The Ising Spin Glass in dimension four; non-universality

The Ising Spin Glass in dimension four; non-universality

P. H. Lundow Department of Mathematics and Mathematical Statistics, Umeå University, SE-901 87, Sweden    I. A. Campbell Laboratoire Charles Coulomb, Université Montpellier II, 34095 Montpellier, France
Abstract

Extensive simulations are made on Ising Spin Glasses (ISG) with Gaussian, Laplacian and bimodal interaction distributions in dimension four. Standard finite size scaling analyses near and at criticality provide estimates of the critical inverse temperatures , critical exponents, and critical values of a number of dimensionless parameters. Independent estimates are obtained for and the exponent from thermodynamic derivative peak data. A detailed explanation is given of scaling in the thermodynamic limit with the ISG scaling variable and the appropriate scaling expressions. Data over the entire paramagnetic range of temperatures are analysed in order to obtain further estimates of the critical exponents together with correction to scaling terms. The Privman-Fisher ansatz then leads to compact scaling expressions for the whole paramagnetic regime and for all sample sizes . Comparisons between the d ISG models show that the critical dimensionless parameters characteristic of a universality class, and the susceptibility and correlation length critical exponents and , depend on the form of the interaction distribution. From these observations it can be deduced that critical exponents are not universal in ISGs, at least in dimension four.

pacs:
75.50.Lk, 05.50.+q, 64.60.Cn, 75.40.Cx

I Introduction

The universality of critical exponents is an important and remarkably elegant property of standard second order transitions, which has been explored in great detail through the Renormalization Group Theory (RGT). The universality hypothesis states that for all systems within a universality class the critical exponents are rigorously identical and do not depend on the microscopic parameters of the model. However, universality is not strictly universal; there are known “eccentric”models which are exceptions and violate the universality rule in the sense that their critical exponents vary continuously as functions of a control variable. The most famous example is the eight vertex model solved exactly by Baxter baxter:71 (). The exceptional physical conditions which apply in this case were discussed in detail in Ref. kadanoff:71 ().

For Ising Spin Glasses (ISGs), the form of the interaction distribution is a microscopic control parameter. It has been assumed tacitly or explicitly that the members of the ISG family of transitions obey standard universality rules, following the generally accepted statement that “Empirically, one finds that all systems in nature belong to one of a comparatively small number of universality classes”stanley:99 (). One should underline the word “empirically”; we know of no formal proof that universality must hold in ISGs. It was found thirty years ago that the -expansion for the critical exponents gardner:84 () in ISGs is not predictive since the first few orders have a non-convergent behavior and higher orders are not known. This can be taken as an indication that a fundamentally different theoretical approach is required for RGT at spin glass transitions. Indeed ”classical tools of RGT analysis are not suitable for spin glasses” parisi:01 (); castellana:11 (); angelini:13 (), although no explicit theoretical predictions have been made so far concerning universality.

ISG transition simulations are much more demanding numerically than are those on, say, pure ferromagnet transitions with no interaction disorder. The traditional approach in ISGs has been to study the temperature and size dependence of observables in the near-transition region and to estimate the critical temperature and exponents through finite size scaling (FSS) relations after taking means over large numbers of samples. Finite size corrections to scaling must be allowed for explicitly which can be delicate. From numerical data, claims of universality have been made repeatedly for ISGs bhatt:88 (); katzgraber:06 (); hasenbusch:08 (); jorg:08 () even though the estimates of the critical exponents have varied considerably over the years (see Ref. katzgraber:06 () for a tabulation of historic estimates).

On this FSS approach the critical inverse temperature estimates are very important for deducing reliable values for the critical exponents notebeta (). Here we also obtain independent estimates for and for the exponent through the thermodynamic derivative peak (pseudocritical temperature) technique. The estimates for and for from this analysis can be considered to be independent of each other and the correction to scaling only plays a minor role. With the best estimates for in hand the numerical data in the thermodynamic limit (ThL) regime are then analysed with the scaling variable appropriate to ISGs, together with scaling expressions which cover the entire paramagnetic temperature regime rather than being limited to the narrow critical region campbell:06 (). Once values for all critical parameters have been obtained by combining information from FSS, pseudocritical temperature, and ThL data, through the Privman-Fisher ansatz privman:84 () compact scaling expressions can be obtained covering the entire paramagnetic temperature range and all sizes .

