Relaxational dynamics in 3D randomly diluted Ising models

Relaxational dynamics in 3D randomly diluted Ising models

Martin Hasenbusch, Andrea Pelissetto, and Ettore Vicari Dip. Fisica dell’Università di Pisa and INFN, I-56127 Pisa, Italy Dip. Fisica dell’Università di Roma “La Sapienza” and INFN,
I-00185 Roma, Italy
Martin.Hasenbusch@df.unipi.it, Andrea.Pelissetto@roma1.infn.it, Ettore.Vicari@df.unipi.it
Abstract

We study the purely relaxational dynamics (model A) at criticality in three-dimensional disordered Ising systems whose static critical behaviour belongs to the randomly diluted Ising universality class. We consider the site-diluted and bond-diluted Ising models, and the Ising model along the paramagnetic-ferromagnetic transition line. We perform Monte Carlo simulations at the critical point using the Metropolis algorithm and study the dynamic behaviour in equilibrium at various values of the disorder parameter. The results provide a robust evidence of the existence of a unique model-A dynamic universality class which describes the relaxational critical dynamics in all considered models. In particular, the analysis of the size-dependence of suitably defined autocorrelation times at the critical point provides the estimate for the universal dynamic critical exponent. We also study the off-equilibrium relaxational dynamics following a quench from to . In agreement with the field-theory scenario, the analysis of the off-equilibrium dynamic critical behavior gives an estimate of that is perfectly consistent with the equilibrium estimate .

1 Introduction

Randomly diluted uniaxial antiferromagnets, for instance, FeZnF and MnZnF,   have been much investigated experimentally and theoretically [1, 2, 3, 4]. For sufficiently low impurity concentration , these systems undergo a second-order phase transition at . The critical behaviour is approximately independent of the impurity concentration and definitely different from the one of the pure system. These results have been successfully explained by the field-theoretical (FT) renormalisation group (RG), which predicts the presence of a single universality class associated with the paramagnetic-ferromagnetic transition that occurs in Ising systems with quenched random dilution. Monte Carlo (MC) results have been contradictory for a long time, finding model-dependent critical exponents. In [5] this apparent non-universality was shown to be an effect of strong scaling corrections. They are slowly decaying due to the fact that the leading correction-to-scaling exponent is quite small: (see A).

The analyses significantly gain accuracy when using improved Hamiltonians, for which the leading scaling corrections are suppressed for any thermodynamic quantity, and improved estimators, which are such that the leading scaling correction is suppressed for any model in the same universality class. MC simulations of different improved Hamiltonians [6, 7] confirmed that the static critical behaviour is model-independent, in agreement with the FT description, and provided accurate estimates of the static critical exponents, and [6, 8, 5]. They are in good agreement with the FT perturbative results [9] and obtained by the analysis of high-order (six-loop) perturbative expansions (similar results are obtained at five loops [10]). The apparent non-universality observed in previous numerical works was mainly due to the fact that scaling corrections were neglected. As a consequence, previous studies did not really observe the asymptotic critical behaviour and only determined effective exponents depending on all parameters of the investigated model.

In this paper we extend the analysis to the critical dynamics. We consider a purely relaxational dynamics without conserved order parameters, also known as model A [11], as appropriate for uniaxial magnetic materials. Experimental results are reported in [12, 13, 14]. According to the FT RG (see, e.g., [15, 16, 17]), the dynamic behaviour should be the same in all RDIs systems, as is the case for the static criticality. Moreover, the leading scaling corrections appearing in dynamical quantities should be associated with the same RG operators that control the nonasymptotic behaviour of static quantities and thus, they should be characterized by the same exponents as in the static case, i.e., by and . As a consequence, in the case of improved Hamiltonians, leading scaling corrections should also be absent in dynamical quantities. Therefore, the most precise estimates of dynamic universal quantities should be obtained in improved models, as in the static case.

