Superuniversality in phase-ordering disordered ferromagnets

Superuniversality in phase-ordering disordered ferromagnets

Malte Henkel Laboratoire de Physique des Matériaux (LPM)111Laboratoire associé au CNRS UMR 7556, Nancy Université, CNRS,
B.P. 239, F – 54506 Vandœuvre lès Nancy Cedex, France
   Michel Pleimling Department of Physics, Virginia Polytechnic Institute and State University,
Blacksburg, Virginia 24061-0435, USA
July 12, 2019
Abstract

The phase-ordering kinetics of the ferromagnetic two-dimensional Ising model with uniform bond disorder is investigated by intensive Monte Carlo simulations. Simple ageing behaviour is observed in the single-time correlator and the two-time responses and correlators. The dynamical exponent and the autocorrelation exponent only depend on the ratio , where describes the width of the distribution of the disorder, whereas a more complicated behaviour is found for the non-equilibrium exponent of the two-time response as well as for the autoresponse exponent . The scaling functions are observed to depend only on the dimensionless ratio . If the length scales are measured in terms of the time-dependent domain size , the form of the scaling functions is in general independent of both and . Conditions limiting the validity of this ‘superuniversality’ are discussed.

pacs:
64.60.Ht,75.10.Nr, 05.70.Ln

I Introduction

Aging phenomena have become one of the paradigmatic examples which are used to study fundamental aspects of non-equilibrium criticality, besides their practical use in materials processing. If some physical system is brought rapidly out of equilibrium by a sudden change of an external control parameter (an often-used device is to quench the system by lowering its temperature rapidly from the disordered high-temperature phase to its ordered low-temperature phase where at least two thermodynamically stable states co-exist) one often finds simultaneously the following three properties: (i) slow (i.e. non-exponential) dynamics, (ii) breaking of time-translation invariance and (iii) dynamical scaling, which together are said to constitute ageing behaviour. These features were first observed together in the mechanical properties of many polymeric materials by Struik Struik78 () and it has since been understood that the broad characteristics of ageing can be found in many different types of non-equilibrium systems. Besides spin glasses Cugliandolo02 (); Vincent07 (), other well-studied examples may be found in the phase-ordering kinetics of a ferromagnet quenched below its critical temperature Bray94 () or in granular media Dauchot07 ().

The analysis of phase-ordering as it occurs in ferromagnets quenched from an initially disordered state into its coexistence phase with temperature is particularly simple to formulate. The motion of the domain walls is driven by the surface tension between the ordered domains. The typical time-dependent length scale is related to the linear size of these ordered domains which grows as where is the dynamical exponent. For a non-conserved order-parameter, it is well-known that Bray94 (). Because of the simple algebraic scaling of the linear domain size , one expects the following scaling behaviour for the single-time correlation function

(1)

for sufficiently large times (with a microscopic reference time such that is of the order of the lattice constant). Similarly, for the two-time correlation and response functions (in the ageing regime, where the observation time and the waiting time satisfy and ):

(2)
(3)

Here is the space-time-dependent order-parameter, whereas is the conjugate magnetic field (spatial translation-invariance of all averages will be assumed throughout this paper) and and are ageing exponents. The scaling functions for which defines the autocorrelation exponent and the autoresponse exponent . For phase-ordering kinetics in pure systems, it is generally admitted that and simple scaling arguments show that . For an initial high-temperature state and for pure ferromagnets, is independent of the known equilibrium exponents Bray94 (); Godreche02 (); Cugliandolo02 (); Chamon06 (); Chamon07 (); Mazenko06 (); Picone04 (). The scenario just described is referred to as simple ageing. The conditions for the onset of ageing and the possible scaling forms have been carefully discussed in Zippold00 (); Andreanov06 ().

