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Ising Model Universality for

arXiv:hep-lat/9310025v1 25 Oct 1993preprint UAB-FT-318hep-lat/9310025Ising Model Universality forTwo-Dimensional Lattices∗Wolfhard Janke1, Mohammad Katoot2 and Ramon Villanova2,31 Institut f¨ur Physik, Johannes Gutenberg-Universit¨at Mainz55099 Mainz, Germany2 Department of Physical Sciences, Embry-Riddle Aeronautical UniversityDaytona Beach, Florida 32114, USA3 Grup de F´ısica Te`orica and IFAE, Facultat de Ci`encies, Universitat Aut`onoma de Barcelona08193 Bellaterra, SpainAbstractWe use the single-cluster Monte Carlo update algorithm to sim-ulate the Ising model on two-dimensional Poissonian random latticesof Delaunay type with up to 80 000 sites.By applying reweightingtechniques and finite-size scaling analyses to time-series data near crit-icality, we obtain unambiguous support that the critical exponents forthe random lattice agree with the exactly known exponents for regularlattices, i.e., that (lattice) universality holds for the two-dimensionalIsing model.∗Work supported in part by The Florida High Technology and Industry Council underContract FHTIC-15423ERAU.

1IntroductionIn numerical simulations of many physical systems random lattices [1, 2] area useful tool to discretize space without introducing any kind of anisotropy.Recent applications of various types of random lattices can be found in a greatvariety of fields, such as quantum field theory or quantum gravity [2, 3, 4],the statistical mechanics of membranes [5], diffusion limited aggregation [6],or growth models of sandpiles [7], to mention a few. As a consequence of thepreserved rotational (or more generally, Poincar´e) invariance, spin systemsor field theories defined on random lattices are expected to reach the infinitevolume or continuum limit faster than on regular lattices.

Implicit in thisapproach is the assumption of (lattice) universality which states that sys-tems defined on lattices of different type should exhibit the same qualitativebehaviour once the physical length scale is much larger than the average lat-tice spacing. While this assumption is known to be true for spin systems ondifferent regular lattices, previous numerical work [8, 9] on random latticescould only give weak evidence that universality holds in this case as well.In this note we reconsider the Ising model defined on two-dimensionalPoissonian1 random lattices constructed according to the Voronoi/Delaunayprescription [2, 4].

In previous work on this model, Espriu et al. [9] haveused standard Metropolis Monte Carlo (MC) simulations on lattices withN = 10 000 sites to study the approach of criticality in the low- and high-temperature phase.

Here we report high-statistics simulations in the veryvicinity of the phase transition, using considerably larger lattices of size up toN = 80 000. To achieve the desired accuracy of the data we made extensivelyuse of recently developed greatly refined MC simulation techniques, such ascluster update algorithms [11, 12] and reweighting methods [13].

As a resultof finite-size scaling (FSS) analyses of our data we obtain very strong supportfor (lattice) universality in this model.2ModelAs partition function we takeZ =X{si}e−KE;E = −X⟨ij⟩sisj;si = ±1,(1)1For alternative site distributions see, e.g., Refs. [6, 10].1

where K = J/kBT > 0 is the inverse temperature in natural units and ⟨ij⟩denote nearest-neighbour links of the Delaunay random lattices, computedaccording to the (dual) Voronoi cell construction as described, e.g., in Ref. [4].Following Ref.

[9] we thus take the relative weights of the links to be constant.The lattice sizes studied are N = 5 000, 10 000, 20 000, 40 000, and 80 000,with three replicas for each of the two smallest lattices, and two replicas forN = 20 000. We always employed periodic boundary conditions, i.e., thetopology of a torus.

In this case Euler’s theorem implies q = 6, where q isthe lattice average of the local coordination numbers q that vary for Poisso-nian random lattices between 3 and ∞. All our lattices satisfy this rigorousconstraint, and also the distributions P(q) agree well with numerical evalua-tions of exact integral expressions [14].

The highest coordination number weactually observed in our simulations was q = 13 in the N = 80 000 lattice.To update the spins si we employed the single-cluster update algorithm[12] which is straightforward to adopt to random lattices. From comparativestudies [15] on regular lattices the single-cluster update is expected to bemore efficient than the multiple cluster variant [11].

All runs were performedat K = 0.263, the estimate of the critical coupling Kc as quoted by Espriu etal. [9].

After discarding from 50 000 to 150 000 clusters to reach equilibriumfrom an initially completely disordered state, we generated a further 4 × 106clusters and recorded every 10th cluster measurements of the energy perspin, e = E/N, and the magnetization per spin, m = Pi si/N in a time-series file. From analyses of the autocorrelation functions of e and m2 weobtained at the scale of our measurements integrated autocorrelation timesof ˆτe ≈0.8 −1.3 and ˆτm2 ≈0.7 −0.9, respectively.

