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Critical Exponents of the Three Dimensional

arXiv:cond-mat/9303045v1 25 Mar 1993Critical Exponents of the Three DimensionalRandom Field Ising ModelH. Rieger* and A. Peter YoungPhysics DepartmentUniversity of CaliforniaSanta Cruz, CA 95064The phase transition of the three–dimensional random field Ising model with adiscrete (±h) field distribution is investigated by extensive Monte Carlo simulations.Values of the critical exponents for the correlation length, specific heat, suscepti-bility, disconnected susceptibility and magnetization are determined simultaneouslyvia finite size scaling.

While the exponents for the magnetization and disconnectedsusceptibility are consistent with a first order transition, the specific heat appears tosaturate indicating no latent heat. Sample to sample fluctuations of the susceptibiltyare consistent with the droplet picture for the transition.PACS numbers: 75.10.H, 05.50, 64.60.C.Submitted to Europhysics LettersTypeset Using REVTEX∗Present address: Institut f¨ur Theoretische Physik, Universit¨at zu K¨oln,5000 K¨oln 41, Germany.1

The three dimensional ferromagnetic Ising model with a random field (RFIM) shows aphase transition to long range order at a critical temperature for small enough field strength[1]. However, the nature of this transition is still unclear; even the question of whether itis first [2] or second [3,4] order remains unsettled.

The droplet theory of Villain [5] andFisher [6] (see also Bray and Moore [7]) develops a self–consistent picture of the transitionas well as a set of scaling relations among the critical exponents. Existing numerical studieshave been unable to test the validity of these scaling relations because not all the exponentswere calculated for any of the relations.

The aim of this paper is to determine all criticalexponents within a single numerical simulation in order to test the scaling relations predictedby the droplet picture.The droplet picture makes other predictions which are relevant to our simulations. Oneis that at and below the transition temperature Tc, the susceptibility is expected to havelarge sample to sample fluctuations [9].

We therefore need to average over a large numberof samples to get a good statistics. Another prediction is thermally activated dynamicalscaling [6,5] resulting in a dramatic slowing down in the critical region.

This means verylong equilibration times. For these reasons we had to confine ourself to modest lattice sizesand the critical exponents will be obtained via finite size scaling.The Hamiltonian of the system is given byH = −X⟨ij⟩SiSj −XihiSi ,(1)where Si = ±1 are Ising spins and the first sum runs over all nearest neighbor pairs on anL × L × L simple cubic lattice with periodic boundary conditions.

The random fields hiin the second sum, running over all sites, take random values with the discrete probabilitydistributionP(hi) = 12δ(hi −hr) + 12δ(hi + hr) . (2)The Monte Carlo (MC) simulations were performed on a transputer array using 40 T414transputers.

We were able to obtain high performance by using a multi-spin coding algo-rithm described in [10] in which each transputer simulates 32 physically different systemsin parallel, each with a different random field realization. This is somewhat different from2

the implementation of multi-spin coding which was applied to the RFIM in [11]. For eachrun at fixed temperature, T, and field–strength, hr, we performed a disorder average over1280 samples.

An average over such a large number of samples is necessary because thesusceptibility is highly non self–averaging, as mentioned before. The simulations were donefor fixed ratio hr/T at different temperatures.To check equilibration we simulated two replicas of the system: one starting from aninitial configuration with all spins up and one with all spins down.

We assumed that thesystem has reached equilibrium when the magnetization measured for both replicas is thesame (within the errorbars). The time needed for equilibration of all 1280 systems variedmuch with the system size and temperature — in case of the largest size (L = 16) we usedup to 0.5·106 MC–steps for equilibration and 1.5·106 MC–steps for measurements.

All of thesamples were equilibrated for L < 10. For larger sizes the number of nonequilibrated samplesgenerally varied between 1% and 3%.

The contribution of these samples was estimated tobe less than the error bars in the points so so no significant error was made by includingthem. The only exception to this was for L = 16, hr/T = 0.5 for which 5% of the sampleswere not equilibrated which gave a significant error in the susceptibility, though not for theother quantities.

We therefore ignored this data point when analyzing the susceptibilty.For each sample and each replica (a,b) we recorded the average magnetization per spin⟨Ma,b⟩, its square ⟨M2a,b⟩, the average energy per spin ⟨Ea,b⟩and its square ⟨E2a,b⟩.Theangular brackets, ⟨. .