Previous MC studies [18, 19, 20, 21, 22, 23] of equilibrium and off-equilibrium dynamics apparently have not confirmed the FT general predictions. They have mainly focused on the dynamic critical exponent , which characterizes the divergence of the autocorrelation times when approaching the critical point. In most of the cases they have found that is model dependent and have provided estimates which range from to , depending apparently on the method, the favoured values of the dilution parameter , whether it is determined from equilibrium or off-equilibrium simulations, etc. In [20, 22] the universality of was verified, obtaining , but the leading scaling-correction exponent was not consistent with the static one, as predicted by the FT approach. Moreover, this result is inconsistent with the perturbative FT estimate obtained from analyses of the perturbative expansions [24, 25, 26, 15, 27, 28, 29] at two and three loops, which suggest .

In this paper we study three disordered Ising systems whose static critical behaviour belongs to the 3D RDIs universality class: the randomly site-diluted Ising model (RSIM), the randomly bond-diluted Ising model (RBIM), and the Ising model along the paramagnetic-ferromagnetic transition line. Their static critical behaviour was carefully investigated in [6, 7]. In particular, the value of the dilution parameter corresponding to an improved model was determined for each of them. We simulate these models by using the Metropolis algorithm (with a suitable modification in the case of the RSIM and RBIM to avoid ergodicity problems, see B), which does not satisfy any conservation law, and thus allows us to investigate the model-A dynamics. We consider cubic lattices of size with .

The main purpose is to check whether the dynamic critical behaviour is consistent with the FT RG, that is with the existence of a unique model-A universality class for RDIs systems. We focus on the dynamic critical exponent , and determine it in the RSIM, the RBIM, and the Ising model. We find that the autocorrelation times extracted from the autocorrelation function of the magnetic suspectibility at behave as

(1)

with a universal value of the dynamic exponent . Moreover, and are consistent with the static scaling-correction exponents and . We obtain the estimates , , and , respectively for the RSIM, the RBIM, and the Ising model at . They are in good agreement, strongly supporting universality. Results for other values of , both larger and smaller than , are consistent with the estimates of obtained at . We consider

(2)

as our best estimate of for the dynamic model-A universality class of RDIs systems. These results confirm the general picture that comes out of the FT analysis. However, from a quantitative point of view, our estimate significantly differs from the perturbative result at three loops [28, 29]. Apparently, perturbative FT expansions at this order are not able to predict accurately the exponent .

The exponent can also be determined by performing off-equilibrium simulations, since the approach to equilibrium is controlled by the same FT model [30, 31, 16]. As a further check of our result (2), we have performed off-equilibrium MC simulations of the RSIM at , quenching configurations to . The results show that the relaxation to equilibrium is controlled by the same dynamic exponent obtained in equilibrium simulations, i.e. . Moreover, the large-time corrections are consistent with what is predicted by the FT RG, which relates them to the static leading and next-to-leading scaling-correction exponents and . Our results therefore confirm the FT analysis of the off-equilibrium relaxational dynamics [30, 31, 16].

The paper is organized as follows. In Sec. 2 we define the disordered Ising models that are considered in the paper. In Sec. 3 we define the quantities that are measured in the MC simulation and discuss the FT predictions. In Sec. 4 we report the finite-size scaling (FSS) analysis of equilibrium MC simulations of the RSIM, the RBIM, and the Ising model. In Sec. 5 we study the off-equilibrium relaxational critical behaviour of the RSIM, in a quench from to . Finally, we draw our conclusions in Sec. 6. In A we refine the estimate of the leading scaling correction exponent, obtaining . Some details on the MC algorithm are discussed in B.

2 Models

We consider the randomly site-diluted Ising model (RSIM) with Hamiltonian

(3)

where the sum is extended over all nearest-neighbour sites of a simple cubic lattice, are Ising spin variables, and are uncorrelated quenched random variables, which are equal to 1 with probability (the spin concentration) and 0 with probability (the impurity concentration). We also consider the randomly bond-diluted Ising model (RBIM) in which the disorder variables are associated with links rather than with sites. It is defined by the Hamiltonian

(4)

where the couplings are uncorrelated quenched random variables, which take values 0,1 with probability distribution

(5)

Note that the exchange interaction is ferromagnetic in both models.

