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CRITICAL BEHAVIOUR OF THE 3D GROSS-NEVEU

arXiv:hep-lat/9310020v1 19 Oct 1993BI-TP-93/31AZPH-TH/93-19SPhT 93/053CRITICAL BEHAVIOUR OF THE 3D GROSS-NEVEUAND HIGGS-YUKAWA MODELSL. K¨arkk¨ainena, R. Lacazeb, P. Lacockc, and B. PeterssoncaDepartment of Physics, University of Arizona,Tucson, AZ 85721, USAbService de Physique Th´eorique de Saclay,91191 Gif-sur-Yvette Cedex, France.cFakult¨at f¨ur Physik, Universit¨at Bielefeld,Postfach 100131, 33501 Bielefeld, GermanyAbstractWe measure the critical exponents of the three dimensional Gross-Neveu model withtwo four-component fermions.

The exponents are inferred from the scaling behaviourof observables on lattice sizes 83, 123, 163, 243, and 323. We find that the model has asecond order phase transition with ν = 1.00(4) and 2 −η = γ/ν = 1.246(8).

We alsocalculate these exponents, through a second order ǫ-expansion around four dimensions,for the three dimensional Higgs-Yukawa model, which is expected to be in the sameuniversality class, and obtain γ/ν = 1.237 and ν = 0.948, while recent second order1/Nf-expansion calculations give γ/ν = 1.256 and ν = 0.903. We conclude that theequivalence of the two models remains valid in 3 dimensions at fixed small Nf values.BI-TP-93/31AZPH-TH/93-19SPhT 93/053July 1993

1IntroductionThe Gross-Neveu (GN) model describes fermions with four-fermion interaction [1, 2].It has a global discrete chiral symmetry, which can break down spontaneously to forma chiral condensate. This can be seen as a composite scalar particle that gives a non-zero mass for the fermions.

Due to its simplicity, the GN model has been studiedextensively. In two dimensions it is perturbatively renormalisable and asymptoticallyfree.

In addition, the chiral symmetry is broken.In three dimensions (3d) it is renormalisable if the number of flavours, Nf, is largeenough. Hence, it is also the first renormalisable model known not to be perturbativelyrenormalisable.

It has been proven to be renormalisable and non-trivial in dimensionsbetween 2 and 4 by means of a 1/Nf-expansion [2].It has been suggested that a model with four-fermion interactions, which leads toa spontaneously broken global chiral symmetry with a chiral condensate, could be acandidate for the Higgs particle of the standard model [3]. Near four dimensions (4d)it has been related to Higgs-Yukawa (HY) type models [4, 5].

In fact, in 4d both thestandard electroweak and the GN model (with certain modifications) are trivial andcan be mapped onto each other [4] and in dimensions 2 ≤d ≤4 the GN model andthe HY model are equivalent in the framework of 1/Nf-expansion [5].The purpose of this work is to enlighten the connection between the composite andfundamental Higgs scenarios in a case where the models are not trivial, namely in3d. The 1/Nf expansions for the HY model and the GN model are order by orderequivalent [5].

In order to go beyond the 1/Nf perturbation theory, we ask whetherthis equivalence persists at small Nf’s. To answer this we study the critical propertiesof GN model at small Nf by means of a Monte Carlo (MC) simulation.

To preservediscrete chiral symmetry, the restoration of which we are interested in, we use staggeredfermions on the lattice and choose the smallest possible value of Nf, that is Nf = 2. Afinite-size analysis of the numerical simulation gives the critical coupling and criticalexponent values to be compared to the exponents of the Higgs-Yukawa model.

In orderto obtain a meaningful comparison, one has to go beyond the order ǫ results in the HYmodel [5]. We have therefore extended the calculation and obtain the critical exponentsto order ǫ2.

The relevance of the comparison can furthermore be evaluated by usingrecent results from a 1/Nf-expansion of the GN model [6, 7, 8, 9, 10] to order 1/N2f .The paper is divided into two major parts: section 2 is devoted to the analyticalresults of the Gross-Neveu and Higgs-Yukawa models with the ǫ-expansion while insection 3 we describe the Monte Carlo runs performed. Section 4 is devoted to thecomparison of numerical and analytical results.