In going from phase-ordering kinetics in simple ferromagnets to glassy systems (usually modelled by spin systems with disorder and frustration), one expects more complicated growth laws which describe a cross-over from the domain growth of essentially pure systems (as long as is small compared to the typical distance between disorder-created defects) to a late-time regime with a slower growth and dominated by the defect structure. For disordered ferromagnets without frustration, this can be studied through generalisations of the Allen-Cahn equation, which attempt to describe how the pinning of the domain walls created by the disorder should be overcome by thermal activation Huse85 (); Lai88 (). In addition, it was suggested that once the unique reference length scale is chosen to be , the resulting scaling functions should become superuniversal in the sense that they should be independent of the disorder Fisher88 (). This superuniversality has indeed been confirmed for the single-time spin-spin correlator Bray91 (); Puri91 (); Hayakawa91 (); Iwai93 (); Biswal96 (); Aron08 (). Furthermore, superuniversality is in qualitative agreement with the experimental observation in several distinct polymers and metals that the linear response to a small mechanical stress can be described in terms of an universal master curve which is independent of the material studied Struik78 (). The universal scaling functions of the pure case can be calculated from the theory of local scale-invariance Henkel02 (). Tests in non-integrable systems include the auto- and space-time-responses in the two- and three-dimensional Ising models Henkel03 (), the autocorrelation function in the two-dimensional Ising model Henkel04 (), the same quantities in the two-dimensional -states Potts model with Lorenz07 () as well as the autoresponse function in several other cases, see Henkel07c () for a recent review. Therefore, with the help of superuniversality, if confirmed, the scaling functions describing non-equilibrium relaxation of quite complex systems would become analytically treatable.

In this work, we shall consider a two-dimensional ferromagnetic Ising model with quenched bond disorder. The nearest-neighbour Hamiltonian is given by Paul04 (); Rieger05 ()

(4)

The random variables are uniformly distributed over where . The model has a second-order phase transition at a critical temperature between a paramagnetic and a ferromagnetic state. It is thought that should not depend strongly on . Using heat-bath dynamics with a non-conserved order-parameter and starting from a fully disordered initial state, phase-ordering occurs and there is evidence which suggests that the characteristic length scale should scale algebraically and where the dynamical exponent should depend continuously on the temperature and the disorder . Indeed, generalising the Huse-Henley heuristic argument Huse85 () by considering the case when the disorder-created energy barriers for the motion of the domain walls are distributed logarithmically with respect to the domain size , the form Paul04 (); Paul05 ()

(5)

was proposed, where the constant parametrises the barrier height. Simulations of the linear domain size Paul04 () seemed to confirm this, with the empirical identification and are also consistent with the results of field-theoretical studies in the Cardy-Ostlund model Schehr05 (). Data from the thermoremanent magnetisation were used to estimate the exponent and, assuming , also looked consistent with (5) and , at least for values for and for which did not become too large Henkel06a (). Superuniversality has been confirmed recently for the hull enclosed area Sicilia07 (). On the other hand, the conclusion of a simple ageing reached in Paul04 (); Paul05 (); Rieger05 () has been questioned by more recent simulations for the random-site Ising model Paul07 (). In that work, a scaling form with was considered, where is a fit parameter. In the limit , one recovers the simple ageing scenario described above (the stationary part merely represents an irrelevant correction to the leading scaling behaviour). The case is called sub-ageing, and the case is called super-ageing. In Paul07 (), systematic deviations from the dynamical scaling of simple ageing were observed in the random-site Ising model quenched to below . A data collapse could be achieved, however, by allowing to vary and values in the range were reported Paul07 (). This finding was interpreted as to suggest the presence of a slight super-ageing effect Paul07 ().

In the following, we shall present new data on the single-time and two-time correlations as well as on two-time response functions. As we shall see in section II, our data are fully compatible with the simple ageing scenario and furthermore, looking at a larger range of values of , we find that the dynamical exponent depends on the control parameters in a more complicated way than suggested in eq. (5). These conclusions are also valid for the two-time response function. The dependence of the various non-equilibrium exponents on both and will be studied. We also show evidence that the scaling functions only depend on the ratio of the control parameters and finally confirm the generic superuniversality of the scaling functions of correlation and response functions. However, we also find two conditions which must be satisfied for superuniversality to hold. Our conclusions are given in section III.

