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Universality and the five-dimensional Ising model

Henk W. J. Blöte, Erik Luijten

Published Jan 1, 1997• 23 citations
OpenAlex

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 = #. 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...

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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

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

Anisotropy and universality in finite-size scaling: Critical Binder cumulant of a two-dimensional Ising model

Boris Kastening

We reanalyze transfer-matrix and Monte Carlo results for the critical Binder cumulant U* of an anisotropic two-dimensional Ising model on a square lattice in a square geometry with periodic boundary conditions. Spins are coupled between nearest-neighboring sites and between next-nearest-neighboring sites along one of the lattice diagonals. We find that U* depends only on the asymptotic critical long-distance features of the anisotropy, irrespective of its realization through ferromagnetic or antiferromagnetic next-nearest-neighbor couplings. We modify an earlier renormalization-group calculation to obtain a quantitative description of the anisotropy dependence of U*. Our results support our recent claim towards the validity of universal finite-size scaling for critical phenomena in the presence of a weak anisotropy.

arXiv:1209.0105•Physical Review E•Apr 19, 2013•9 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 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

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 = #. 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...
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.
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 reanalyze transfer-matrix and Monte Carlo results for the critical Binder cumulant U* of an anisotropic two-dimensional Ising model on a square lattice in a square geometry with periodic boundary conditions. Spins are coupled between nearest-neighboring sites and between next-nearest-neighboring sites along one of the lattice diagonals. We find that U* depends only on the asymptotic critical long-distance features of the anisotropy, irrespective of its realization through ferromagnetic or antiferromagnetic next-nearest-neighbor couplings. We modify an earlier renormalization-group calculation to obtain a quantitative description of the anisotropy dependence of U*. Our results support our recent claim towards the validity of universal finite-size scaling for critical phenomena in the presence of a weak anisotropy.
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.
. 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...
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.