Within the uncertainties associated onthe one hand with the statistics of the numerical simulations and on the other handwith the still short series expansion in ǫ and 1/Nf, we confirm the equivalence of thetwo 3d models at small Nf.1

2Analytical ResultsHere we recall a few known properties of the models under study, and present our newanalytical calculations, in particular the fixed Nf ǫ2 expansion of the HY model expo-nents. The 1/Nf expansions of the latter are then compared with the 1/Nf expansionsof the GN model.2.1Continuum Gross-Neveu modelThe continuum GN model with Nf fermion flavours is defined by the LagrangianL =NfXα=1ψα(x) ̸∂ψα(x) + g22NfXα=1ψα(x)ψα(x)2.

(1)Usually an auxiliary scalar field σ is introducedL =NfXα=1ψα(x)[̸∂−gσ(x)]ψα(x) −12σ2(x),(2)which is formally equivalent to (1) upon integration over σ field.The GN model has a discrete chiral symmetryψ →γ5ψ,ψ →−ψγ5,σ →−σ,(3)which in 3d is spontaneously broken at small couplings.The critical exponents for d = 3 to order 1/N2f are [8, 9, 10]1ν = 1 −323π2N + 64(27π2 + 632)27π4N2,(4)γν = 1 +643π2N + 64(27π2 −304)27π4N2,(5)where N = Nf Tr 1 is the total number of fermionic variables.2.2The discretised Gross-Neveu modelWe consider the GN model defined on a 3d symmetric lattice. The discretisation of thecontinuum Lagrangian has to be implemented in such a way as to reproduce the correctsymmetries in the continuum limit [11].

We use staggered fermions, which means thatin 3d there are 8 doublers. Using 4 component spinors, we can assign the componentsto the corners of the cube, leaving 8/4 = 2 continuum flavours from 1 staggered latticefermion per site.

The discretised action readsS = NLXnσ2n2λ +Xn,mNLXα=1χαn(Dnm + Σnm)χαm. (6)2

The staggered fermion matrix D is given byDnm = 12Xjηn,j(δn,m+ˆj −δn,m−ˆj),(7)where the sum j is over the directions (j = 1,2,3) and ηn,j is the staggered fermionphase factorηn,j = (−1)n1+...+nj−1. (8)with periodic boundary condition in d-1 dimensions and antiperiodic in the last onefor a finite lattice.The mass matrix Σ is diagonalΣnm = ¯σδnm(9)and depends on σ field.

We choose a discretisation in which ¯σ is the average of σ atthe six nearest neighbours of the lattice site n.The coupling λ is connected to the continuum coupling g by λ = g2Nf, with Nf = 2NLas explained.One may integrate over the Grassmann variables χαx, ¯χαx to express the partitionfunction in terms of an effective actionSeff = NL [Xxσ2x2λ −Tr ln(D + Σ) ]. (10)The NL = ∞critical value λ0c, where chiral symmetry is restored, can be obtainedfrom the saddle point equation at σ = 0 [6] asλ0c = 0.989.

(11)Taking into account the quadratic fluctuations [12] we can obtain the one loop valueof λc including 1/NL corrections. As we are interested to rather small NL values, wechoose to solve the gap equation of the one-loop effective potential.

Solving this gapequation on the lattice with linear sizes L = 32 and 64 we estimateλ1c(NL = 1) = 0.800(12)λ1c(NL = 6) = 0.938(13)This last value agrees with a direct calculation of the perturbative correction in theinfinite lattice limit.2.3The Higgs-Yukawa modelThe Lagrangian L′ of the HY model has a fourth order interaction term and a kineticterm added for the σ field,L′ =NfXα=1ψα(̸∂+ gσ)ψα + 12m2σ2 + 12(∂µσ)2 + λ44! σ4.

(14)3

This model becomes renormalisable in four dimensions. It has been argued that theterms added are irrelevant for the number of dimensions less than 4 and marginal atd=4.