Ii Results

The simulations are carried out as follows. For the integrated response we simulated systems with spins using the standard heat-bath algorithm. Prepared in an uncorrelated initial state corresponding to infinite temperatures, the system is quenched to the final temperature in the presence of a random binary field with strength , following the well-established method of Barrat Barrat98 () (using a random field avoids a bias which would drive the system rapidly out of the scaling regime). Turning off the random field after a waiting time , the thermoremanent magnetisation is measured at time . We averaged over at least different runs with different initial states and different realizations of the noise. We point out that the data discussed in this paper are of much higher quality than our earlier data Henkel06a () for the autoresponse. For the autocorrelation function, we considered systems with up to spins in order to avoid the appearance of finite-size effects for the times accessed in the simulations. The data discussed in the following have been obtained after averaging over at least 5000 different runs with different random numbers. Our main focus was on , , and where we considered for every case at least four different temperatures. In addition, some runs where also done for other values of . The total study took approximately CPU hours on Virginia Tech’s System X supercomputer composed of Dual 2.3 GHz PowerPC 970FX processors.

Figure 1: Scaling of the autocorrelation for (a) , , (b) , and (c) , , for several values of . Here and in the following error bars are smaller than the sizes of the symbols, unless explicitly stated otherwise.

Our first question is about the scaling form to be used. In figure 1, we show data for the autocorrelation , plotted over against , for several typical values of and . A nice data collapse is seen, which is fully consistent with simple ageing. Our scaling plots also imply that the exponent , analogously to what is found in the phase-ordering of pure systems. In Paul07 (), a ‘super-ageing’ scaling form was considered for the random-site Ising model where the exponent is fitted to the data. Simple ageing is recovered in the limit. However, the values of reported in Paul07 () are so close to unity that a careful study on possible finite-time corrections to scaling appears to be required before such a conclusion could be accepted footnote1 (). For all , our data for the random-bond model show no hint for a ‘super-ageing’ behaviour footnote2 (), in contrast to the findings in Paul07 (). Note that the observed scaling form of simple ageing would be incompatible with a non-power-law form of in the range of times considered. We also remark that for and larger temperatures finite-time corrections to simple ageing are observed, see Henkel06a (); Henkel07a ().

Figure 2: Left panel: Dynamical exponent , determined from the condition , as a function of . The dashed line corresponds to the equation (5). Right panel: ageing exponent , as determined from the scaling of . In some cases we have more than one data point for a given value of . These data points correspond to different values of and with constant, see table 1, showing that is not simply a function of the ratio but that it depends in a more complicated way on both and .
       
           
           
           
           
           
           
           
           
           
           
       
           
           
           
       
         
Table 1: Dynamical exponent , non-equilibrium response exponent , the autocorrelation exponent and the autoresponse exponent for different values of and .

Having in this way checked that the relevant length scale should indeed scale algebraically with time, we next determined the dynamical exponent from the criterion Sicilia07 () involving the single-time correlator

(6)

The results for are shown in figure 2 and listed in table 1. First, we observe that the values of obtained in fact only depend on the dimensionless ratio , to within our numerical accuracy. Second, we see that the function is non-linear and only becomes an approximately linear function in a relatively small region of values of . We are confident that our results are more reliable than earlier ones since they do not just describe the scaling of a single quantity, but rather will be needed for the correct scaling description of several other observables, as we shall show below. We stress that only the values of as given in table 1 will lead to a good scaling according to simple ageing without having to consider possible corrections to scaling.

Figure 3: Scaling of the thermoremanent magnetisation for several values of , with (a) , , (b) , and (c) , .

In the same way, in figure 3 we show data for the scaling of the thermoremanent magnetisation, expected to be of the form

(7)

The exponent is obtained in the usual way Henkel05 () by plotting the thermoremanent magnetisation as a function of the waiting time for fixed values of the ratio . The resulting power-law decay, see eq. (7), then yields the value of the exponent . The numerical values and error bars given in table 1 and shown in figure 2 are obtained after averaging over the values obtained for five different values of , namely 3, 5, 7, 10, and 15. Looking at these values (see figure 2), we observe that the estimates scatter considerably more than those for . We consider this scatter to be large enough to conclude that cannot be reduced to a function of the single variable . Furthermore, considering in detail the numerical values from table 1, we see that the relation , known from the phase-ordering of pure ferromagnets Cugliandolo02 (); Bouchaud00 (); Henkel03a (), is no longer valid.