MC simulations [6, 7] have provided strong numerical evidence that the static critical behaviour of the RSIM (for , where is the site-percolation point, on a simple cubic lattice [32]) and of the RBIM (for , where is the bond-percolation point, on a simple cubic lattice [33]) belong to the same universality class. The most precise estimates of the static critical exponents have been obtained by MC simulations: [6, 8, 5] and . These estimates are in good agreement with the perturbative FT results [9, 10] and , and with experiments [1, 2]. Also the leading and next-to-leading correction-to-scaling exponents have been computed. Here we shall obtain a precise estimate of the leading exponent , , by a combined analysis of the data obtained in [6] and those obtained in the present work; see A for details. As for the next-to-leading exponent, we quote the FT estimate obtained in [6], .

We also consider the Ising model, defined by Hamiltonian (4) with exchange interactions which take values with probability distribution [34]

(6)

Unlike the RSIM and the RBIM, the Ising model is frustrated for any . Nonetheless, the paramagnetic-ferromagnetic transition line that occurs in this model for and also belongs to the RDIs universality class [7]. Here is the location of the magnetic-glassy Nishimori multicritical point, which has been recently computed in [35]: .

In this work we consider a relaxational dynamics without conserved order parameters, i.e. the so-called model A. In lattice systems this dynamics is usually realized by using the Metropolis algorithm. In the case of the RSIM and of the RBIM however, if a sequential updating scheme is used, the Metropolis algorithm with the standard acceptance probability min is not ergodic and thus it does not provide the correct dynamics. An ergodic dynamics is obtained by introducing a simple modification which is described in B. In the Ising model we use the standard Metropolis algorithm with a sequential updating scheme. In this model the specific problem we observed in the RSIM and in the RBIM is not present (note, however, that, to our knowledge, a rigorous proof of ergodicity is lacking for this updating scheme; this is also the case of the pure Ising model).

Note that the algorithm with sequential updating does not satisfy detailed balance and hence does not strictly correspond to a reversible dynamics.111 The Metropolis update is obtained from a single-site update. If is the transition matrix for the update of site , satisfies the detailed-balance condition . However, this does not imply that the dynamics is reversible. Indeed, if lattice sites are updated sequentially, the transition matrix for a full sweep is , where is the number of lattice sites. does not satisfy the detailed-balance condition since the matrices , for nearest neighbours and do not commute. For a more detailed discussion, see, e.g., [36]. Detailed balance is satisfied only if the spins are updated in random order. It is commonly accepted that these two dynamics belong to the same universality class: these violations of detailed balance are irrelevant in the critical limit.

3 Autocorrelation times: definitions and critical properties

We consider the two-point correlation function

(7)

where the overline indicates the quenched average over the disorder probability distribution and indicates the thermal average. Near the critical point correlations develop both in space and time. They can be characterized in terms of the equal-time second-moment correlation length and of an autocorrelation time . In the infinite-volume limit the correlation length can be defined as

(8)

where is the Fourier transform of with respect to the variable and

(9)

is the static magnetic susceptibility. On a finite lattice with periodic boundary conditions, we define as

(10)

where , . To define the autocorrelation time, we consider the autocorrelation function of a long-distance quantity. Then, we define the integrated autocorrelation time

(11)

Here is the Metropolis time and one time unit corresponds to a complete lattice sweep.

In the critical limit and the autocorrelation time diverge. If and is the critical temperature, for we have in the thermodynamic limit

(12)

where is the usual static exponent and is a dynamic exponent that depends on the considered dynamics.

The correlation function is the quantity of direct experimental interest and thus we could take . However, for the determination of the dynamic critical exponent , it is computationally more convenient to use a different quantity. We consider the autocorrelation function of the magnetic susceptibility

(13)

Using (11) we could determine the autocorrelation time and then, we could use it to determine . However, the determination of this quantity requires the knowledge of the large- behaviour of . Since it is difficult to determine it precisely, is unsuitable for a high-precision study. We now introduce a new time scale which is particularly convenient numerically. Let us define

(14)

where is a fixed integer number. A linear interpolation can be used to extend to all real numbers. Then, for any positive , we define an autocorrelation time as the solution of the equation

(15)

This definition is based on the idea that, if were a pure exponential, i.e., , then for all and thus for any .

