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arXiv:cond-mat/0412150 · Physical Review B

Finite-size scaling of the correlation length above the upper critical dimension in the five-dimensional Ising model

Jeff L. Jones, A. P. Young

Published May 31, 2005• 51 citations
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Abstract

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

DOI
10.1103/physrevb.71.174438

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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.
We present a unified view of finite-size scaling (FSS) in dimension d above the upper critical dimension, for both free and periodic boundary conditions. We find that the modified FSS proposed some time ago to allow for violation of hyperscaling due to a dangerous irrelevant variable applies only to k=0 fluctuations, and "standard" FSS applies to k≠0 fluctuations. Hence the exponent η describing power-law decay of correlations at criticality is unambiguously η=0. With free boundary conditions, the finite-size "shift" is greater than the rounding. Nonetheless, using T-T(L), where T(L) is the finite-size pseudocritical temperature, rather than T-T(c), as the scaling variable, the data do collapse onto a scaling form that includes the behavior both at T(L), where the susceptibility χ diverges like L(d/2), and at the bulk T(c), where it diverges like L(2). These claims are supported by large-scale simulations on the five-dimensional Ising model.
Finite-size scaling above the upper critical dimension is a long-standing puzzle in the field of statistical physics. Even for pure systems various scaling theories have been suggested, partially corroborated by numerical simulations. In the present manuscript we address this problem in the even more complicated case of disordered systems. In particular, we investigate the scaling behavior of the random-field Ising model at dimension D=7, i.e., above its upper critical dimension D_{u}=6, by employing extensive ground-state numerical simulations. Our results confirm the hypothesis that at dimensions D>D_{u}, linear length scale L should be replaced in finite-size scaling expressions by the effective scale L_{eff}=L^{D/D_{u}}. Via a fitted version of the quotients method that takes this modification, but also subleading scaling corrections into account, we compute the critical point of the transition for Gaussian random fields and provide estimates for the full set of critical exponents. Thus, our analysis indicates that this modified version of finite-size scaling is successful also in the context of the random-field problem.
We progress finite-size scaling in systems with free boundary conditions above their upper critical dimension, where in the thermodynamic limit critical scaling is described by mean-field theory. Recent works show that the correlation length is not bound by the system's physical size, a belief that long held sway. Instead, two scaling regimes can be observed — at the critical and pseudo-critical temperatures. We demonstrate that both are manifest for free boundaries. We use numerical simulations of the d = 5 Ising model to analyse the magnetization, susceptibility, magnetization Fourier modes and the partition function zeros. While some of the response functions hide the dual finite-size scaling, the precision enabled by the analysis of Lee–Yang zeros allows this be brought to the fore. In particular, finite-size scaling of leading zeros at the pseudo-critical point confirms recent predictions coming from correlations exceeding the system size. This paper is dedicated to Jaroslav Ilnytskyi on the occasion of his 60th birthday.
Field-theoretical calculations predict that, at the upper critical dimension dc=4{d}_{c}=4dc​=4, the finite-size scaling (FSS) behaviors of the Ising model would be modified by multiplicative logarithmic corrections with thermal and magnetic correction exponents $({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{t},{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{h})=(1/6,1/4)$. Using high-efficient cluster algorithms and the lifted worm algorithm, we present a systematic study to the FSS of the four-dimensional Ising model at criticality in the Fortuin-Kasteleyn (FK) bond and loop representations. In the FK representation, the size of the largest cluster is observed to scale as ${C}_{1}\ensuremath{\sim}{L}^{3}{(lnL)}^{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{h}}$, while the size of the second-largest cluster scales as ${C}_{2}\ensuremath{\sim}{L}^{3}{(lnL)}^{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{h2}}$ with ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{h2}=\ensuremath{-}1/4$ a new correction exponent not yet predicted from field theory. In the loop representation, we observe that the size of the largest loop cluster scales as ${F}_{1}\ensuremath{\sim}{L}^{2}{(lnL)}^{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{t}}$, and the specific heat scales as ${c}_{\mathrm{E}}\ensuremath{\sim}{(lnL)}^{2{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{t}}$. This clarifies the long-standing open question that whether the specific heat for the critical Ising model at dc=4{d}_{c}=4dc​=4 diverges logarithmically.
. 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...