In order to understand this finding, let us briefly reconsider how the relation may be derived for pure ferromagnets. Consider a pure ferromagnet in an external oscillating magnetic field of angular frequency . The dissipative part of the linear response is given as the imaginary part of the dynamic susceptibility and reads (see e.g. Cugliandolo02 ())

(8)

where the last relation follows from the usually assumed scaling (3) of the autoresponse function . On the other hand, motivated from the physical picture that the dynamics in phase-ordering should only come from the motion of the domain walls between the ordered domains, one would expect to find Bouchaud00 ()

(9)

from which one may identify the stationary and the ageing part with the terms in eq. (8) coming from the scaling analysis. Since only the domain boundaries contribute to the dynamics, the leading time-dependent part should be proportional to the surface area of the domain divided by the total volume, hence to which accounts for the factor in (9). Comparison of eqs. (8,9), together with , then gives .

Our empirical observation that suggests that the above argument should no longer apply to random ferromagnets. Since (8) only depends on the dynamical scaling assumption (3), and given that our numerical results appear to be compatible with it, we expect that (8) should remain valid for disordered ferromagnets. Since also for disordered ferromagnets, the contribution to the ageing behaviour should come from the boundary region between ordered domains (this is also suggested by looking at the microscopic spin configurations, see e.g. Henkel07a ()), we expect it to be proportional to where denote the number of mobile spins in the bulk and at the domain boundaries, respectively. While one should still have , disorder may cause the domain boundary to become fractal and, hence, with the fractal dimension (and for the pure case). Then eq. (9) would be replaced by

(10)

and comparison with (8) would now imply

(11)

Our empirical results (table 1) imply that , that is, the disorder should modify the domain boundaries into fractal curves. From eq. (11), since depends on the dynamical exponent as well as the fractal dimension , it may appear more natural that cannot be written as a function of the single variable .

Figure 4: (a) Scaling behaviour of the single-time correlator for and and several values of , as a function of the scaling variable . (b) Scaling of the space-time-dependent correlator for and for several values of and two fixed values of , as a function of the scaling variable . (c) Scaling of the space-time-dependent thermoremanent magnetisation for and for several values of , with . The lines are guides to the eye.

In figure 4 we show the scaling behaviour of the space- and time-dependent correlation and response functions for various values of the waiting time . For the selected typical values of and , a simple ageing behaviour is observed, in agreement with the observed scaling behaviour of the autocorrelation and of the autoresponse. We have found completely analogous results for all other values of and of which we considered. In fact, the scaling is much cleaner than for the quantities and for none of the studied cases a sizeable correction to scaling could be identified. Obviously, space- and time-dependent quantities are much better suited for an investigation of the scaling forms than quantities that only depend on time. Similar conclusions have recently been drawn from a study of nonequilibrium growth models Rot06 ().

Using the scaling forms (2,3) in the limit of large for the autocorrelation and autoresponse functions (where ), we have also extracted the exponents and and list our results in table 1. In contrast to the pure case, where for fully disordered initial conditions one may show that Bray94 (); Picone04 (), the values of the autocorrelation exponent are different from those of the autoresponse exponent . In particular, we find that within our numerical accuracy, is a function of the single variable , at least for , while cannot be expressed in this way. Our data suggest that and they are consistent with the rigorous Yeung-Rao-Desai inequality Yeung96 (). Furthermore, we observe that should be practically constant (again for and with a value in the range ).

In conclusion, our data are clearly consistent with simple ageing of the single-time and two-time correlation functions, as well as for the thermoremanent magnetisation, and fully confirm the anticipated scaling forms (1,2,3) with .

Figure 5: (a) Space-time correlator for (filled symbols) and (open symbols) for several waiting times and the values and . The full curves are guides to the eye. (b) Autocorrelator for (filled symbols) and (open symbols) for several waiting times . (c) Scaled space-time thermoremanent magnetisation for (filled symbols) (with ) and (open symbols) (with ) for several waiting times and the values and . The full curves are guides to the eye.