Let us now consider the thermodynamic limit with (high-temperature phase) and let us prove that, if the autocorrelation functions decay faster than any power of in the critical limit, then behaves as as any “good” autocorrelation time. More precisely, we show that is finite and nonzero in the critical limit for any finite . Since is an autocorrelation function of a long-range quantity, close to the critical point it obeys the scaling law

(16)

In the critical limit and for fixed , we have . Thus, we can expand

(17)

If we now define , we obtain in the critical limit the equation

(18)

It is a simple matter to show that, if decays faster than any power of ( for and any ), there is always (at least) one strictly positive solution of (18).222 Proof. The function is expected to be positive and strictly decreasing, so that and for any . Since for and , decreases for large values of . Therefore, we have . This implies for large enough. Since can be arbitrarily large, this implies for . To end the proof, define . For we have . For , we have (here we use the result for ). The function is therefore negative for small and positive for large . Since it is continuous, must vanish at a finite nonvanishing value of . Thus, we have proved that, for any , the ratio is finite and strictly positive in the critical limit. It follows that diverges as in the critical limit.

The condition that decays faster than any power of is obviously satisfied if decays exponentially, i.e. if for large , where is some exponent. While an exponential decay of the correlations is obvious in pure ferromagnetic systems for temperatures , in the case of random systems some discussion is needed. Indeed, in dilute systems one expects a non-exponential relaxation for large values of [37], due to the presence of rare compact clusters without vacancies that are fully magnetized at temperatures that are below the critical temperature of the pure system (the same clusters are responsible for the weak Griffiths singularities in the high-temperature free energy [38]). For instance, in Ising systems the infinite-volume spin-spin autocorrelation function is expected to decay as [37, 39, 40, 41]

(19)

for . In the infinite-volume limit also may show a non-exponential behavior for large in the high-temperature phase. However, note that this does not necessarily imply that the scaling function defined in (16) decays non-exponentially. On the contrary, one can argue [37] that the Griffiths tail (19) becomes irrelevant in the critical limit. This is essentially due to the fact that and that appear in (19) are expected to be smooth functions of the temperature that approach finite constants as . Thus, in the critical limit, , at fixed , the non-analytic contribution simply vanishes.333 This phenomenon can be easily understood if one imagines to have the form . The first term is the critical contribution, while the second one is the non-exponential Griffiths tail. The second term dominates for , where is the value of at which the two terms have the same magnitude. In the critical limit we have . Since the critical limit is taken at fixed, the relevant quantity is , which diverges as , as . This means that, for any fixed value of , sufficiently close to the critical temperature, always satisfies the condition , i.e. belongs to the region in which the non-exponential tail (19) is negligible. These considerations also indicate that one should limit oneself to times in studies of the infinite-volume critical behavior in the high-temperature phase. Therefore, one should always choose so that for all considered systems. Otherwise, the extrapolated critical behavior would be incorrect.

In the above-presented discussion, represents an infinite-volume autocorrelation time determined in the high-temperature phase. A similar discussion applies if we consider the FSS behavior. For instance, at we have

(20)

where is the lattice size. The function decays exponentially for any (this is rigorously true for an aperiodic dynamics in a discrete spin system). This fact does not necessarily imply that decays exponentially (a non-exponential behavior could occur if the exponential decay sets in for , ), though the discussion presented above makes this possibility quite unlikely. In any case, if decays faster than any power of , the previous proof indicates that is finite in the critical limit for any finite , and thus is a good autocorrelation time.

Beside the integrated autocorrelation time one can also define an exponential autocorrelation time:

(21)

This quantity is well defined in a finite volume since decays exponentially, but, as a consequence of (19), it diverges in the infinite-volume limit for all . As a consequence, in the infinite-volume limit at fixed temperature, diverges as . However, the decoupling of the non-exponential tail in the critical limit implies that

(22)

is finite and related to the decay rate of for large . Of course, the two limits in (22) cannot be interchanged.