Our samples thus consisteffectively of about 200 000 statistically independent data. The statisticalerrors are estimated by deviding the time series into 20 blocks, which arejack-knived to avoid bias problems in reweighted data.3ResultsTo determine the transition point Kc and the correlation length exponent νwe first concentrated on the magnetic Binder parameter [16],UL(K) = 1 −⟨m4⟩3⟨m2⟩2,(2)2

where L ≡√N is defined as the linear length of the lattice in natural units.It is well known [16] that the curves UL(K) for lattices of size L and L′should intersect in points (K×(L, L′), U×(L, L′)) which approach (Kc, U∗)for large L,L′, and the derivatives U′L ≡dUL/dK at these points should scaleasymptotically with L1/ν. Our results for UL(K) obtained from reweightingthe time-series data at K = 0.263 are plotted in Fig.

1. For the small latticesthe curves are an average over the different replicas [17].Taking as estimate for Kc the average of the K×(L, L′) for the threelargest lattices, we obtainKc = 0.2630 ± 0.0002,(3)where the (rough) error estimate reflects also the fluctuations between differ-ent replicas.

The value (3) is in very good agreement with high-temperatureseries expansion analyses (Kc ≈0.26303) [9] and MC simulations in thedisordered phase (Kc = 0.2631(3)) [9].At the critical coupling (3), UL(K) varies only little and an average overall lattice sizes givesU∗= 0.6123 ± 0.0025. (4)At K = Kc −0.0002 and K = Kc + 0.0002 we obtain U∗= 0.6054(25)and U∗= 0.6183(28), respectively.

The value (4) is in very good agreementwith MC estimates for the regular simple square (sq) lattice which are U∗=0.615(10) [18] and U∗= 0.611(1) [19]. This agreement may be taken as afirst indication of lattice universality.To get an estimate for the exponent ν we have computed the effectiveexponentsνeff=ln(L′/L)ln (U′L′(K×)/U′L(K×))(5)for all possible combinations of L and L′.

Since we do not observe any definitetrend of νeffas a function of L and L′, we quote as our final result for ν theaverage over all combinations,ν = 1.008 ± 0.022,(6)where the error estimate is the standard deviation of the νeff. If we consideronly the crossing points with the N = 80 000 curve, the estimate for ν evensharpens to ν = 1.0043 ± 0.0036.

We can thus conclude that our estimate3

of the exponent ν for the random lattice is fully consistent with the exactregular lattice value of ν = 1.The ratio of exponents γ/ν follows from the scaling of the maxima,χ′max(L) ∝Lγ/ν, of the (finite lattice) susceptibilityχ′(K) = K N(⟨m2⟩−⟨|m|⟩2). (7)The curves of χ′(K) obtained by reweighting the primary data at K = 0.263are shown in Fig.

2. It is then straightforward to determine the maximaχ′max for each lattice size L, and a straight line fit through all data points ina log-log plot of χ′max vs L givesγ/ν = 1.7503 ± 0.0059,(8)with a goodness-of-fit parameter [20] of Q = 0.035.

This is again in perfectagreement with the exact value for regular lattices, γ/ν = 1.75. We canthus conclude that universality also holds as far as the exponent ratio γ/ν isconcerned.The locations of the susceptibility maxima, Kχ′max, should scale for largeL according to Kχ′max = Kc + aL−1/ν, where a is a non-universal constant.Assuming ν = 1 and performing a linear fit through the Kχ′max of the threelargest lattices we obtain Kc = 0.262947(77) with Q = 0.24, in good agree-ment with our earlier estimate from the intersection points of the parameterUL.Having estimated ν and γ, all other exponents can in principle be calcu-lated by scaling or hyperscaling relations, e.g., 2β/ν = d−γ/ν, where d is thedimension.

To get an independent estimate for the exponent ratio β/ν wehave considered the FSS behaviour of the magnetization ⟨|m|⟩at its point ofinflection, which is given by ⟨|m|⟩|inf(L) ∝L−β/ν. From a linear fit throughall data points we obtainβ/ν = 0.1208 ± 0.0092,(9)with Q = 0.10.

Also this result is perfectly compatible with the exact valuefor regular lattices, β/ν = 0.125, thus supporting the hyperscaling hypothesisfor random lattices as well.Furthermore, from the asymptotic scaling of the points of inflection,K⟨|m|⟩inf= Kc + a′L−1/ν, we can get another estimate for the critical cou-pling. Assuming again ν = 1, a fit through the points of the three largest4

lattices yields Kc = 0.26304(14) with Q = 0.60, thus confirming our previousestimates.Let us finally consider the specific heat,C = K2N(⟨e2⟩−⟨e⟩2),(10)and the associated critical exponent α. Here hyperscaling predicts α = 2−dν.Since we already know that ν ≈1 we thus expect α ≈0 for two-dimensionalrandom lattices.