.⟩, denote a thermal average for a single random field configuration.From these data we get the specific heat per spin, C, the susceptibility χ, the disconnectedsusceptibility, χdis, and the order parameter, m, as follows:[C]av = Nn[⟨E2⟩]av −[⟨E⟩2]avo/T 2 ,[m]av= [|⟨M⟩|]av ,[χ]av = Nn[⟨M2⟩]av −[⟨M⟩2]avo/T ,[χdis]av = N[⟨M⟩2]av ,(3)where [. .

. ]av denotes the average over different random field configurations.The procedure we used to extract the critical exponents is the following: Let t = T −Tc(the deviation from the critical temperature), then the finite size scaling functions for theabove quantities read:3

T 2[C]av = Lα/ν ˜C(tL1/ν) ,[m]av= L−β/ν ˜m(tL1/ν) ,T [χ]av = L2−η ˜χ(tL1/ν) ,[χdis]av = L4−η ˜χdis(tL1/ν) ,(4)where α is the specific heat exponent, β the order parameter exponent, ν the correlationlength exponent and η and η describe the power law decay of the connected and disconnectedcorrelation functions, see e.g. [1].

Note that the susceptibility exponent is given by γ =(2−η)ν. The scaling function ˜C(x) has a maximum at some value x = x∗.

For each lattice–size we estimate the temperature T ∗(L), where T 2[C]av is maximal. Since t∗(L) L1/ν = x∗weobtain in this way the critical temperature Tc and the correlation length exponent ν from:t∗(L) ≡T ∗(L) −Tc = x∗L−1/ν .

(5)We denote the value of T ∗(L)2C at this temperature T ∗(L) by C∗and similarly forthe other quantities in Eq. (4).In the vicinity of x∗the scaling function ˜C(x) can beapproximated by a parabola.

Therefore three temperatures near the maximum of the specificheat are enough to determine the values of T ∗(L) as well as [C∗]av etc. Our results for theexponents obtained in this way are summarized in Table 1.

For illustration, we show theresults for T ∗(L), [C∗]av, [χ∗]av, [m∗]av and [χ∗dis]av for hr/T = 0.35 in Fig. 1a–d.

Severalcomments have to be made:1) The higher the field strength the harder it is to equilibrate the samples. However,the lower the field the less pronounced is the random field behavior for small lattice sizesbecause of crossover from pure Ising model behavior.

Therefore the investigation of larger aswell as smaller ratios hr/T did not seem to be advisable to us. If the transition is of secondorder and no tricritical point occurs along the critical line (Tc, hc) the exponents should beuniversal, i.e.

independent of the value of hr/T.2) The shift of T ∗(L) with respect to Tc becomes smaller for low field strength, so it isharder to determine the exponent ν. In case of hr/T = 0.25 it was not possible to performan acceptable fit for T ∗(L) according to equation (5).

The values of ν obtained for the otherratios hr/T are somewhat higher than that obtained in [4], where ν = 1.0 ± 0.1.3) We did not find any indication of a divergence, even logarithmic, of the specificheat, so α is negative.This is different from what is found experimentally [1,8], wherethe specific heat diverges logarithmically, corresponding to α = 0. Furthermore, in our4

simulations α seems to get more negative with increasing ratio hr/T. This may indicatethat it is difficult to determine α when α is negative because non-singular (but temperaturedependent) background terms can give a significant contribution to the specific heat.4) The order parameter [m∗]av shows only a very small size dependence, and does notapproach zero but limL→∞[m∗]av ≈0.52, 0.50 and 0.47 for hr/T = 0.5, 0.35 and 0.25,respectively (see the inset of fig.

1d). This indicates that β = 0 and that the transition isfirst order.

This seems to contradict our results for the specific heat since the specific heat isexpected to diverge as Ld [13] at a first order transition, because of the latent heat, whereasour specific heat data seem to saturate for large L. Perhaps the coefficient of Ld is zero(though we see no symmetry reason for this) or is so small that Ld behavior would only beseen for larger sizes.5) For the exponent η we get a best estimate that is slightly higher than 1/2, which is thevalue obtained below Tc and also the value at Tc if the transition is first order [9]. However,the value η = 0.5 is not excluded by our data.