On a finite lattice of size , is always well defined. Nonetheless, this does not imply that is a good autocorrelation time. On the contrary, at we expect to diverge as . Indeed, for each , is always given by the decay rate of the autocorrelation function for the slowest sample, however small is the amplitude of this contribution to the autocorrelation function (note that, for finite values of , the disorder average is a finite sum). As a consequence, is the exponential autocorrelation time for a pure Ising system in the low-temperature phase, which is expected to increase faster than any power of , as [if tunnelling events dominate . Therefore, as . The irrelevance of the Griffiths phenomenon in the critical limit should however imply that

(23)

is finite and related to the decay rate of . This is the finite-volume analogue of (22).

In the definition (14) the integer can be taken arbitrarily. However, the asymptotic critical behaviour is observed only if , see (17). Therefore, in practice should not be too large. It is also convenient to take not too small, since this avoids computing too frequently in the MC simulations. Note also that, when decreases, the errors on increase since and are close. The effect is however small, because of statistical correlations that also increase as decreases. In our work we have always considered values of much smaller than (typically ) and we have verified that the estimate of the autocorrelation times are independent of the chosen (small) value of .

Definition (14) provides an effective exponential autocorrelation time at a finite time scale. In the same spirit, one can also define truncated integrated autocorrelation times. Define

(24)

for any integer , and for any real by linear interpolation. Then, we can define an autocorrelation time as the solution of the equation

(25)

For any , this definition provides a good autocorrelation time, which converges to for . This definition is similar to that proposed in [42]; note, however, the completely different spirit in the two definitions. In [42] the method was proposed as a practical self-consistent method for the determination of and for this reason had to be large (in practice was usually taken between 5 and 10). Instead, if one is not interested in determining but only in computing , can be taken at will.

In this paper we compute the exponent from the volume dependence of an autocorrelation time at the critical temperature. Including scaling corrections, we expect a behaviour of the form

(26)

where and are the leading and next-to-leading critical exponents. As in [6, 7] we also consider the dynamical behaviour at a fixed value of a renormalized coupling constant. Also in this case autocorrelation times behave as in (26).

In order to determine it is crucial to have some knowledge of the correction-to-scaling exponents that appear in (26). RG predicts that the static correction-to-scaling exponents also occur in dynamic quantities. For instance, if behaves as at criticality for , then a correction term decaying as is also expected in for any . However, dynamics gives also rise to new scaling corrections and they may decay slower than the static ones (for instance, this occurs in the model-C dynamics, see Sec. 6). In this paper we make the assumption that no new scaling corrections with exponent less than appear, as indicated by the FT description of the model-A dynamics. As we shall see, this will be confirmed by our numerical analysis. Thus, in (26) and should be identified with the static scaling-correction exponents.

In our analysis, we make use of improved models, which are such that the leading scaling correction with exponent vanishes. Since ratios of leading scaling-correction amplitudes are universal (both in static and in dynamic correlation functions), this cancellation also occurs in dynamic quantities. Improved models have been determined in [6, 7]: the RSIM at , the RBIM at , and the Ising model at are improved. In these models the scaling corrections proportional to vanish, so that the leading correction-to-scaling exponent is . Therefore, numerical studies of improved models are expected to provide the most precise estimates of universal quantities. Of course, this is true only if the usual model-A FT description is correct; otherwise, there could be corrections with a new dynamic exponent , which do not cancel and may give rise to large corrections even in models that are improved for static quantities. A stringent check of this picture should be the fact that the three different improved models we consider give consistent results.

4 Equilibrium estimate of the dynamic critical exponent

4.1 Monte Carlo simulations

We perform MC simulations of the RSIM, the RBIM, and the Ising models for various values of , close to the critical temperature on cubic lattices of size with and periodic boundary conditions. We use the Metropolis algorithm with multispin coding as described in B.

