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Reexamination of scaling in the five-dimensional Ising model

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M.A. Sumour, D. Stauffer, M.M. Shabat, A.H. El-Astal

Statistical Mechanics · Submitted 3 Mar 2006 · 11 references

Pith's verdict: Magnetization confirmation is solid; the new susceptibility claim is not supported by the data as presented.

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arXiv IDcond-mat/0603087 (version 1)
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CitationA.H. El-Astal, D. Stauffer, M.A. Sumour, M.M. Shabat. "Reexamination of scaling in the five-dimensional Ising model." arXiv:cond-mat/0603087 (2006). https://pith.science/paper/TMIFBTYC
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The paper

The paper claims

In the 5D Ising model, scaling exponents shift with the path to the critical point.

The reading

The paper tries to establish that, in the five-dimensional Ising model, finite-size scaling exponents are not fixed by universality alone but depend on the line along which the lattice approaches its critical point. The authors test this by Monte Carlo simulation on lattices of linear size 10 to 71, holding the combination $A = L^{2}(T-T_c)/T_c$ constant while varying $L$. They report slopes for the average absolute magnetization of about $-1$, $-5/4$ and $-3/2$ for $A = -1$, $0$, $+1$, and susceptibility slopes of about $2$, $5/2$ and $2$. The lower and critical branches double the magnetization slopes, while the above-critical branch is closer to $2$ than to the expected $3$. They also report that the above-$T_c$ susceptibility data are the least reliable. A sympathetic reader would care because it says standard finite-size scaling laws, taken for granted below the upper critical dimension, become path-dependent above four dimensions.

What carries the argument

The load-bearing object is the scaling variable $A = L^{2}(T-T_c)/T_c$, which fixes the line in the $(T-T_c,\,1/L^{2})$ plane along which the system size is changed. The argument is that log-log plots of $\langle |M| \rangle$ and of the susceptibility $\chi = \langle M^{2}\rangle - \langle M\rangle^{2}$ against $L$ at fixed $A$ have slopes that depend on the sign of $A$, and the paper reads those slopes as the effective finite-size exponents. The ratio of the susceptibility above $T_c$ to that below $T_c$ is also used to show that the standard expectation of a constant ratio fails as $|T-T_c|$ grows.

Extended reading notes

Core claim

The central finding is that in five dimensions, at the critical coupling $K_c = 0.113915$ used in the paper, the apparent critical exponents of a finite lattice change when the critical point is approached along lines of constant $A = L^{2}(T-T_c)/T_c$ with $A = -1$ ($T < T_c$), $A = 0$ ($T = T_c$), and $A = +1$ ($T > T_c$). For the magnetization the fitted log-log slopes are $-1$, $-5/4$ and $-3/2$ respectively, and for the susceptibility the slopes are $2$, $5/2$ and $2$; the $A = +1$ branch is noted as fitting better to $2$ than to the value $3$ expected from naive doubling. This is presented as a confirmation of the predicted path-dependent scaling and as an extension of earlier magnetization-only results to the susceptibility. The paper notes that the susceptibility data above $T_c$ are problematic, so the $A = +1$ susceptibility result is the least certain part of the claim.

Load-bearing premise

The results assume that after discarding the first half of the 500,000 Monte Carlo sweeps the measured magnetization and susceptibility are equilibrated, and that lattices of size 10 to 31 already show the asymptotic power-law behavior, so the fitted slopes are the true exponents.

Editorial extensions

If this is right

  • Finite-size scaling exponents reported for the five-dimensional Ising model must be accompanied by the approach path, since different constant-$A$ lines give different slopes.
  • The susceptibility follows the same path dependence as the magnetization, with slopes roughly double for the lower and critical branches, so scaling relations that tie the susceptibility to the magnetization hold along a fixed path.
  • The above-$T_c$ branch deviates from the naive doubling expectation, with a fitted slope near $2$ rather than $3$, so the susceptibility above the critical temperature needs its own scaling description.
  • Simulations above the upper critical dimension that quote a single set of critical exponents are comparing quantities measured along different paths unless the path is fixed.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same constant-$A$ analysis could be applied to the high-dimensional percolation problem mentioned in the introduction; the paper motivates that check but does not perform it.
  • If the path dependence is asymptotic rather than a finite-size artifact, published high-dimensional exponents that do not specify the path may need to be re-read with the path in mind.
  • A direct extension would be to measure the fourth-order cumulant of the magnetization along the same three constant-$A$ lines, since the theory that motivated this study concerned that quantity while this paper reports only magnetization and susceptibility.
Original abstract

