Random Field Potts model with dipolar-like interactions: hysteresis, avalanches and microstructure
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t Random Field Potts model with dipolar-like interactions: hysteresis, avalanches andmicrostructure
Benedetta Cerruti ∗ and Eduard Vives Departament d’Estructura i Constituents de la Mat`eria, Universitat de BarcelonaDiagonal 647, Facultat de F´ısica, 08028 Barcelona, Catalonia (Dated: November 4, 2018)A model for the study of hysteresis and avalanches in a first-order phase transition from a singlevariant phase to a multivariant phase is presented. The model is based on a modification of theRandom Field Potts model with metastable dynamics by adding a dipolar interaction term truncatedat nearest neighbors. We focus our study on hysteresis loop properties, on the three-dimensional(3D) microstructure formation and on avalanche statistics.
PACS numbers: 75.60.Ej, 75.50.Lk, 81.30.Kf, 64.60.Cn
I. INTRODUCTION
Microstructure formation in first-order phase transi-tions is a phenomenon that has been studied by physi-cists, mathematicians and engineers [1, 2, 3]. It is im-portant not only from a fundamental point of view, butalso for applications, due to its relationship with materialproperties. Microstructures occur in ferroic systems (fer-romagnetic, ferroelectric and/or ferroelastic) which aredriven through a first-order phase transition (FOPT) inwhich some symmetry operations of the parent phase arelost. The product phase (usually less symmetric) mayappear in energetically equivalent variants which are re-lated by the symmetry operations that have been lost inthe transition. The obtained microstructures correspondto the arrangement of such equivalent variants and aredecided by the interplay of many energetic terms: inter-face energy, surface energy and long-range interactions.Until now many of the studies on microstructures havefocused on the determination of the optimal variant con-figuration minimizing a certain thermodynamic potentialthat takes into account the above factors and externalconditions. Nevertheless, in many cases, when real ma-terials are studied, such optimal microstructures are notobserved. This is mainly due to two important factors:(i) the existence of disorder sources of very different na-tures both in the bulk and on the surfaces and also (ii)the athermal character of the phase transition dynam-ics: when the temperature is not very high, the ener-getic barriers that separate the optimal solutions fromthe parent phase cannot be overcome. Thus, the systemevolves following metastable paths which locally optimizethe system energy, but are far from the trajectories ob-tained from a global minimization principle. An interest-ing suggestion on the behavior of microstructure forma-tion comes from the glass-jamming transition framework(see Refs. 4, 5 and references therein), which is asso-ciated with the so-called kinetically constrained model. ∗ Electronic address: [email protected]
These models are stochastic lattice gases with hard coreexclusion, with the addition of some local constraints,which mimic the geometric constraints on the possiblerearrangements in physical systems. Similar behavior isquite likely to arise in microstructure formation.The use of continuum models derived from elasticitytheory [6, 7, 8, 9, 10] has been proposed as another ap-proach to microstructure formation. Some of these mod-els have been successful in explaining microstructures,hysteresis and avalanches. Nevertheless, they are verytime consuming from a computational point of view. Forthis reason, in many cases only 2D problems have beenaddressed, and even this item presents difficulties con-nected with large statistics and with the scanning of themodel parameters in order to study their influence. Ouraim here is to find a statistical mechanical lattice model,easy to simulate and which allows for the study of statis-tics of microstructure sequences dynamically generatedin athermal systems, under the influence of disorder.The Random-Field Ising model (RFIM) [11] withmetastable dynamics is one of the simplest models forthe study of the combined effects of disorder and ather-mal evolution. It is formulated in a magnetic languagefor a spin reversal transition, driven by an external mag-netic field H . The only degrees of freedom are spin vari-ables defined on a lattice ( i = 1 , ..., N ), which take values S i = ± i -th lattice site. The RFIM enables com-putation of hysteresis loops M ( H ) corresponding to thebehavior of the order parameter M = P i S i as a func-tion of H as well as the analysis of the intermediate statesbetween the negatively saturated initial state { S i = − } and the final positively saturated state { S i = +1 } , and vice versa . In particular, the RFIM has been successfulin understanding the avalanche dynamics (Barkhausennoise [12]) that joins the intermediate metastable statesand shows absence of characteristic scales for a criticalamount of disorder. Moreover, the RFIM displays sev-eral other interesting properties [11, 13]: it exhibits awell-defined rate-independent trajectory, it shows returnpoint memory, it satisfies the abelian property and, froma computational point of view, it is fast to simulate tra-jectories