Next, we shall compare the form of the scaling functions, for several values of the control parameters and . Since the dynamical exponent , see figure 2, one might expect that the scaling functions themselves should only depend on the ratio , rather than on and separately. In figure 5, we test this idea by comparing data for with those for , for the three cases of (a) the space-time-dependent correlation , (b) the autocorrelation and (c) the space-time-dependent thermoremanent magnetisation . In all cases, there is a clear scaling behaviour consistent with simple ageing footnote3 () and the scaling functions nicely superpose (for the integrated response, the data for were multiplied by 1.17 in order to take into account the well-known presence of the non-universal numerical prefactor). This result, namely that the form of the scaling function only depends on the ratio , goes beyond the standard scaling form (1,2,3), yet it does not require to re-scale the length by the typical domain size , as it would be required for a test of superuniversality. More systematic tests of this result would be welcome. We point out that the findings of figure 5 are consistent with our earlier observation that . The more complicated dependence of on both and would only appear if in plots such as figure 5c one would concentrate on the region .

Figure 6: (Colour online) Test of the superuniversality for the single-time correlation function. In panel (a), data for are shown to fall onto the same rescaled curve. In panel (b), we show that the data for describe a curve different from the one of panel (a). The green curve is the scaling function of the pure system. In panel (c), data for and different times are compared with the scaling curves as measured for (lower green line) and (upper green line).

After these preparations, we are ready for a test of superuniversality. Superuniversality of the single-time correlator is tested in figure 6. First, we show in figure 6a data for several values of and , where the values of may be read off from figure 2 and table 1. The times were chosen sufficiently large such that a clear scaling behaviour has set in. Using the typical length scale as determined earlier from eq. (6), we see that indeed all data, including the one for the case without disorder, collapse nicely onto a single curve, within the numerical accuracy. This is a clear confirmation of superuniversality, very much in agreement with earlier studies Bray91 (); Puri91 (); Hayakawa91 (); Iwai93 (); Biswal96 (); Sicilia07 (); Aron08 (). However, when considering the case , which is shown in figure 6b, a different picture emerges. Clearly, the scaling curves for as obtained for several values of again collapse onto each other, but, as the comparison with the scaling function of the pure case shows, the scaling function is no longer the same as the one of the pure case. Therefore, if taken at face value, the case might represent a distinct superuniversality class. In order to get a better understanding on this point, we show in figure 6c data for a relatively large value of . One sees that for moderately large times, the data are very close to the curve found for , but when the time is made very large, a cross-over to the curve of the pure case is observed. Qualitatively, the cross-over time increases when and becomes so large that a cross-over is no longer detectable for the times (and the lattice sizes, which must be increased to large times in order to avoid finite-size effects) reachable with our numerical methods. We did not see any sign for a cross-over in our data with , but purely numerical techniques cannot distinguish between a very large and an infinite cross-over time .

We extend the test of superuniversality to the case of space-time-dependent two-time correlators in figure 7 and similarly for the space-time-dependent two-time response in figure 8. Qualitatively, we arrive at essentially the same conclusion as for the single-time correlator. In both cases, panel (a) demonstrates a superuniversal behaviour for values of . We show here our data for , but the same behaviour is observed for other values of accessed in this study, namely . However, closer inspection also shows that superuniversality is no longer true for relatively small spatial distances . This is shown in the insets of the panels (a) in figures 7 and 8. Indeed, systematic deviations are observed for small spatial distances, the largest deviations being observed for the autocorrelation and autoresponse functions with . The value of where the deviations set in seems to depend slightly on the value of , but a larger range of values than accessed in the present study is needed for a more quantitative discussion of this point. That means that although dynamical scaling does hold true even down to the autocorrelators and autoresponses, correlators and responses taken over a spatial distance of at least a typical cluster size show yet a larger degree of universality.