From a comparison of the critical values for dimensionless parameters and for the critical exponent values, each of which is characteristic of a universality class, we conclude that the Gaussian, Laplacian and bimodal ISGs in dimension four are not in the same universality class. This counter example to the general rule implies that universality does not hold in ISGs. It is relevant that it has already been shown experimentally that in Heisenberg spin glasses the critical exponents depend on the strength of the Dzyaloshinski-Moriya interaction campbell:10 ().

Ii Ising Spin Glass simulations

The Hamiltonian is as usual

 H=−∑ijJijSiSj (1)

with the near neighbor symmetric distributions normalized to . The Ising spins live on simple hyper-cubic lattices with periodic boundary conditions. We have studied the bimodal (Jd) model with a interaction distribution, Gaussian (Gd) model with a interaction distribution, and Laplacian (Ld) model with a ) interaction distribution, all in dimension . We will compare with published measurements on d ISGs.

The simulations were carried out using the exchange Monte-Carlo method for equilibration using so called multi-spin coding, on (up to ) or for larger individual samples at each size. An exchange was attempted after every sweep with a success rate of at least 30%. At least 40 temperatures were used forming a geometric progression reaching down to for J4d, for G4d and for L4d. This ensures that our data span the critical temperature region which is essential for the FSS fits. Near the critical temperature the step length was at most . The various systems were deemed to have reached equilibrium when the sample average susceptibility for the lowest temperature showed no trend between runs. For example, for this means about sweep-exchange steps.

After equilibration, at least measurements were made for each sample for all sizes, taking place after every sweep-exchange step. Data were registered for the energy , the correlation length , for the spin overlap moments , , , and the corresponding link overlap moments. In addition the correlations between the energy and observables were also registered so that thermodynamic derivatives could be evaluated using the relation where is the energy ferrenberg:91 (). Bootstrap analyses of the errors in the derivatives as well as in the observables themselves were carried out.

Iii Finite size scaling

For the present analysis we have first observed the FSS behavior of various dimensionless parameters, not only the familiar Binder cumulant

 g(β,L)=12(3−[⟨q4⟩][⟨q2⟩])2) (2)

and the correlation length ratio , but also other observables showing analogous critical behavior. One alternative dimensionless parameter

 W=1π−2(π⟨|m|⟩2⟨m2⟩−2) (3)

was introduced in the Ising ferromagnet context in lundow:10 (). In the ISGs can be replaced by so

 Wq=1π−2(π[⟨|q|⟩]2[⟨q2⟩]−2) (4)

In the same spirit we have also introduced the dimensionless parameter

 h(β,L)=1√π−√8(√π[⟨|q3|⟩][⟨q2⟩]3/2−√8) (5)

We have also registered the non-self averaging parameter , the kurtosis of the spin overlap distribution, and the moments of the absolute spin overlap distribution, together with the variance and kurtosis of the link overlap distribution lundow:13 (); lundow:12a (). Only a fraction of these data are reported here.

Analyses with the traditional technique of estimating crossing point temperatures , defined through , have disadvantages. The statistical errors in both sizes and contribute to the uncertainty of the crossing temperature; the scaling correction to the smaller size dominates and is combined with the numerical difficulty in equilibrating at the larger size . Instead we interpolate using the data points for each so as to obtain sets of data for a few fixed in the critical region, after making a first rough estimate of . (It is important to span the range of temperatures on both sides of the true ). We then make a global fit with the standard FSS expression, valid in the critical region if there is a single dominant scaling correction term :

 U(β,L)=U(βc,∞)+AL−ω+B(β−βc)L1/ν (6)

The fit uses all the FSS region data, and gives output estimates, with error bars, for and the critical exponents and . The parameters of Eq. (6), and their error bars at the 95% level, were found by using Mathematica’s built-in routines for nonlinear model fitting. The data points for each were obtained at some 20 fixed near , using cubic spline interpolation. Thus the six parameters are based on some 200-300 data points from different and . The quality of the fit is checked by looking at not only the adjusted R-square index, which is always extremely close to 1 in our fits. A better test is to see whether the distribution of deviations between data and fitted model, so called standardized residuals, can be considered zero by using Mathematica’s built-in zero-location tests, using the sign-test or T-test. The fitted models reported here pass these tests at the 5% level.