at
8 1 7.311(5) 7.946(8) 7.946(8) 8.342(15) 8.535(25)
12 2 18.016(10) 19.827(17) 19.826(17) 20.88(3) 21.35(5)
16 2 34.783(20) 38.42(3) 38.42(3) 40.57(6) 41.57(10)
24 4 88.47(5) 98.21(8) 98.20(8) 103.78(14) 106.51(24)
32 6 172.25(9) 191.64(16) 191.61(16) 202.8(3) 207.9(5)
48 16 442.4(3) 494.0(6) 493.8(6) 523.1(1.0) 538.5(1.7)
64 30 864.0(1.1) 966.4(2.0) 966.0(2.0) 1024(3) 1052(7)
Table 1: MC estimates of for the RSIM at and for various values of at . For we also report estimates of extrapolated to . We also report the value of that enters in the definition (14).
RSIM RSIM RSIM RBIM RBIM
8 7.595(7) 7.946(9) 10.343(10) 9.410(19) 12.853(14)
12 18.322(13) 19.826(17) 30.79(3) 22.746(22) 33.30(3)
16 34.564(24) 38.42(3) 67.55(6) 42.64(4) 65.41(5)
24 84.94(5) 98.20(8) 204.66(22) 103.45(6) 169.03(11)
32 161.15(8) 191.61(16) 447.7(7) 193.56(10) 331.38(21)
48 398.0(4) 493.8(6) 1326(3) 468.5(5) 853.2(1.0)
64 756.6(1.4) 966.0(2.0) 2846(12) 874.5(1.9) 1676(5)
Table 2: MC estimates of for the RSIM at and , and , and and , and for the RBIM at and , and and .
Is Is Is
8 10.260(22) 6.507(3) 6.035(8)
10 10.918(5)
12 29.73(5) 16.590(7) 14.882(10)
14 23.705(9)
16 63.94(11) 32.300(17) 28.471(16)
18 42.353(23)
20 54.13(3)
22 67.47(4)
24 188.7(3) 82.53(8) 70.84(6)
28 118.02(17)
32 407.1(8) 161.50(20) 135.37(19)
40 271.1(4)
48 413.5(7) 336.9(6)
56 592.9(1.6)
64 2540(15) 813(3) 652(3)
Table 3: MC estimates of for the Ising model at various values of and at fixed .

For each lattice size we consider disorder samples, with decreasing with increasing , from for to for the largest lattice . Note that these numbers of samples are much larger than those typically considered in previous numerical studies. For each disorder sample, we thermalize the system by using a mixture of Metropolis and Wolff cluster updates in the case of the RSIM and of the RBIM, while in the case of the Ising model we only used the Metropolis algorithm.444The presence of rare disorder instances characterized by large compact clusters with no vacancies—those that give rise to the Griffiths tail—might be a serious problem for the thermalization if only the Metropolis algorithm is used. If a fixed thermalization schedule (independent of the disorder sample) is employed, the system may be thermalized on average, but in a few rare cases the sampling may begin much before the equilibrium state has been reached. However, the considerations presented in Sec. 3 indicate that these contributions are irrelevant for the critical behavior. Moreover, their probability is quite low. For instance, in the RSIM at , for the probability of a cube of size () without vacancies is of order (), which should be compared with . Then, at equilibrium, we perform runs of approximately Metropolis sweeps, where is the typical autocorrelation time. The averages over disorder are affected by a bias due to the finite number of measures at fixed disorder [43, 6]. A bias correction is required whenever one considers the disorder average of combinations of thermal averages. We use simple generalizations of the formulas reported in App. B of [6].555 In App. B of [6] we discuss the case of uncorrelated data. In our case correlations are relevant and thus we must somehow modify those expressions. For instance, in order to compute , we use where is a generic observable, are the corresponding MC estimates, is the number of samples, is the number of measures in equilibrium for each sample, and a suitable number. We have usually taken . The bias is of the order , where is the integrated autocorrelation time of the variable , and the corresponding autocorrelation function. Similar expressions are used in other cases. Errors are computed from the sample-to-sample fluctuations and are determined by using the jackknife method.

We considered the RSIM at (which are the same values considered in [6]) and also at . For the RBIM we worked at . These runs provided new data for the static quantities that were merged with the old ones [6] and with the results obtained in some additional cluster MC simulations at the largest lattices. They allowed us to obtain a new estimate of (see A) and new estimates of . Repeating the analysis presented in [6] we obtain , , and for the RSIM at , respectively, and for the RBIM at , respectively. For each and we usually considered two values of very close to and determined the autocorrelation times at by linear interpolation. For the RSIM at , runs were performed at and subsequently extrapolated at (see below). For the model we did not perform additional simulations and used the results of [7]. They allowed us to determine at . In the case of the RSIM we also determined at fixed . The results are very similar to those obtained at , and therefore we do not consider them in the following.