For hr/T = 0.5 we had to exclude the sizeL = 16 from the analysis since 5% of our samples were not equilibrated and the contributionof these samples was larger than the error bars. Our estimates for η are consistent with thatobtained in [3]: η = 0.5 ± 0.1.6) The exponent η for the disconnected susceptibility turns out to be equal to one, sothat the scaling relationβ = (d −4 + η)ν/2(6)is fulfilled (as indicated in the table).

We also see that the Schwartz–Soffer [12] inequalityη ≤2η ,(7)holds as an equality within the error bars. In [4] it was found that η = 1.1 ± 0.1.7) In the droplet picture [5,6] θ = 2−η+η is called the violation of hyperscaling exponent.The hyperscaling relations then have the spatial dimension, d, replaced by d −θ, e.g.2 −α = (d −θ)ν .

(8)As indicated in the table this equality seems not to be fulfilled though the error bars arequite large and our estimate for α might be affected by temperature dependent background5

terms, as discussed above. The estimates of both sides of this equation cannot be madewithout knowing both α and ν but for hr/T = 0.25 we only determined the ratio.

Theentries in the table for 2 −α and (d −θ)ν are therefore left blank for hr/T = 0.25.8) One of the main predictions of the droplet picture [5,6] is a long tail in the distributionof the susceptibility χ for samples of size L at T = Tc [9]. An analysis of this distributionextracted from our results for the 1280 samples confirms the existence of this long tail.Figure 2 shows the histograms for the probability distribution P(χ) close to the temperatureT ∗(T=3.80 for L=8 and T=3.75 for L=16) for hr/T=0.35.

The second moment of thisdistribution [χ∗2]av, shown in the inset of fig. 1c, scales like Lζ, with ζ = 3.8 ± 0.1 (forhr/T = 0.35), which is larger than the square of the mean L4−2η ∼L2.9, but somewhatsmaller than the predicted value ζ = 6 −η −η ≈4.4 [9].

We attribute this difference in ζ tothe number of samples being too small to catch a sufficient number of rare samples whichdominate the higher moments.To conclude, while the data for the magnetization and disconnected susceptibility in-dicate a first order transition fairly convincingly, the specific heat seems to saturate to afinite value so there is no detectable latent heat. It is interesting to ask if the order of thetransition might be different for a different random field distribution, since mean field theorypredicts [14] that the transition becomes first order for large fields for the ±h distribution,but not, for example, for the Gaussian distribution.

Since the multi-spin coding techniquethat we used does not work for a continuous distribution of fields, the answer to this questionneeds an even larger computing effort. Nevertheless we are currently attempting to carryout similar calculations for the Gaussian distribution.

Our results are consistent with theSchwartz-Soffer inequality, Eq. (7), being satisfied as an equality, and support the scalingrelation, Eq.

(6). The scaling relation involving the specific heat, Eq.

(8), does not seem tobe satisfied, though our values for α may only be effective exponents particularly since wefind α is negative and so a more detailed determination of non-singular background termsmight be necessary to determine α accurately. Our results do support the prediction of thedroplet theory that there are large sample to sample variations in the susceptibility at Tc.We would like to thank D. P. Belanger for helpful discussions.

One of the authors (HR)would like to thank the HLRZ at the KFA J¨ulich (Germany) for allocation of computer6

time for the L = 24 run and acknowledges financial support from the DFG (DeutscheForschungsgemeinschaft). The work of APY was supported in part by the NSF grant no.DMR-91-11576.7

REFERENCES[1] For a recent review see T. Nattermann and J. Villain, Phase Transitions 11, 817 (1988)and D. P. Belanger and A. P. Young, J. Magn.

Magn. Mat.

100, 272 (1991). [2] A. P. Young and M. Nauenberg, Phys.

Rev. Lett.

54, 2429 (1985). [3] A. T. Ogielski and D. A. Huse, Phys.

Rev. Lett.

56, 1298 (1986). [4] A. T. Ogielski, Phys.

Rev. Lett.

57, 1251 (1986). [5] J.

Villain, J. Physique 46, 1843 (1985). [6] D. S. Fisher, Phys.

Rev. Lett.

56, 416 (1986). [7] A. J. Bray and M. A. Moore, J. Phys.