Hence, the critical behaviour of HY model and GN model should be identical.Due to its renormalisability, the HY model can be studied by means of an ǫ-expansion.The two-loop β functions and anomalous dimension ησ can be obtained from [13] andwe have computed the mass anomalous dimension ηm.βλ4 = −ǫλ4+1(4π)2(3λ24+2Nλ4g2−12Ng4)+1(4π)4(−173 λ34−3Nλ24g2+7Nλ4g4+96Ng6),(15)βg = −ǫ2g +1(4π)2(N/2 + 3)g3 +1(4π)4(−9 + 12N4g5 −2λ4g3 + 112λ24g),(16)ησ =1(4π)2Ng2 +1(4π)4(16λ24 −52Ng4),(17)ηm = −1(4π)2λ4 +1(4π)4(λ24 + λ4Ng2 −2Ng4) −ησ,(18)with d = 4 −ǫ and N = Nf Tr 1.The corresponding fix points to order ǫ2 areg∗2(4π)2 =1(N + 6)ǫ + (N + 66)√N2 + 132N + 36 −N2 + 516N + 882108(N + 6)3ǫ2(19)λ∗4(4π)2 =−N + 6 +√N2 + 132N + 366(N + 6)ǫ+[−√N2 + 132N + 36 (3N3 −43N2 −1545N −1224)+3N4 + 155N3 + 2745N2 −2538N + 7344]54(N + 6)3√N2 + 132N + 36ǫ2. (20)The anomalous dimensions at these fix points give the critical exponents 1/ν = 2 +ηm(g∗, λ∗4) and γ/ν = 2 −ησ(g∗, λ∗4)1ν = 2−5N + 6 +√N2 + 132N + 366(N + 6)ǫ−[√N2 + 132N + 36(3N3 + 109N2 + 510N + 684)−3N4 −658N3 −333N2 −15174N + 4104]54(N + 6)3√N2 + 132N + 36ǫ2,(21)γν = 2 −NN + 6ǫ −(11N + 6)√N2 + 132N + 36 + 52N2 −57N + 3618(N + 6)3ǫ2.

(22)4

2.4Gross-Neveu and Higgs-Yukawa ComparisonThe ǫ = 4 −d expansions of the Gross-Neveu exponents [8, 9, 10] are1ν |GN = 2−ǫ + (−6ǫ + 132 ǫ2 −38ǫ3 + · · ·) 1N+(396ǫ −11252ǫ2 −1140ζ(3) −4018ǫ3 + · · ·) 1N2,(23)γν |GN = 2−ǫ + (6ǫ −72ǫ2 −118 ǫ3 + · · ·) 1N+(−36ǫ + 512 ǫ2 + 192ζ(3) + 2818ǫ3 + · · ·) 1N2,(24)while the 1/N expansion of Eqs. (21 and 22) gives1ν |HY = 2−ǫ + (−6N + 396N2 −26136N3+ · · ·)ǫ+( 132N −11252N2 + 48951N3+ · · ·)ǫ2,(25)γν |HY = 2−ǫ + ( 6N −36N2 + 216N3 + · · ·)ǫ+(−72N + 512N2 + 1215N3 + · · ·)ǫ2.

(26)Up to order ǫ2 and 1/N2 the two models agree as expected. In the GN 1/N-expansion,the ǫ2 terms are comparable to the ǫ ones and the ǫ3 is relatively small in 1/ν and of thesame magnitude in γ/ν.

In contrast, the HY ǫ-expansion shows that the coefficientsof the 1/N expansion are always rapidly increasing, in particular in the case of 1/ν.Thus a resummation for the GN ν and the HY γ/ν has to be made to improve thecorresponding estimates. The necessity of such a resummation is also manifest fromthe importance, at low N, of the 1/N2f contribution for the GN ν, and the ǫ2 one forthe HY γ/ν (about 20% for N=8).Because of the lack of information on asymptotic behaviour, we use a simple Pad´e-Borel resummation [14] with arbitrary choice of function, instead of the more sophis-ticated Borel resummation [15].

For an expansionA(x) = 1 + a1x + a2x2 + O(x3),(27)we writeA(x) = 1xZ ∞0dte−t/x[1 −a1t −(a2/2 −a21)t2]−1. (28)5

These formulae are directly used for the GN ν obtained from Eq. (4) [10], while wefirst expand the HY γ/ν of Eq.

(22) around the N = ∞point asγν |HY = (2 −ǫ)(1 + a1ǫ + a2ǫ2) + O(ǫ3),(29)and resum with Eq. (28) only the second bracket.The comparison of the resulting critical exponents for the two 3d models as a functionof the fermion number N is summarised in Fig.

1, where the dotted lines representthe computation of the GN 1/ν and the HY γ/ν without the resummation proceduredescribed above. The difference between the two models is small except for ν at lowN.