The corresponding FSS prediction for the maxima of C isthenCmax(L) = B0 + B1 ln L,(11)with non-universal constants B0 and B1. The semi-log plot in Fig.

3 clearlydemonstrates that our data is consistent with this prediction. A linear fitthrough all data points gives B0 = 0.346(52) and B1 = 0.391(12) withQ = 0.84.

On the other hand, we cannot claim unambiguous support forlogarithmic scaling. In fact, we can even fit the data with a pure power-lawAnsatz, Cmax ∝Lα/ν, yielding α/ν = 0.1824(53) with a similar goodness-of-fit parameter, Q = 0.93, as for the logarithmic fit.

We also tried a non-linearthree-parameter fit to the more reasonable Ansatz Cmax = b0 + b1Lα/ν. Eventhough the exponent ratio α/ν = 0.17(16) then comes out consistent withzero, the errors on all three parameters are much too large to draw a firmconclusion from such a fit.

By means of exact results for the sq lattice [21],we have checked [17] that for the regular lattice the specific heat behaves verysimilar. In both cases one would need much larger lattice sizes to discriminatebetween logarithmic and power-law scaling.As before the peak locations KCmax should scale like KCmax = Kc+a′′L−1/ν.Assuming again ν = 1, we obtain from a fit to the data for the three largestlattices Kc = 0.26295(33) with Q = 0.95, in agreement with the previousestimates.4ConclusionIn summary, we have performed a fairly detailed finite-size scaling studyof the Ising model on two-dimensional Poissonian random lattices of theDelaunay type.Our estimate for the critical coupling derived from theintersection points of the Binder parameter is Kc = 0.2630(2), the inflec-tion points of the magnetization yield asymptotically Kc = 0.26304(14), and5

from the peak locations of the suceptibility and specific heat we extrapolateKc = 0.262947(76) and Kc = 0.26295(33), respectively. These values are ingood agreement with previous simulations in the disordered phase and withanalyses of high-temperature series expansions by Espriu et al.

[9].As usual the specific-heat maxima are difficult to analyze, since theasymptotic finite-size scaling behaviour sets in only for extremely large latticesizes. Our data is consistent with a logarithmic scaling, i.e., with a criticalexponent α = 0, but not yet sufficient to exclude a power-law scaling withα ̸= 0 on a statistically firm basis.

Precisely the same situation is encoun-tered, however, for the (exactly known) specific heat of the regular sq lattice.We take this observation as further support that also for this quantity thereis no violation of universality.Our results for the critical exponents ν, γ and β are much easier tointerpret. They clearly indicate that these exponents have the same values asfor regular lattices, i.e., here we obtain strong support for lattice universalityin the two-dimensional Ising model.As a future project it would be interesting to repeat this study for dy-namical random lattices that satisfy the Voronoi/Delaunay construction atall times [22].

The important question would be whether the critical be-haviour is still governed by the critical exponents of the static random (orregular) lattice considered here, or by the critical exponents predicted bymatrix model theory [23]. For standard dynamically triangulated lattices,which do not satisfy the Voronoi/Delaunay construction, strong numericalevidence for the second alternative was reported recently in Ref.[24].AcknowledgementsR.V.

is supported by a fellowship from the “Centre de Supercomputaci´o deCatalunya”, and W.J. thanks the Deutsche Forschungsgemeinschaft for aHeisenberg fellowship.

Some of the simulations were performed on the SCRIcluster of fast RISC workstations.6

References[1] J.L. Meijering, Philips Res.

Rep. 8, 270 (1953); R. Collins, in PhaseTransitions and Critical Phenomena, Vol. 2, ed.

C. Domb and M.S.Green (Academic Press, London, 1972), p. 271; J.M. Ziman, Models ofDisorder (Cambridge University Press, Cambridge, 1976); C. Itzyksonand J.-M. Drouffe, Statistical Field Theory (Cambridge University Press,Cambridge, 1989), Vol.

2. [2] N.H. Christ, R. Friedberg, T.D.

Lee, Nucl. Phys.

B202, 89 (1982); Nucl.Phys. B210 [FS6], 310, 337 (1982).

[3] C. Itzykson, Fields on Random Lattices, in Progress in Gauge Field The-ories, proceed. Carg`ese Summer School, ed.

G.’t Hooft (Plenum Press,New York, 1983). [4] R. Friedberg and H.-C. Ren, Nucl.

Phys. B235[FS11], 310 (1984); H.-C.Ren, Nucl.

Phys. B235[FS11], 321 (1984).