Iv Thermodynamic derivative peaks analysis

The thermodynamic derivative (pseudocritical temperature) analysis can be an efficient method for analyzing data in a ferromagnet or an ISG. Near criticality in a ferromagnet for many observables the heights of the peaks of the thermodynamic derivatives scale for large as ferrenberg:91 (); weigel:09 ()

 max[∂U(β,L)∂β]∝L1/ν(1+aL−ω/ν) (7)

A peak height against log-log plot tends linearly to at large and so provides an estimate for directly without the need for a value of as input. Corrections to scaling only play a minor role.

The temperature location of the derivative peak also scales as

 βc−βmax(L)∝L−1/ν(1+bL−ω/ν) (8)

so as both and vary as at large , a plot of the peak locations against the inverse peak heights tends linearly to at large . This estimate is independent of the FSS estimate.

The observables used for ferrenberg:91 () can be for instance the Binder cumulant , the logarithm of the finite size susceptibility , or the logarithm of the absolute value of the magnetisation . Each of these data sets can give independent estimates of and without any initial knowledge of either parameter. For a ferromagnet, Ferrenberg and Landau ferrenberg:91 () find this form of analysis significantly more accurate than the standard Binder cumulant crossing approach.

All plots of this type with different observables should extrapolate consistently to the true , with the confluent correction only appearing as a small modification to the straight line. Provided that the peaks for the chosen observable fall reasonably close to , these data are in principle much simpler to analyse than those from the Binder crossing technique where one must estimate simultaneously and together with the strength and the exponent of the leading correction term. For ISGs very much the same thermodynamic differential peak methodology can be used as in the ferromagnet. As far as we are aware this analysis has not been used previously in the ISG context.

V Thermodynamic limit scaling variables and expressions

We will give a detailed discussion of scaling in ferromagnets and in ISGs in the thermodynamic limit (ThL) regime, using the scaling variable in ferromagnets and in ISGs campbell:06 (), as this approach has been widely misunderstood or simply ignored.

The ThL regime concerns all the data for each which obey the condition where is the infinite sample size correlation length for inverse temperature . In this regime all observables are independent of and so are equal to their infinite size values as measured for instance by long series HTSE sums. A rule of thumb for this regime is . In order to estimate critical exponents, an extrapolation to criticality at must complete the overall fit to ThL data taken at finite .

In the literature, numerical data on critical transition phenomena are almost always analysed using as the scaling variable, following the standard critical regime prescription of the Renormalization Group Theory (RGT). In particular scaling has been used in recent ISG studies hasenbusch:08 (); banos:12 (); baity:13 (). However with this variable, explicit analyses of numerical data are limited to the finite size scaling regime within a narrow critical region in temperature; diverges at infinite temperature, so when is used outside the critical temperature region, corrections to scaling inevitably proliferate.

In ferromagnets, for well over fifty years the scaling variable has been used for the analysis of the high temperature scaling expansion (HTSE) for the susceptibility; in particular the scaling variable was used in the original discussion of confluent corrections to scaling in ferromagnets by Wegner wegner:72 () which led to the important ThL susceptibility expression

 χ(β,∞)=Cχτ−γ(1+aχτθ+bχτ+⋯) (9)

The first correction term is the confluent correction and the second an analytic correction.

In the critical region and become equivalent, but as at infinite temperature (where ) instead of diverging, the expression is well controlled over the whole paramagnetic regime. In principle there can be many correction terms in Eq. (9) but in practice to high precision a leading term and one single further effective correction term are generally sufficient for an analysis at the level of precision of numerical data. This is because the leading terms in the HTSE provide strict closure conditions on the Eq. (9) series. Thus for the canonical Ising ferromagnet in dimension two, there are five or more further well identified correction terms gartenhaus:88 () but in practice with one single weak effective correction term beyond the leading term, the expression Eq. (9) represents the temperature dependence of the ThL susceptibility to high precision from criticality to infinite temperature campbell:08 (). The same remark holds for Ising ferromagnets in dimension three campbell:11 (); lundow:11 (). In ferromagnets the use of as the scaling variable leads to a “crossover”to a high temperature mean field behavior luijten:97 (), which is a pure artefact lundow:11 (). Note that for ferromagnets with zero temperature ordering can be a suitable variable katzgraber:08 ().