The data point at N = 48 comes from Ref. [6], while those at N = 8 result fromthe simulation described in the next section.3Numercial ResultsHere we present our simulation and the analysis leading to estimates of the criticalindices for the 3d Gross-Neveu model at N = 8.3.1Simulation of the lattice Gross-Neveu modelFor the numerical simulation we consider the effective action Eq.

(10) with NL = 1which corresponds to N = 8, and use an exact Hybrid Monte Carlo algorithm. It hasa point update of 8µs-14µs on a Cray Y-MP, increasing with lattice size.

This is dueto the fact that more conjugate gradient steps are needed to invert the fermion matrixfor large lattices. We perform runs on lattice sizes 83, 123, 163, 243 and 323.

We use 20time steps of length 0.2, except for the largest lattice where the time step is reduced to0.05. As a rule, measurements are carried out every 5th trajectory.

Details regardingthe runs are listed in Table 1. The integrated autocorrelation time τint quoted is thatfor ⟨σ2⟩.To analyse the data we use a variant of multihistogram reweighting analysis whichdoes not require the binning of data [16].

This is used to obtain the values of observ-ables in between the simulated data points. These points are close enough, and thesimulations long enough, to produce bosonic energy distributions that fill the wholecoupling range of interest.3.2The scaling and critical exponents from MC dataLet us first consider the critical coupling and the critical exponent ν, which describesthe behaviour of the correlation length near the phase transition.

We define the renor-malised coupling gR asgR ≡⟨σ2⟩2⟨σ4⟩. (30)6

Table 1: Statistics of the simulations.SizeλTrajectoriesτint830.78750001100008830.815625011000011830.8437500190000181230.78750009000081230.8156250430000111230.843750090000151630.78750009000091630.8156250282000111630.8437500100000172430.8156250432240142430.818437519000043230.81675002671605This expression for gR lacks a constant factor and is the inverse of the usual definition,but this does not affect its scaling properties. The scaling of gR is extremely simple:gR = f(L1/νt),(31)where f denotes a universal scaling function, L is the linear extent of the lattice andt = (λ−1 −λc−1)λc is the reduced coupling (”temperature”).

The subscript c refers tothe infinite volume critical coupling.We can determine λc by noting that, according to Eq. (31), the curves of gR fordifferent lattice sizes should cross at λc, up to scaling violations visible on too smalllattices.

The simulation results are shown in Fig. 2.

The reweighting analysis givescrossings at λc = 0.820(2) for 83 and 123, λc = 0.817(1) for 123 and 163, λc = 0.815(1)for 163 and 243 and λc = 0.817(3) for 243 and 323. We thus conclude that the GNmodel has a second order phase transition λc = 0.815(3).

The renormalised couplingat λc is (gR)c = 0.473(4).The usual way to avoid refering to λc in the critical exponents determination is touse thermodynamic quantities that peak in the scaling region. This is possible since,according to the scaling ansatz, L1/νt is constant at the maxima.

Unfortunately, thereare no quantities whose scaling behaviour provide a direct estimate of the exponent ν.Therefore, if one tries to measure ν one also has to specify the value of t, and thus λc.We can relax this requirement by noting that the scaling formula derived above isindependent of the critical coupling and is valid in the whole critical region as long asthe scaling violations can be neglected. Hence, we can perform a scaling analysis toquantities which do not necessarily peak at the critical coupling.

Moreover, the latterneed not even be specified: invering Eq. (31), L1/νt can be expressed as a function ofgR and the finite size analysis can be made at constant value of gR [17].

We have to pay7

a price, though, since measuring gR can be demanding. It is also crucial to eliminatet, since its uncertainty contributes a lot to the errors in the exponents.In order to extract the critical exponent ν, we consider the logarithmic derivative ofgR with respect to the reduced couplingD ≡∂ln(gR)∂t= L1/ν f ′(L1/νt)f(L1/νt) ≡L1/νF(L1/νt),(32)with F a new universal scaling function.

Inverting the scaling equation of gR for L1/νtwe obtainD = L1/νG(gR),(33)where G is a scaling function.On the other hand, D is a correlator of powers of the σ-field and the bosonic energyS = 1/2 Pn σ2n from the definition of average quantities :D = ⟨S⟩+ ⟨Sσ4⟩⟨σ4⟩−2⟨Sσ2⟩⟨σ2⟩,(34)which can thus be measured numerically (i.e. by means of MC simulations).From this measurement, we determine ν from a fit at constant gR toln D|gR=const = 1/ν ln [L] + const(35)In Fig.