Estimates of for the RSIM at are reported in Table 1. In the table we report the data at and, for , also the extrapolations at . Note that the correction due to the small change in is significantly smaller than the statistical error. Estimates of for at for all models and several values of are reported in Tables 2 and 3.

4.2 Results for the RSIM

Figure 1: The effective exponent vs with for the RSIM at , as obtained from . The dotted lines correspond to the estimate obtained by using the RSIM data at .

In order to determine , we define an effective exponent

(27)

which, for and , behaves as

(28)

see (26). In Fig. 1 we show as obtained from the estimates of for the RSIM at , reported in Table 2. The raw data show significant scaling corrections and it is far from clear that their limit for is independent of .

In the following we present a detailed analysis of the MC data for the RSIM. First, we analyse the data at . If the FT description is correct, we should observe a fast convergence to the infinite-volume limit, with corrections proportional to . The results presented in Sec. 4.2.1 confirm this prediction. In particular, there is no evidence of a correction-to-scaling exponent smaller than in dynamic quantities. These results support the general FT scenario which predicts that the two leading correction-to-scaling exponents are the static ones and . Then, we assume the FT scenario and perform a consistency check, verifying that the large differences observed in Fig. 1 can be explained by scaling corrections. To make the check more quantitative, we introduce an improved estimator for the exponent (we use here the same strategy employed in [6] for static quantities) and show that it converges to the same value obtained for with the expected scaling corrections. This allows us to confirm universality, i.e. the independence of the dynamic critical behaviour.

4.2.1 Analysis at .

Figure 2: The effective exponent and the corresponding improved quantities and defined in (52) versus with . Results for the RSIM at . The effective exponents are obtained by using with . The dotted lines correspond to the results of fits to with .

Let us first analyse for the RSIM at . If the standard FT description of the model-A dynamics holds, the static correction-to-scaling exponents are the most relevant ones. Since the RSIM at is improved, the scaling corrections are suppressed and therefore we expect the dominant scaling corrections to be proportional to with . In Fig. 2 we plot as obtained from versus . The data with clearly fall on a line. To determine we assume to behave as

(29)

for , and perform fits of the form

(30)

with , which correspond to . Results for are shown in Fig. 3 versus , the smallest lattice size used in the fit. They are independent of for , with (DOF is the number of degrees of freedom of the fit). For example, for and , we obtain and for , and and for . For , , and , we obtain , and . One can also estimate by fitting to . If we determine from , we obtain for and for . All results are perfectly consistent. From these analyses we obtain the estimate

(31)

which is the result of the fit of with and . The error in brackets gives the variation of the estimate as varies within one error bar.

Figure 3: Estimates of the dynamic exponent for the RSIM at obtained from fits of to (30) and of the effective exponents to . We report results corresponding to different values of : (some data are slightly shifted along the axis to make them visible). The dotted lines correspond to the estimate (55), .

In the above-reported determination we have implicitly assumed that the RSIM at is exactly improved so that there are no leading scaling corrections. However, is only known approximately and thus some residual scaling correction are still present. To determine their relevance, we exploit the fact that ratios of leading scaling-correction amplitudes are universal and use the bound [6]

(32)

which holds for any quantity , be it static or dynamic, computed in the RSIM at and ( is the amplitude of the correction appearing in the large- behaviour of ). Bound (32) shows that is exactly improved (the leading correction proportional to exactly cancels) for some satisfying . Thus, an upper bound on the systematic error due to the scaling corrections is obtained by analyzing

(33)

instead of . The estimate of varies by , which represents the systematic error due to the residual corrections. The final result is therefore

(34)

Figure 4: Estimates of , as obtained from for several values of , for the RSIM at . The dotted lines correspond to the result , see (55).