Following a protocol well-established in ferromagnets kouvel:64 (); butera:02 () one can define a temperature dependent effective exponent for the susceptibility

 γ(τ)=−∂lnχ(τ)∂lnτ (10)

tends to the critical as , and to exactly as , where is the number of nearest neighbors so in simple [hyper]-cubic lattices.

For the ThL regime can be written

 γ(τ)=γ−aχθτθ+bχyτy1+aχτθ+bχτy (11)

including the leading order confluent scaling term and a further effective higher order correction term. The exact infinite temperature HTSE condition on the fit parameters is

 γ−aχθ+bχy1+aχ+bχ=2dβc (12)

The estimates for the critical and should be consistent with those from FSS and from pseudocritical peak temperature analyses.

Again in Ising ferromagnets, the analogous ThL expression for the second moment correlation length is campbell:06 ()

 ξ(τ)=Cξβ1/2τ−ν(1+aξτθ+⋯) (13)

The factor arises because the generic infinite temperature limit behavior is . The temperature dependent effective exponent is then

 ν(τ)=−∂ln(ξ(τ)/β1/2)∂lnτ (14)

so with a two correction term relation:

 ν(τ)=ν−aξθτθ+bξyτy1+aξτθ+bxiτy (15)

tends to the critical as , and to as for ferromagnets on simple [hyper]-cubic lattices.

Another effective exponent for ferromagnets is which tends to the critical at the large critical temperature limit. Plotted against the plot of this exponent is defined numerically without reference to (as against the plots of and ). In the analogous ISG plots is replaced by and by , see Figs. 6, 13 and 20. The extrapolation giving the estimate for is independent of the estimate for .

In ISGs, because the effective interaction energy parameter is and not , the appropriate inverse “temperature”parameter is not , so the appropriate scaling variable is , (or for the bimodal ISG case). This ISG scaling variable (or ) was used for the analysis of the ThL ISG susceptibility immediately after the introduction of the Edwards-Anderson ISG model singh:86 (); klein:91 (); daboul:04 (); campbell:06 (). With this the ThL susceptibility relations Eqs. (9),(10) and (11) are formally the same as in the ferromagnet.

 χ(τ)=Cχτ−γ(1+aχτθ+⋯) (16)

and tends to the critical as . The exact high temperature limit from HTSE is as in simple [hyper]-cubic lattices.

The appropriate ISG correlation length expression is

 ξ(τ)=Cξβτ−ν(1+aξτθ+⋯) (17)

The factor arises from the generic form of the ISG high temperature series campbell:06 (). The temperature dependent ISG effective exponent is

 ν(τ)=−∂ln(ξ(β)/β)∂lnτ (18)

We know of no HTSE calculations of the second moment correlation length in ISGs which would lead to an exact limit for in ISGs. As an empirical rule deduced from the HTSE analysis of Ref. daboul:04 (), in simple hyper-cubic lattices of dimension this limit is where is the kurtosis of the interaction distribution.

FSS analyses rely mainly on the size dependance of the critical behavior of observables and their derivatives . For the dimensionless parameters such as the cumulant or alternatively the Binder parameter , and the correlation length ratio , the form of the critical size dependence

 U(βc,L)=Uβc,∞+KUL1/ν(1+aUL−ω+⋯) (19)

and the critical derivative expression

 [∂U(β,L)∂β]βc=KU′L1/ν(1+aU′L−ω+⋯) (20)

can be retained unaltered with scaling for very small both for ferromagnets and for ISGs.