3, the value of ν is shown as a function gR, together with the corresponding χ2value of the fit. The scaling behaviour is realized for the entire fitting rangeχ2 < 0.5(36)for three degrees of freedom.

The errors on ν are coming from a fit which uses theerrors of the original data. These were obtained by a jackknifed reweighting analysis.The estimate obtained with gR = (gR)c = 0.473 isν = 1.00(4).

(37)Notice that the dependence on (gR) is indead weak, and that we did not have to specifyλc in our fits. To see how well our data is actually scaling we display gR versus tL1/νwith our MC estimates of ν and λc in Fig.

4.Using the hyperscaling relations, only two exponents are independent. As a secondexponent we choose 2 −η = γ/ν.

This governs the behaviour of the susceptibility χnear the critical pointχ =Dσ2E−⟨σ⟩2 = Lγ/νg(L1/νt),(38)where g denotes a scaling function.However, in numerical simulations there is aproblem concerning the susceptibility: on finite lattices the average of the σ field isalways zero.8

The use of absolute values in definition (38) could distort the scaling behaviour andmay lead to wrong exponents.To overcome this we used the susceptibility on thesymmetric side [18] λ > λc whereχ =Dσ2E= Lγ/νg(L1/νt). (39)Eliminating again t as in the case for D (and gR), we obtainlnDσ2E|gR=const = γν ln L + const.

(40)and get from a fit to the measured L dependenceγν = 1.246(8).(41)Fig. 5 shows the results of the fit.

Notice that the fit is valid for the whole criticalrange (χ2 < 0.4 for 3 d.o.f) and the deviation as a function of gR is very small asexpected. The value we quote is taken at (gR)c = 0.473.

From Fig. 6 one can see thatthe scaling of the data is excellent with our values of exponents: within error bars allthe data from different lattice sizes lie on the same curve.As a check of consistency with hyperscaling, we can measure other critical exponents.The expectation value of the sigma field acts as an order parameter for the discretechiral symmetry Eq.

(3), which is preserved on the lattice. On a finite lattice, theabsolute value of σ yields an estimate for the combination β/ν, which is shown in Fig.7.

At gR = (gR)c = 0.473 we getβν = 0.877(4),(42)with χ2 < 0.3 for 3 d.o.f. With this value of exponents the quality of scaling is againexcellent, as one can see from Fig.

8.The combination β/ν is connected to γ/ν through the hyperscaling relationβν = 12(d −γν ). (43)Using the value of γ/ν given in (41) this gives β/ν = 0.877(4), which is in completeagreement with estimate obtained above (42).

This agreement is noteworthy since weused the absolute value of σ, which can lead to a distortion of the scaling relation. Atleast in our case we see that it does not.

Also, the definition of the susceptibility withthe absolute value of σ leads to identical results. However, for the standard method ofmeasuring the critical exponents from the scaling behaviour of thermodynamic quan-tities at their peak values, the usage of the absolute value ⟨σ⟩may result in a changein the position of the peaks and thus make the scaling analysis dubious.The heat capacity should give the combination α/ν.

The hyperscaling relation pre-dicts a value of −1. This means that heat capacity does not diverge at the critical9

Table 2: The critical exponents obtained from different methods.The numbers with a star are obtained with resummation.ExponentMCǫǫ21/Nf1/N2fν1.00(4)0.95450.94801.1350.903∗γ/ν1.246(8)1.42851.237∗1.2701.2559point. In fact, it is dominated by its regular part which makes it impossible to extractα/ν.

In order to do this we would need the second derivative of the heat capacity withrespect to t. This quantity would diverge as t−(α/ν+2) ∼t−1. Unfortunately the qualityof the MC data deteriorates as higher order derivatives of the free energy are taken:the 4th derivative is out of reach in the present simulation.All of the previous analysis relied heavily on reweighting the data from a finite setof couplings to a very dense set of couplings.