Figure 7: (Colour online) Test of the superuniversality of the two-time correlation function, with and analogous to Figure 6. The data shown are for . The green curves show the data of the pure system. The inset in panel (a) is a magnification of the area within the dashed box. The black lines are guides to the eye.
Figure 8: (Colour online) Test of the superuniversality of the two-time thermoremanent magnetisation, with and analogous to Figure 6. The data shown are for . In order to compare the data for different and , we have to take care of the fact that the response contains a non-universal multiplicative factor. The data have been multiplied by a constant in such a way that the different curves coincide with the curve for and for large values of . The green curves show the data of the pure system. The inset in panel (a) is a magnification of the area within the dashed box. The black lines are guides to the eye.

This requirement appears to be consistent with the known numerical values of the exponents and as listed in table 1. Superuniversality at would have required that their values should have been equal to those of the pure case , but we have rather seen that they depend on and .

On the other hand, the case again stands apart, as we illustrate in the panels (b) in both figure 7 and 8. Comparing the data for with the scaling functions found for (see panels (a)), we find small but systematic deviations. We stress that although these deviations are not very large, they are well outside the error bars of our high-quality data. As for the single-time correlator, it remains a possibility that for enormous times there might occur a cross-over to the scaling function of the pure case, but the relevant cross-over time is far larger than the time scales reached by our simulation.

Iii Discussion

In this work, we have studied the non-equilibrium scaling behaviour of a disordered Ising model without frustration, in an attempt to appreciate better the role of disorder by considering its effects in a system which is no longer identical to a pure system but which yet does not show the full complexity of a spin glass. Our conclusions are as follows:

  1. When quenched to a temperature from a totally disordered state, the two-dimensional bond-disordered Ising model undergoes phase-ordering kinetics. The typical length scale of the ordered domains scales algebraically with time, where the dynamical exponent depends continuously on the dimensionless ratio of the control parameters.

    Quantitatively, this dependence can be read off from figure 2, and does not agree with earlier proposals of a linear relation such as in eq. (5). We have seen that this value of correctly describes the dynamical scaling of not only single-time correlators, but of the two-time correlators and responses as well.

  2. Our data are completely compatible with simple ageing.

  3. While the non-equilibrium exponent of the correlation function is un-modified with respect to pure phase-ordering systems, the non-equilibrium exponent which describes the scaling of the response function is no longer simply related to the dynamical exponent , see table 1 and figure 2.

    We propose to account for this finding in terms of a postulated fractal structure of the domain walls, which has led us to eq. (11). Further tests of this idea would be welcome.

    The autocorrelation and autoresponse exponents and are distinct from each other, in contrast to the pure case.

  4. Our data suggest that the form of the scaling functions only depends on the dimensionless ratio . It remains to be seen to what extent this observation can be extended to different systems.

  5. In general, our data appear to confirm the superuniversality hypothesis, that is when all length scales are expressed in terms of , the form of the scaling function is independent of both the disorder and the temperature .

    However, we have also found two important qualifications:

    1. Superuniversality does not hold for sufficiently small spatial distances .

    2. For , although we find throughout a similar data collapse, the form of the scaling functions no longer coïncide with the ones of the pure case. It is not understood whether the data presented here should be viewed as giving evidence for a distinct superuniversality class or else if there is a cross-over to the scaling functions of the pure scale at time scales much larger than the ones reached in our study.

    A better understanding of superuniversality will require an explanation of these conditions.

What can these findings tell us on the behaviour of real materials ? Indeed, it has been shown recently, in a comparative study of the three-dimensional random-field Ising model and the three-dimensional Edwards-Anderson spin glass Aron08 (), that superuniversality is apparently satisfied in the former case (in which the disorder is ‘weak’ such that the ground state is still ferromagnetically ordered) while in the latter it is not frust (). Our own result is in qualitative agreement with this, but it raises the question how to explain the celebrated universality of the scaling functions for the linear response found in largely different materials Struik78 ().

Acknowledgements

We thank Leticia Cugliandolo for useful discussions. The simulations have been done on Virginia Tech’s System X. MH thanks H. Park for kind hospitality at the KIAS Seoul, where the writing-up of this work was finished.

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