It has been pointed out on general grounds katzgraber:06 (); hasenbusch:08 () that the logarithmic derivative of the susceptibility has the form

 ∂χ(β,L)/∂βχ(β,L)=KχL1/ν(1+aχL−ω+⋯)+K1 (21)

No evaluation was proposed for the constant term in Ref. katzgraber:06 (); hasenbusch:08 (). From the leading order scaling finite size expression for campbell:06 () it is easy to show that in a ferromagnet campbell:11 (). In an ISG the constant term in is . As pointed out in Ref. katzgraber:06 (), for many years the published estimates of the exponent in ISGs were wrong by factors of the order of because a constant term was not included in the FSS susceptibility analyses.

As stated above, the ThL regime is limited for each by a condition for which a rule of thumb is , and an extrapolation to criticality must complete the overall fit to all the ThL data in order to estimate critical exponents. The accuracy of this extrapolation depends on a figure of merit, the minimum value of for which the ThL condition still holds for samples of size . This figure of merit in ISGs can be taken to be

 τmin∼(L/6βc)−1/ν (22)

In dimension with and the condition implies if the largest size used is . This corresponds to so to a temperature within of the critical temperature. It should be underlined that in dimension with the appropriate parameters for ISGs, , , to reach would require sample sizes to , well beyond the maximum sizes which have been studied numerically so far in d ISGs. When fitting to obtain the extrapolation, no a priori assumption is made as to the value of the dominant scaling correction exponent, which can be that of the confluent correction (as in the d ferromagnet campbell:11 ()), of an analytic correction (as in the d ferromagnet campbell:08 ()), or of a high order effective correction if the prefactors of the low order terms happen to be very weak. The exponent values and the prefactor for the leading term are obtained from the fit. It can be noted that an exactly equivalent procedure is followed in the traditional fits to FSS data, where extrapolations are made from finite to infinite size. The FSS correction exponent and the ThL correction exponent are related by . This rule provides an important consistency test for FSS and ThL analyses on each system. The strength of the leading correction prefactor is an important parameter for both FSS and ThL analyses, which is rarely quoted explicitly in publications on numerical work on ISGs.

Vi Overall scaling plot for Ising spin glasses

One method of showing ThL and derivative peak data together in ISGs (following a suggestion by K. Hukushima) is to plot against , see Figs. 1, 8 and 15. (One can also plot against ). These plots are purely displays of raw measured data and do not require or any other parameter as input.

Each individual curve consists of data for a given . Following the derivative peak discussion above, on this plot the set of minima points for large extrapolate linearly to the critical point at .

The envelope curve corresponding to the data which for each are in the ThL regime and its extrapolation to the critical point can be fitted by an expression :

 ∂β2∂lnχ(β)=(β2−β2c)(1+aχτθ)γ+(γ−θ)aχτθ (23)

A further correction term can be readily included if needed. The intercept of the fit curve occurs at the critical point where and the initial slope starting at the intercept is . The fit parameters are , , and . The fit must obey the condition that at , , so as is known from the minima point and corrections beyond the leading one are negligible, there are just two free fit parameters. As a bonus, all the data used for the estimates come from temperatures above the critical temperature, where equilibration is easier to achieve than at and below criticality. Thus from an analysis of susceptibility derivative data alone, it is possible to estimate all the principal critical parameters () for each model.

Vii Analytic corrections to scaling in ISGs

It will be noted that the ThL data analyses presented here show no evidence for the presence of an analytic correction term proportional to , which if it exists should become dominant as criticality is approached when the confluent is greater than .

In a ferromagnet the leading analytic term is due to the field dependence of the analytic part of the free energy. We know of no a priori estimate of the strength of such terms in the ISG context, but it is plausible that they are intrinsically weak because the field is only present at higher order. HTSE analyses, particularly the M and M techniques daboul:04 (), should be sensitive to the presence of analytic terms. In the HTSE measurements of Klein et al. klein:91 () an explicit test was made for an analytic correction in the d bimodal ISG. No evidence was found for such a term. In all the more extensive HTSE analyses of Ref. daboul:04 () the leading ThL correction term effective exponent was always significantly greater than .