This enabled us to accurately explore thedependence of the variables on the renormalised coupling gR. We note that both ourmethod of analysis and the number of trajectories used allow us to achieve a betterdetermination of the critical exponents than was achieved in a comparable analysis forNf = 12 [6].4ConclusionsWe have performed a high statistics simulation of the 3d GN model with two flavours of4-spinors.

We show that it has a second order phase transition at λ = 0.815(3). Hence,it leads to a continuum field theory, which is characterised by critical exponents whichwe have measured.

This proves numerically that the GN model is renormalisable inthree dimensions, even for a small number of flavours. The transition point is closeto the 1/Nf expectation λ1c = 0.80, but the 1/N2f correction can be as significant asin 2d calculation [12].

Table 2 displays the results from our simulations together withestimates obtained by other methods: the ǫ-expansion is for the HY model to firstorder by Zinn-Justin [5], and to second order as presented above, the 1/Nf-expansionfor the GN model calculated to one loop by Hands et al. [6] and to order 1/N2 byGracey [7, 8, 10] and by Derkachov et al.

[9]. The second order contributions of the HYγ/ν and the GN ν are large and the corresponding expressions have been resummedas explained in Sect.

2.4.The striking feature of the data is that the HY ǫ-expansion results at the two-looplevel are in very good agreement with the simulation values. Even without resumma-tion, which can be found quite arbitrary, the direct result of γ/ν is not very far fromthe data point as seen in the Fig.

1. The GN second order 1/Nf-expansion works verywell for γ/ν which has a small 1/N2f correction.

Concerning the GN ν, even thoughthe resummed value is not too far offfrom our numerical result, the discrepancy does10

suggest that higher order terms may be important. However, the agreement with theHY ν shows that no new phenomenon appears at small N.As a whole, our results strongly support the conjecture that these models are equiv-alent even in three dimensions, where they are not trivial, and that the propertiesinferred from perturbation theory are valid at low fermion number.AcknowledgmentsWe thank John Gracey for discussions and informing us of his second order resultsprior to publication and Jean Zinn-Justin and Andr´e Morel for fruitful conversations.One of us (R.L.) thanks the hospitality of Crete University where the ǫ2 expansion wasinitiated and E. G. Floratos for discussions and pointing out Ref.

[13]. This project wassupported by the Deutsche Forschungsgemeinschaft and by the U.S. Dept.

of Energygrant No. DE-FG02-8SER40213.

The MC simulations were performed at the HLRZ,J¨ulich, and at Saclay.References[1] D.J. Gross and A. Neveu, Phys.

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Lett. B297 (1992) 293 and private communication.

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Kivel, A.S. Stepanenkoand A.N. Vasil’ev, On calculation of1/n Expansions of Critical Exponents in the Gross-Neveu Model with the Confor-mal Technique, SPhT-93/016.

[10] J.A. Gracey Computation of β′(gc) at O(1/N2) in the O(N) Gross Neveu modelin arbitrary dimensions, Liverpool preprint LTH-312.

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Figure captionsFigure 1.Comparison of critical exponents obtained with different means asfunction of the effective fermion number N. Solid lines for the ǫ2 Higgs-Yukawa model,dashed lines for the 1/N2 Gross-Neveu model, dotted lines for HY γ/ν and GN ν directresults (without resummation). The data point at N = 48 is for ν from Ref.

[6], thosefor N = 8 result from our simulation.Figure 2.The renormalised coupling gR as function of the coupling λ for latticesizes 83, 123, 163, 243 and 323 in order of increasing slopes. The results of simulationswithout the reweighting are shown as circles.Figure 3.The critical exponent ν as function of the value of gR.

The correspondingχ2 plot gives the quality of the fit.Figure 4.The renormalised coupling gR as function of tL1/ν for different latticesizes (83 is labeled with plus, 123 with octagons, 163 with squares, 243 with circles and323 with diamonds). The ν and λc have the measured MC values.Figure 5.The critical exponent γ/ν as function of the critical value of gR.

The χ2gives the quality of the fit.Figure 6.The combination σ2L3−γ/ν and gR as a function of tL1/ν for differentlattice sizes (83 is labeled with plus, 123 with octagons, 163 with squares, 243 withcircles and 323 with diamonds). The ν, γ/ν and λc have the measured MC values.Figure 7.As in Fig.

5, but for the critical exponent β/ν.Figure 8.As in Fig. 6, but for the combination |σ|Lβ/ν.13

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