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arXiv:cond-mat/9901042 · The European Physical Journal B

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

Published May 1, 1999• 65 citations
arXivHTMLPDF
DOI
10.1007/s100510050768

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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
We show numerically that correlation length at the critical point in the five-dimensional Ising model varies with system size LLL as L5∕4{L}^{5∕4}L5∕4, rather than proportional to LLL, as in standard finite-size scaling (FSS) theory. Our results confirm a hypothesis that FSS expressions in dimension ddd greater than the upper critical dimension of 4 should have LLL replaced by Ld∕4{L}^{d∕4}Ld∕4 for cubic samples with periodic boundary conditions. We also investigate numerically the logarithmic corrections to FSS in d=4d=4d=4.
. 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...
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.