It can be noted that for dimensional reasons, FSS corrections have exponents related to the equivalent ThL exponents through . So for ISGs in dimension with hasenbusch:08 (); baity:13 (), the leading analytic correction term would have an FSS exponent . The leading correction term estimated from extensive d bimodal ISG numerical data analysis hasenbusch:08 (); baity:13 () has an exponent, , implying a dominant confluent ThL correction with exponent . There is no mention in these publications of any analytic FSS term with an exponent . We conclude empirically that quite generally in ISGs the ThL analytic correction terms proportional to are small or negligible, presumably due to vanishingly weak prefactors.

Working with the scaling variable for ISGs as in hasenbusch:08 (); banos:12 () means that information on critical exponents coming from data temperatures well above criticality is lost. Comments which have been published such as ”The difference between the [ scaling] expressions and the standard expressions is only in the corrections to scaling.” katzgraber:06 () or ”[The scaling] approach might partly take into account the scaling corrections…” hasenbusch:08 () are incorrect and follow from a misunderstanding. They refer to the initial lowest order form of scaling campbell:06 () where corrections to scaling were explicitly left out of the analysis. Full expressions including the Wegner correction terms have been used in the analysis of ferromagnets campbell:08 (); campbell:11 (); lundow:11 () and are used here for ISGs. It is helpful to note that because of exact infinite temperature closure conditions on , , and from HTSE, a potentially infinite set of high temperature corrections can generally be grouped together into a single effective correction.

Viii Privman-Fisher scaling

The Privman-Fisher FSS ansatz Eq. (24) was originally presented privman:84 () in terms of scaling near criticality, with as the scaling variable. With the scaling expressions the ansatz can be applied successfully over the entire paramagnetic temperature range (see Ref. campbell:11 () for a ferromagnet case). Once estimated values for (and possible higher order correction terms if necessary) have been obtained by fits to the ThL regime and extrapolations to criticality, one has explicit expressions for and for the entire paramagnetic regime. Privman-Fisher ansatz privman:84 () scaling plots then can be made for all and all the paramagnetic regime :

 χ(τ,L)χ(τ,∞)=Fχ[L/ξ(τ,∞)]+a(ω,χ)LωGχ[L/ξ(τ,∞)] (24)

and

 ξ(τ,L)ξ(τ,∞)=Fξ[L/ξ(τ,∞)]+a(ω,ξ)LωGξ[L/ξ(τ,∞)] (25)

Scaling, with , should be ”perfect” for all and over the whole paramagnetic temperature range including the critical FSS regime if the critical parameter estimates have been chosen correctly. This overall scaling can be considered to provide an ultimate validation of the coherence of ThL and FSS fits.

In the case of the Privman-Fisher procedure with scaling applied to the cubic Ising ferromagnet susceptibility campbell:11 (), a simple explicit form for the principal Privman-Fisher scaling function was proposed as a further ansatz:

 Fχ(x)=(1−exp(−bx(2−η)/a))a (26)

where , and and are fit parameters. This extremely compact form automatically fulfils the limit conditions for large and small . If this ansatz turns out to be generally applicable, the parameters and as well as should be characteristics of a universality class. For the scaling plot the fit ansatz becomes even simpler :

 Fξ(x)=(1−exp(−bx1/a))a (27)

with different and parameters. The same approach will be applied below to the ISGs in dimension four.

Ix The Gaussian ISG in dimension 4

We will now address the question of specific ISGs in dimension four, starting with the Gaussian interaction distribution. Simulation measurements up to were published on the d Gaussian ISG, together with a d bimodal ISG with diluted interactions ( of the interactions being set to ) jorg:08 (). The critical temperature for the 4d Gaussian ISG was estimated from Binder parameter and correlation length ratio measurements to be so , in agreement with earlier simulation estimates parisi:96 (); ney:98 () and consistent with a high temperature series expansion (HTSE) estimate , i.e. daboul:04 (). The simulation analyses jorg:08 () led to essentially identical exponents for the two systems : and and so through scaling rules jorg:08 (). The HTSE critical exponent estimates were and daboul:04 (). The two systems of Ref. jorg:08 () happen to show small (for the Gaussian) and almost negligible (for the diluted bimodal) corrections to scaling for the Binder parameter, rendering the estimates particularly reliable.