C 18, L927 (1985). [8] D. P. Belanger, A. R. King, V. Jaccarino and J. L. Cardy, Phys.

Rev. B 28, 2522 (1983).

[9] I. Dayan, M. Schwartz and A. P. Young, to be published. [10] H. Rieger, J. Stat.

Phys. 70, 1063 (1993).

[11] D. Stauffer, C. Hartzstein, A. Aharony and K. Binder, Z. Phys. B 55, 325 (1984).

[12] M. Schwartz and A. Soffer. Phys.

Rev. Lett.

55, 2499 (1985). [13] M. Nauenberg and B. Nienhuis, Phys.

Rev. Lett.

33, 944 (1974). [14] A. Aharony, Phys.

Rev. B 18, 3318 (1978).8

FIGURESFIG. 1.

The results of least square fits to data obtained by the procedure described in the textfor hr/T = 0.35. The points indicated by diamonds (⋄) correspond to lattice sizes L=4, 6, 8, 10,12, 16 — from right to left in (a) and (b) and left to right in (c) and (d).

With the exceptionof the susceptibility we also inserted data for L = 24 with squares (✷), which we obtained byusing a the same algorithm but on a CRAY Y-MP instead of the transputer array and which areaveraged over only 64 samples. The L = 24 data were not used for the least square fits.

(a) Thetemperature T ∗(L) of the specific heat maximum versus L−1/ν with ν = 1.64 and Tc = 3.552, seeEq. (5).

(b) Specific heat [C∗]av ≡Lα/ν ˜C(x∗) versus L−1/ν with α = 1.04 and ν as in (a), see Eqs. (4) and (5).

The specific heat appears to saturate for L →∞at a value of limL→∞[C∗]av ≃25.3. (c) Susceptibility [χ∗]av ≡L2−η ˜χ(x∗) in a log–log plot.

The slope of the straight line is 2 −ηwith η = 0.53. The inset shows the second moment [χ∗2]av of the probability distribution P(χ)in a log–log plot.The slope of the straight line is ζ = 3.82.

(d) Disconnected susceptibility[χ∗dis]av ≡L4−η ˜χdis(x∗) in a log–log plot. The slope is 4 −η with η = 1.0.

The insert shows themagnetization [m∗]av ≡L−β/ν ˜m(x∗) as a function of a L (the scale of the x–axis is logarithmic).The straight line is the extrapolation to [m∗]av(L = ∞), which is clearly nonzero and so β = 0.FIG. 2.

The histograms for the probability distribution P(χ) of the susceptibility for differentL = 8 and 16 with hr/T = 0.35. The temperatures are chosen to be as close to T ∗(L) as possible:T = 3.80 for L=8 and T = 3.75 for L=16.The y–axes of the inserts are scaled differently to emphasize the long tail of the distribution.

Thisfeature originates in the rare samples with the extremely large values of the susceptibility scalingwith the volume of the system (since 4 −η ≈3).9

TABLESTABLE I. The critical exponents obtained via finite size scaling according to the proceduredescribed in the text.hr/T0.250.350.5Tc3.9±0.13.55±0.053.05±0.05νααν =–0.50±0.051.6–1.0±0.3±0.31.4–1.5±0.2±0.3η0.60±0.030.56±0.030.6±0.1η0.97±0.081.00±0.061.04±0.08θ = 2 −η + η1.6±0.11.6±0.11.6±0.1β000(d −4 + η)ν/20±0.050±0.050±0.05(d −θ)ν2.3±0.52.0±0.62 −α3.0±0.33.5±0.310

3.553.63.653.73.753.83.853.93.9544.0500.050.10.150.20.250.30.350.40.45T∗(L)L−1/νCritical Temperature✸✸✸✸✸✸✷Fig. 1a11

1416182022242600.050.10.150.20.250.30.350.40.45[C∗]avLα/νSpecific Heat✸✸✸✸✸✸✷Fig. 1b12

1010010[χ∗]avLSusceptibility✸✸✸✸✸✸10010001000010000010[χ∗2]avL✸✸✸✸✸✸Fig. 1c13

1010010001000010[χ∗dis]avLDisconnected Susceptibility✸✸✸✸✸✸✷0.460.480.50.520.5410[m∗]avL✸✸✸✸✸✸✷Fig. 1d14

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