[5] F. David and J.-M. Drouffe, Nucl. Phys.

B (Proc. Suppl.) 4, 83 (1988).

[6] C. Moukarzel and H.J. Herrmann, J. Stat.

Phys. 68, 911 (1992).

[7] H. Puhl, preprint HLRZ 16/93, J¨ulich (1993). [8] H.-C. Hege, Diploma thesis, FU Berlin (1984), unpublished; J.F.

Mc-Carthy, Nucl. Phys.

B275 [FS17], 421 (1986); W.-Z. Li and J.-B.

Zhang,Phys. Lett.

B200, 125 (1988); H. Koibuchi, Z. Phys. C39, 443 (1988);H. St¨uben, Diploma thesis, FU Berlin (1989), unpublished; H. St¨uben,H.-C. Hege, and A. Nakamura, Phys.

Lett. B244, 473 (1990); J.-B.Zhang and D.-R. Ji, Phys.

Lett. A151, 469 (1990).

[9] D. Espriu, M. Gross, P.E.L. Rakow, and J.F.

Wheater, Nucl. Phys.B265[FS15], 92 (1986).

[10] G. Le Ca¨er and J.S. Ho, J. Phys.

A23, 3279 (1990). [11] R.H. Swendsen and J.-S. Wang, Phys.

Rev. Lett.

58, 86 (1987). [12] U. Wolff, Phys.

Rev. Lett.

62, 361 (1989).7

[13] A.M. Ferrenberg and R.H. Swendsen, Phys. Rev.

Lett. 61, 2635 (1988);and Erratum, ibid.

63, 1658 (1989). [14] J.-M. Drouffe and C. Itzykson, Nucl.

Phys. B235[FS11], 45 (1984).

[15] U. Wolff, Phys. Lett.

B228, 379 (1989);C.F. Baillie and P.D.

Coddington, Phys. Rev.

B43, 10617 (1991). [16] K. Binder, Z. Phys.

B43, 119 (1981). [17] W. Janke, M. Katoot, and R. Villanova, in preparation.

[18] K. Binder and D.P. Landau, Surf.

Sci. 151, 409 (1985).

[19] D.W. Heermann and A.N. Burkitt, Physica A162, 210 (1990).

[20] W.H. Press, B.P.

Flannery, S.A. Teukolsky, and W.T. Vetterling, Nu-merical recipes – the art of scientific computing (Cambridge UniversityPress, Cambridge, 1986).

[21] A.E. Ferdinand and M.E.

Fisher, Phys. Rev.

185, 832 (1969). [22] K.B.

Lauritsen, C. Moukarzel, and H.J. Herrmann, preprint HLRZ26/93, J¨ulich (1993).

For technical details, see also K.B. Lauritsen, H.Puhl, and H.-J.

Tillemanns, preprint HLRZ 30/93, J¨ulich (1993). [23] V.A.

Kazakov, Pis’ma Zh. Eksp.

Teor. Fiz.

44, 105 (1986) [English trans-lation in JETP Lett. 44, 133 (1986)]; Phys.

Lett. A119, 140 (1986); D.V.Boulatov and V.A.

Kazakov, Phys. Lett.

B186, 379 (1987); Z. Burdaand J. Jurkiewicz, Acta Physica Polonica B20, 949 (1989). [24] J. Jurkiewicz, A. Krzywicki, B. Petersson, and B. S¨oderberg, Phys.

Lett.B213, 511 (1988); R. Ben-Av, J. Kinar, and S. Solomon, Int. J. Mod.Phys.

C3, 279 (1992); S.M. Catterall, J.B. Kogut, and R.L.

Renken,Phys. Rev.

D45, 2957 (1992); C.F. Baillie and D.A.

Johnston, Mod.Phys. Lett.

A7, 1519 (1992).8

Figure HeadingsFig. 1: The parameter UL(K) vs the inverse temperature K for randomlattices of size L =√N with N = 5 000, 10 000, 20 000, 40 000 and80 000.

The curves are obtained by reweighting the time-series data atK = 0.263 (≈Kc).Fig. 2: The (finite lattice) susceptibility χ′(K) for the same random latticesas in Fig.

1. The curves are obtained by reweighting the time-seriesdata at K = 0.263 (≈Kc).Fig.

3: Finite-size scaling plot of the specific-heat maxima Cmax vs ln L,where L =√N.The solid straight line shows the least-squares fitCmax = B0 + B1 ln L, with B0 = 0.346(52) and B1 = 0.391(12).9

This figure "fig1-1.png" is available in "png" format from:http://arxiv.org/ps/hep-lat/9310025v1

This figure "fig1-2.png" is available in "png" format from:http://arxiv.org/ps/hep-lat/9310025v1

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