We have repeated the Gaussian measurements of the Binder parameter and the correlation length ratio , and have also measured the dimensionless parameters , Eq. (4), and , Eq. (5), in the critical region. Plots of for chosen fixed as functions of for a fixed are shown in Fig. 2, together with fits as described above, Eq. (6). Due to the weakness of the corrections to scaling for Gaussian interactions ( in Table 1) choosing to be , or made little difference to the output optimal fit parameters. The figures for the other dimensionless parameters are similar and are not presented explicitly for space considerations. The overall fit parameters including uncertainties due to are given in Table 1.

We thus obtain consistent estimates, Table 1, , (taking the average values) together with the infinite size limit dimensionless critical parameter values , , and . The estimates from the Binder cumulant are in agreement with those of Ref. jorg:08 () where (no value for was cited explicitly and , were not measured). The present value is rather more accurate mainly because of a more closely spaced temperature grid and better statistics to higher . The FSS scaling rule at criticality is . Using the present estimate a FSS log-log plot of against gives a straight line of slope , consistent with the estimate of jorg:08 (). From the scaling rule we obtain a FSS estimate .

Thermodynamic derivative peak data are shown in the form of peak location against inverse peak height for the derivatives , , and , Fig. 3. The linear extrapolations to lead consistently to , in full agreement with the FSS estimate. Log-log plots of against have limiting slopes of , so from the scaling rule Eq. (7) again consistent with the FSS estimate. This estimate is independent of the estimate for .

The temperature dependent effective exponents and , Eqs. (10), (18) assuming are shown in Figs. 4 and 5. The ThL regime can be recognized by the condition or becoming independent of , or from the figure of merit Eq. (22). The ThL regime correlation length exponent has only a weak temperature variation. The overall fit to the data in Fig. 5 gives a ThL temperature dependent exponent

 ν(τ)=1.032−0.041⋅1.6τ1.6+0.017⋅3τ31+0.041τ1.6+0.017τ3 (28)

or

 ξ(β,∞)=0.95βτ−1.032(1+0.041τ1.6+0.017τ3) (29)

so with and a weak higher order contribution. The correction exponent estimate is compatible with the HTSE value daboul:04 ().

The curve evaluated directly from the HTSE series daboul:04 () is exact at high and moderate temperatures, once is estimated, and is fully consistent with the ThL simulation data in the appropriate range, Fig. 4 (Fig. 1 and Fig. 4 are alternative presentations of the same derivative data). The simulation and HTSE data taken together show that for the leading confluent correction term the prefactor is small, and that there is a weak high order effective correction term, so

 γ(τ)=2.44−0.06⋅1.6τ1.6−0.017⋅8τ81+0.06τ1.6−0.017τ8 (30)

i.e.

 χ(β,∞)=0.96τ−2.44(1+0.06τ1.6−0.017τ8) (31)

Hence , and . From the extrapolated derivative , Fig. 6, . This estimate is independent of the estimate for .

These critical exponents : , , estimated from the ThL data are thus in full agreement with the FSS estimates above, and with the slightly less precise FSS estimations of Ref. jorg:08 () : , , . The observed exponents and are also consistent with the HTSE estimates and daboul:04 (). There is thus excellent overall consistency. The critical temperature and exponent values are particularly reliable in this model because of the accidental weakness of the correction to scaling terms.

With the ThL and expressions in hand we make up the Privman-Fisher susceptibility plot , Fig. 7. The whole data set for the entire paramagnetic temperature region and all (so covering the FSS regime, the ThL regime, and the intermediate regime) shows an excellent scaling, which can be fitted by the ansatz Eq. (26) using the fit parameters , , . The Privman-Fisher correlation length plot can be fitted with parameters and in Eq. (27). There is a small finite size scaling correction which we have not attempted to analyse.

We have no data for the diluted bimodal ISG of Ref. jorg:08 () but in view of the fact that the FSS data in that model displayed the figures show even weaker corrections to scaling than for the Gaussian, the exponent and dimensionless parameter estimates , , , are certainly very reliable also.

