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Annalen der Physik

A finite size scaling study of the five‐dimensional Ising model

Ch. Rickwardt, P. Nielaba, Kurt Binder

Published Jan 1, 1994• 43 citations

Abstract

Abstract For systems above the marginal dimension d*, where mean field theory starts to become valid, such as Ising models in d = 5 for which d* = 4, hyperscaling is invalid and hence it was suggested that finite size scaling is not ruled by the correlation length ξ (∝ |t| −1/2 in Landau theory, t being the distance from the critical point) but by a “thermodynamic length” l (∝ |t| −2/d). Early simulation work by Binder et al. using nearest neighbor hypercubic L5 lattices with L ⩽ 7 yielded some evidence for this prediction, but the renormalized coupling constant gL = −3 + 〈M4〉/〈M2〉2 at Tc was gL ≈ −1.0 instead of the prediction of Brézin and Zinn‐Justin, gL(Tc) = −3 + Γ4(1/4)/(8 π2) ≈ −0.812. In the present work, we try to shed light on this controversy obtaining much more precise Monte Carlo data using multihistogram techniques and lattices as large as L = 17 (i.e., 1419857 Ising spins). While our value of Tc agrees nicely with recent high temperature series work (KBTc/J ≈ 8.7774 ± 0.0035), the coupling at Tc {gL(Tc) ≈ −0.958 ± 0.050} confirms the work for small lattices and disagrees with the analytical prediction. Hence the data are better consistent with a shift of the effective critical temperature Tc(L) ‐ Tc(∝) ∝ L−d/2 rather than with L−(d−2) according to the analytical theory. If the latter behavior is correct, it can hence be seen for extremely large systems only.

DOI: 10.1002/andp.19945060606

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. -- 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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Abstract For systems above the marginal dimension d*, where mean field theory starts to become valid, such as Ising models in d = 5 for which d* = 4, hyperscaling is invalid and hence it was suggested that finite size scaling is not ruled by the correlation length ξ (∝ |t| −1/2 in Landau theory, t being the distance from the critical point) but by a “thermodynamic length” l (∝ |t| −2/d). Early simulation work by Binder et al. using nearest neighbor hypercubic L5 lattices with L ⩽ 7 yielded some evidence for this prediction, but the renormalized coupling constant gL = −3 + 〈M4〉/〈M2〉2 at Tc was gL ≈ −1.0 instead of the prediction of Brézin and Zinn‐Justin, gL(Tc) = −3 + Γ4(1/4)/(8 π2) ≈ −0.812. In the present work, we try to shed light on this controversy obtaining much more precise Monte Carlo data using multihistogram techniques and lattices as large as L = 17 (i.e., 1419857 Ising spins). While our value of Tc agrees nicely with recent high temperature series work (KBTc/J ≈ 8.7774 ± 0.0035), the coupling at Tc {gL(Tc) ≈ −0.958 ± 0.050} confirms the work for small lattices and disagrees with the analytical prediction. Hence the data are better consistent with a shift of the effective critical temperature Tc(L) ‐ Tc(∝) ∝ L−d/2 rather than with L−(d−2) according to the analytical theory. If the latter behavior is correct, it can hence be seen for extremely large systems only.

The authors review exact studies on finite-sized 2 dimensional Ising models and show that the point for an infinite-sized model at the critical temperature is a point of nonuniform approach in the temperature-size plane. They also illuminate some strong effects of finite-size on quantities which do not diverge at the critical point. They then review Monte Carlo studies for 3 dimensional Ising models of various sizes (L = 2--100) at various temperatures. From these results they find that the data for the renormalized coupling constant collapses nicely when plotted against the correlation length, determined in a system of edge length L, divided by L. They also find that {zeta}{sub L}/L {ge} 0.26 is definitely too large for reliable studies of the critical value, g*, of the renormalized coupling constant. They have reasonable evidence that {zeta}{sub L}/L {approx} 0.1 is adequate for results that are within one percent of those for the infinite system size. On this basis, they have conducted a series of Monte Carlo calculations with this condition imposed. These calculations were made practical by the development of improved estimators for use in the Swendsen-Wang cluster method. The authors found from these results, coupled with a reversed limit computation (size increases with the temperature fixed at the critical temperature), that g* > 0, although there may well be a sharp downward drop in g as the critical temperature is approached in accord with the predictions of series analysis. The results support the validity of hyperscaling in the 3 dimensional Ising model.

The four-dimensional Ising model is studied to probe the possibility of observing in Monte Carlo simulations the logarithmic corrections to the mean-field theory near criticality. The finite-size-scaling behavior for the correlation length is proposed. The scaling forms of the finite-size renormalized coupling, susceptibility, and fourth field derivative at the renormalized tree-level approximation are derived. These results are used to analyze simulation data of the simple hypercubic lattices of sizes 4\ensuremath{\le}L\ensuremath{\le}14, near criticality. Our simulation results are in agreement with the presence of logarithmic corrections and recent field-theoretical calculations of the specific heat. The approach to the nonscattering theory is observed and is consistent with the predicted logarithmic finite-size dependence.

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.

Using single cluster flip Monte Carlo simulations we accurately determine new finite size scaling functions which are expressed only in terms of the variable x=${\ensuremath{\xi}}_{\mathit{L}}$/L, where ${\ensuremath{\xi}}_{\mathit{L}}$ is the correlation length in a finite system of size L. Data for the d=2 and d=3 Ising models, taken at different temperatures and for different size lattices, show excellent data collapse over the entire range of scaling variable for susceptibility and correlation length. From these finite size scaling functions we can estimate critical temperatures and exponents with rather high accuracy even though data are not obtained extremely close to the critical point. The bulk values of the renormalized four-point coupling constant are accurately measured and show strong evidence for hyperscaling. \textcopyright{} 1996 The American Physical Society.

We have studied the three-dimensional Ising spin glass with a ± J distribution by Monte Carlo simulations. Using larger sizes and much better statistics than in earlier work, a finite size scaling analysis shows quite strong evidence for a finite transition temperature, Tc, with ordering below Tc. Our estimate of the transition temperature is rather lower than in earlier work, and the value of the correlation length exponent, ν, is somewhat higher. Because there may be (unknown) corrections to finite size scaling, we do not completely rule out the possibility that Tc = 0 or that Tc is finite but with no order below Tc. However, from our data, these possibilities seem less likely.

. -- 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...