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.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,915 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.09 · randomized assignment 0.02 · states quantitative result 0.06 · compares with prior results 0.50 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 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. background 1.00 0.04 0.02 hedged possibility 0.75
s1 Above the upper critical dimension of four, finite-size scaling is more difficult. background 0.96 0.04 0.03 asserted 0.93
s2 Chen and Dohm predicted deviation in the universality of the Binder cumulants for three dimensions and more for the Ising model. background 1.00 0.03 0.07 asserted 0.98
s3 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. background 0.92 0.08 0.12 asserted 0.94
s4 This effect was confirmed by Monte Carlo simulations with Glauber and Creutz kinetics. background 0.91 0.11 0.05 asserted 1.00
s5 Because of the importance of this effect and the unclear situation in the analogous percolation problem, we here reexamine the five-dimensional Glauber kinetics. contribution 0.91 0.33 0.05 asserted 0.77

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 0.92.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Susceptibility versus temperature with different
Figure 2. Ratio of susceptibility above to below Tc, plotted semi-lo￾garithmically versus |Tc − T|/Tc, for L = 10,13,17,23,31, for 10 neighbors up to time = 50000. A = L 2 (T − Tc)/Tc. 3. Simulations and Results From our simulation for different sizes of lattice, by varying the tem￾perature near the critical temperature we get the ratio of susceptibility by dividing the susceptibility of temperature above Tc to the susceptibility be￾low Tc at the same |T −Tc|. Then the ratio of susceptibility was drawn versus |Tc − T|/Tc as shown in figure (2) . It can be seen that the ratio of susceptibility
Figure 3. M| and susceptibility versus |Tc − T|/Tc with fixed size L = 71 of lattice in log-log plot with lines indicating the theoretical slopes -1 and + . ferent L (10,13,17,23,31) for long time (500000). Then the data are obtained as seen in figures 4 and 5. Now if the average of the absolute value of magnetization is taken, and plotted with the size of lattices with all constants A = +1, 0, −1 with log-log plot , the slopes are obtained in figure 4, in agreement with previous theories and simulations [1,8,9]. Then by drawing the susceptibility versus the size L of lattices for the constan
Figure 4. < |M| > versus size L of lattice (10, 13, 17, 23, 31), in log-log plot along constant A = L 2 (T − Tc)/Tc. The upper data correspond to T < Tc (A = −1) with slope –1, the middle data to T = Tc (A = 0) with slope –5/4, and the lower to T > Tc (A = +1) with slope –3/2. DIMENSION IEX(-10:10) BYTE IS(LMAX) DATA TC,MAX,IBM,ISEED/0.113915,500000,1,1/ IBM=2*ISEED -1 CONST=0.0 T=-(TC*CONST/L2)+TC T1=TC/T-CONST C T= TI*(1.0-0.1/(L*L)) PRINT *,L,T,T1,MAX,ISEED LP1=L4+1 L2PL=L5+L4 DO 1 I=1,LMAX 1 IS(I)=1 DO 2 IE=-10,10 5
Figure 5. Susceptibility < M2 > − < M >2 versus size L of lattice (10, 13, 17, 23, 31, (37)), as log-log plot along constant A = L 2 (T −Tc)/Tc, the upper data correspond to T = Tc (A = 0), the middle data to T > Tc (A = 1.0), and the lower data to T < Tc (A = −1). The middle data fit better the indicated slope 2 than the expected slope 3. IBM=IBM*16807 EX=EXP(-2.0*IE*T) 2 IEX(IE)=2147483648.0D0*(4.*EX/(1.0+EX)-2.0)*2147483648.0D0 DO 3 MC=1,MAX DO 4 I=LP1,L2PL IE=IS(I)*(IS(I-1)+IS(I+1)+IS(I-L)+IS(I+L)+IS(I-L2)+IS(I+L2) 1 +IS(I-L3)+IS(I+L3)+IS(I-L4)+IS(I+L4)) IBM=IBM*16807 IF (IBM.LT.IEX(

Figures reproduced from arXiv:cond-mat/0603087 by the authors.

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