The above-presented analysis shows that the estimates of obtained by using with different values of are perfectly consistent, as of course should be expected.666The consistency of the estimates shows also that the potential problems due to the Griffiths tail do not occur at the values of and we consider here. There is therefore little advantage in considering many values of and it is simpler to restrict the analyses to a single . We wish to choose it in such a way to minimize scaling corrections and statistical errors. As is clear from Table 1 statistical errors decrease with decreasing . In Fig. 4 we show as computed from for different values of . Scaling corrections decrease with increasing and are essentially independent of for . Thus, a good compromise between small statistical errors and small scaling corrections is obtained by taking neither too small nor too large. We have thus chosen . The quantities that are analysed in the following sections are always obtained from .

4.2.2 Analysis for and .

Let us now consider the RSIM at and . Since the model is not improved we must include corrections with exponent and at least, i.e. consider correction-to-scaling terms proportional to and , which decrease slower than the leading correction term occurring in improved models. Assuming this type of corrections, we fitted with the ansätze

(35)
(36)

and

(37)

fixing .

Let us first discuss the case . Fits to (35) give (DOF is the number of degrees of freedom of the fit) starting from . For we obtain . Fits to (36) give starting from . For we obtain . Fits to (37) have DOF close to one already for . For and we obtain and , respectively.

The same analysis can be repeated for . If we consider the smallest corresponding to DOF close to 1 for each fit ansatz, we obtain (fit to (35), ), (fit to (36), ), and (fit to (37), ). Again, the results of fits to (37) vary significantly with : for we obtain .

Using the simple power-law ansatz (35) one obtains results that apparently indicate non-universal, -dependent values of . Including the expected corrections to scaling the results for the critical dynamic exponent change rather dramatically, indicating that scaling corrections play a crucial role in the analysis. However, since the results obtained for depend strongly on the number of correction terms included in the fit and also on the minimal lattice size , we cannot obtain a direct accurate estimate of at these values of . Analogously, it is not possible to include the additional correction term , which was important for the analysis at (in this case we should also consider the equivalent correction ). For these reasons, we do not quote a final result for at and .

4.2.3 Correction-to-scaling amplitudes.

Figure 5: Difference versus with for the RSIM at . The dotted lines are the results of fits to in the case of , and to for .

Here we assume that the value of is universal, i.e. that it does not depend on . Based on this assumption, we compute amplitude ratios that involve the correction amplitude defined in (28) and verify that these ratios do not depend on the chosen value of . This provides a consistency check that the dynamic universality class is independent of . This type of analysis is equivalent in spirit to an analysis in which data at different values of are fitted together assuming the same dynamic exponent . For instance, this is what was done in [5]. There are, however, two significant differences: first, we use the static correction-to-scaling exponents (this allows us to consider the leading and the subleading scaling correction); second, we verify that the amplitudes of the leading scaling correction satisfy the constraints imposed by the RG, i.e. we verify the universality of the amplitude ratios.

For this purpose we consider

(38)

For it behaves as

(39)

if the dynamic critical behaviour does not depend on . Since the RSIM at is approximately improved, we have for , so that

(40)

In Fig. 5 we show the difference (39) as obtained from the available data. Fits of to and with provide estimates of . We obtain at and at . As expected, corrections have opposite sign in the two cases and are quite significant at the present values of . Note that at only fits with two corrections give a reasonable , indicating that at least two correction terms must be taken into account.

Then, we consider the static quartic cumulants

(41)

where , at fixed —we call them , , and , respectively. For they behave as

(42)

where [6] , , and . The ratios of the leading scaling-correction amplitudes are universal. In the case of and , we have [6]

(43)

Analogously, the ratio

(44)

is expected to be universal if the dynamic universality class is independent of . The ratios (44) can be directly estimated by considering

(45)

In Fig. 6 we show and for . At a fit of the data with to () gives and . At , the same fit gives and . The agreement is satisfactory, taking also into account that the errors do not take into account several sources of systematic uncertainty. The approximate -independence of the ratios and represents a nontrivial check that the dynamic universality class is independent of . Assuming universality, we obtain for the RSIM

(46)

Note that these ratios are consistent with , cf. (43).

Figure 6: Ratios and defined in (45) for , versus , .

It is interesting to note that the scaling corrections occurring in are significantly larger than those occurring in static quantities. For instance, we have