Two new results are presented here. First, we compare the RG predictions for the first and third absolute moment to MC data. Good agreement and finite-size corrections which are much smaller than for the cumulant are found. Second, we observe that the size dependence of the corrections is consistent with the RG square-root law, which is slow. This, together with a large correction amplitude for the cumulant, provide a possible explanation for prior disagreement.

Europhysics Letters (EPL)•May 20, 1996•36 citations

Reply to Comment on “inite-size scaling of the 5D Ising model"

K. K. Mon

In our recent letter [1], we address the disagreement between renormalization group (RG) analytical prediction [2] and Monte Carlo simulation [3] for the magnetization distribution cumulants of the five-dimensional Ising model. The Monte Carlo data [3] for finite lattices ( L ⩽ 17) did not agree with the RG predictions for the large- L limit [2]. We explore the possibility that this difference can be traced to strong finite-size corrections. Therefore, we calculate numerically the RG predictions for the finite-size corrections [1]. Our numerical RG finite-size corrections can be described very well with a square-root dependence L [( − 1)/(2)] . Such a power law was predicted in the RG paper [2] to be the leading correction term. Then, we compare the numerical RG predictions for the finite-size correction to the deviations of the Monte Carlo data from the RG predictions for the large- L limit. We find good agreement for the first and third absolute moment. For the fourth-order Binder cumulant, only the Monte Carlo data for the larger lattices is in agreement with the numerical RG predictions.

Europhysics Letters (EPL)•Mar 1, 1997•6 citations

Reexamination of scaling in the five-dimensional Ising model

Muneer A. Sumour, Dietrich Stauffer, Mohammed Musa Shabat +1 more

In three dimensions, or more generally, below the upper critical dimension, scaling laws for critical phenomena seem well understood, for both infinite and for finite systems. Above the upper critical dimension of four, finite-size scaling is more difficult. Chen and Dohm predicted deviation in the universality of the Binder cumulants for three dimensions and more for the Ising model. This deviation occurs if the critical point T = Tc is approached along lines of constant A = L*L*(T-Tc)/Tc, then different exponents a function of system size L are found depending on whether this constant A is taken as positive, zero, or negative. This effect was confirmed by Monte Carlo simulations with Glauber and Creutz kinetics. Because of the importance of this effect and the unclear situation in the analogous percolation problem, we here reexamine the five-dimensional Glauber kinetics.

arXiv:cond-mat/0603087•Mar 3, 2006•1 citation

Finite-Size Scaling and Universality above the Upper Critical Dimensionality

Erik Luijten, Henk W. J. Blöte

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.

arXiv:cond-mat/9601130•Physical Review Letters•Mar 4, 1996•98 citations
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 address the long-standing disagreement between renormalization group (RG) and Monte Carlo (MC) study of finite-size scaling for the order parameter distribution function of the 5D Ising model. MC study finds disagreement between the RG and MC on the fourth-order Binder cumulant. Two new results are presented here. First, we compare the RG predictions for the first and third absolute moment to MC data. Good agreement and finite-size corrections which are much smaller than for the cumulant are found. Second, we observe that the size dependence of the corrections is consistent with the RG square-root law, which is slow. This, together with a large correction amplitude for the cumulant, provide a possible explanation for prior disagreement.
In our recent letter [1], we address the disagreement between renormalization group (RG) analytical prediction [2] and Monte Carlo simulation [3] for the magnetization distribution cumulants of the five-dimensional Ising model. The Monte Carlo data [3] for finite lattices ( L ⩽ 17) did not agree with the RG predictions for the large- L limit [2]. We explore the possibility that this difference can be traced to strong finite-size corrections. Therefore, we calculate numerically the RG predictions for the finite-size corrections [1]. Our numerical RG finite-size corrections can be described very well with a square-root dependence L [( − 1)/(2)] . Such a power law was predicted in the RG paper [2] to be the leading correction term. Then, we compare the numerical RG predictions for the finite-size correction to the deviations of the Monte Carlo data from the RG predictions for the large- L limit. We find good agreement for the first and third absolute moment. For the fourth-order Binder cumulant, only the Monte Carlo data for the larger lattices is in agreement with the numerical RG predictions.
In three dimensions, or more generally, below the upper critical dimension, scaling laws for critical phenomena seem well understood, for both infinite and for finite systems. Above the upper critical dimension of four, finite-size scaling is more difficult. Chen and Dohm predicted deviation in the universality of the Binder cumulants for three dimensions and more for the Ising model. This deviation occurs if the critical point T = Tc is approached along lines of constant A = L*L*(T-Tc)/Tc, then different exponents a function of system size L are found depending on whether this constant A is taken as positive, zero, or negative. This effect was confirmed by Monte Carlo simulations with Glauber and Creutz kinetics. Because of the importance of this effect and the unclear situation in the analogous percolation problem, we here reexamine the five-dimensional Glauber kinetics.
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.