X The Laplacian interaction ISG in dimension 4

Simulation data were obtained for the Laplacian interaction ISG, , in dimension with the same method and analyses as for the Gaussian, and with samples for each size up to . The FSS of the data, Fig. 9, showed negligible corrections to scaling, and the other dimensionless parameters showed only very weak corrections. In consequence , , and the critical values of the dimensionless parameters could be estimated rather precisely, Table 2. The thermodynamic peak location data, Fig. 10, extrapolate to a estimate which is consistent with the FSS estimate. The susceptibility and correlation length ThL measurements give estimates of the critical exponents and which are also in full agreement with the FSS and pseudo-critical peak data, see Fig. 11 and Fig. 12. They also show weak correction term factors.

The overall conclusions from the analyses for the Laplacian are : , with critical exponents , , and critical values for the dimensionless parameters , , , and . For all the parameters the correction exponent or values are large, we have used for the fits and error estimates. As these are effective exponents the values are not the same for all the parameters. The overall fit to the data with the leading correction term is . The overall fit to the data with a leading and a high order correction term is

 ξ(τ,∞)=0.90βτ−1.02(1+0.1τ1.7+0.007τ10) (32)

The Privman-Fisher FSS fit Eq. (26), Fig. 14, is

 χ(τ,L)χ(τ,∞)=⎛⎜⎝1−exp⎡⎢⎣−0.38(Lξ(τ,∞))2.31.75⎤⎥⎦⎞⎟⎠1.75 (33)

Xi The bimodal ISG in dimension 4

For the d bimodal ISG, from early simulation measurements up to a critical temperature (i.e. ) was estimated marinari:99 () using the Binder parameter crossing point criterion. However, finite size corrections to scaling were not allowed for. The exponent estimates were and . Extensive domain wall free energy measurements to gave an estimate (i.e. ) hukushima:99 (). Inspection of the raw data hukushima () shows strong finite size corrections; extrapolation to larger leads to an infinite size limit definitely greater than . The HTSE critical temperature and exponent estimates are daboul:04 () (i.e. ), and a leading confluent correction exponent .

From bimodal d simulations with impressive numbers of samples up to and to a maximum inverse temperature , Baños et al. banos:12 () give estimates , (so ), (so ) and , all from a FSS analysis with as scaling variable. We can note that the inverse temperatures for the and crossing points should scale linearly with , Eq. 30 of banos:12 (). However, it can be seen that in the crossing point plot, Fig. 9 of Ref. banos:12 (), with the authors’ preferred values of and the scaling against is far from being linear. Baños et al. obtain fits by discarding the lower points or by introducing strong higher order terms. An alternative explanation could be that despite the precautions taken complete equilibration has not quite been achieved for the largest measurements at the largest size (where equilibration is the most difficult), so the to crossing points could be dropped. In this case the remaining data points appear more consistent and indicate a rather larger critical inverse temperature and a rather larger . Our independent data reported below are compatible with these modified values.

We have made measurements in the critical region of the standard finite size Binder cumulant , the dimensionless parameters and , and the normalized correlation length . The present data are for all from to and span the estimated inverse critical temperature . From fits to the , , and we obtain FSS estimates for the critical temperature and the exponent together with the dimensionless critical values , , see Fig. 16, and . The correlation length ratio has much larger corrections to scaling than the other observables, so the data for this parameter was not readily exploitable. As explained above rather than following the traditional crossing point analysis, the raw dimensionless parameter data are fitted directly to Eq. (6) at fixed temperatures in the critical region. Consistent independent fits could be made to the , and results for the data from to . (To minimize higher order corrections we have eliminated from the fitting process). Fits were made fixing the correction exponent at three alternative values, , and . The fit values with error bars which include the effect of the assumed uncertainty in are given in Table 3.

Estimates for the important critical values of the dimensionless parameters for d bimodal ISG are not quoted explicitly in Ref. banos:12 (), but a limit i.e. can be read off the appropriate figure. From the present bimodal analysis, (or ), and .

Thermodynamic derivative peak data are shown in Fig. 17. The straight line extrapolations of peak locations against the inverse peak strengths for observables show consistently values tending to a in the infinite limit. Log-log plots of versus tend to straight lines with limiting slopes corresponding to