in relatively large systems [14].Nevertheless, the usefulness of the RFIM for the studyof microstructures is almost null. This happens becausethe parent and product phases in the RFIM are the pos-itively magnetized phase and the negatively magnetizedphase. These two phases are single variant and totallyequivalent from a symmetry point of view. Consequently,the obtained hysteresis loops are symmetric under theexchange H → − H and M → − M , there is absence oflatent heat associated with the FOPT and the domainsare spherically symmetric (except for some short-rangecorrelations due to lattice symmetries).Within this framework, the aim of the present workis to explore some modifications that should be intro-duced in the RFIM in order to obtain 3D microstruc-tures without losing, as far as possible, some of the use-ful RFIM properties that we have mentioned above. Afirst step in this direction was done several years agoby defining the Random Field Blume-Emery-Griffithsmodel, in which the ’spin’ variables take three differentvalues ( S i = − , ,
1) with metastable dynamics [15]. Inthis case, the FOPT takes place from a single variant par-ent phase, represented by S i = 0, and a product phasewith two variants S i = +1 and S i = −
1. In the citedwork, the hysteresis loops, phase diagram and avalanchedistribution were studied for this type of simple case. Inthe present paper we would like to go one step forward.This will be done by starting from the Random FieldPotts model [16] with metastable dynamics. In the Pottsmodel the spin variables can take an arbitrary number ofvalues. This model allows phase transitions to be stud-ied from a non-degenerate phase S i = 0 to a multivariantproduct phase. We will explore the effect of an extra in-teraction term (of a dipolar nature, but truncated to thenearest-neighbor approximation), which will be necessaryin order to produce microstructures (for the introductionof dipolar interactions in RFIM, see [17]). It is not ouraim to focus on the detailed analysis of any particulartransition. We will study a model that, from the point ofview of symmetry, would correspond to a transition froma cubic phase (single variant) to a tetragonal phase (withthree equivalent variants), although neither the detailedinteractions nor the external constraints will be tuned forthe particular modeling of such transitions in ferroelasticsystems. This will be the aim of a future work [26].The paper is organized as follows: in section II weintroduce the hamiltonian of the model. In section IIIwe detail the metastable dynamics that has been usedfor the simulations. Section IV is devoted to the discus-sion of the obtained results: in section IV A we presentour analysis on the shape of hysteresis loop cycles as afunction of the model parameter values. In this sectionit is shown that the loops happen to be unsymmetric,in contrast to the RFIM results. The microstructuresare analyzed in section IV B, where we discuss three dif-ferent regimes corresponding to different ranges of theparameters values. Moreover, in section IV C we presentthe statistical analysis of avalanche behavior. Finally, wesummarize and discuss the future perspectives in section V. II. MODEL
The model can be defined on any regular lattice. Wewill consider a simple cubic lattice of size N = L × L × L with periodic boundary conditions. At each latticesite we define a variable S i ( i = 1 , . . . N ), which cantake four different values that we will call 0, x , y and z .We can choose different representations for our variablesbut it is convenient to consider a vector ~S i having threecomponents: we will indicate the four possible values as0 = (0 , , x = (1 , , y = (0 , ,
0) and z = (0 , , M = P i ( ~S i ) , where the sum spans overthe whole lattice. M represents the amount of the sys-tem that has transformed from the cubic to the tetrag-onal phase. By following the analogy with the magneticcase, we will refer to M as the total magnetization of thesystem. Moreover, we define the normalized magnetiza-tion m , as m = M/N . We will drive the system by anexternal field H coupled to M , since we are interested inthe transition from the 0 phase to the multivariant phasewhich will be composed of regions (variants) in the states x , y and z . The field H would correspond to the drivingeffect of the temperature in athermal structural transi-tions. We will start by decreasing H , from the M = 0state. We consider the following hamiltonian: H = − k NN X
1) with zero mean and unitary standard deviation,and h i is a scalar field, again extracted from a gaussiandistribution N (0 , σ and ρ control theamount of quenched disorder in the system.In order to compare our model with the standardRFIM, we define the total amount of disorder σ = σ + ρ . In practice, the two disorder terms can be un-derstood as arising from a local random field ~f i , whosecomponents are correlated, being ~f i = σ ( g ix , g iy , g iz ) + ρ ( h i , h i , h i ) , (3)so that h f ix i = h f ix i = h f ix i = σ and h f ix f iy i = h f ix f iz i = h f iy f iz i = ρ . III. DYNAMICS
There are several possibilities for the choice ofmetastable dynamics. In Fig. 1 we show examples ofhysteresis loops obtained with three possible choices ofdynamics. At first sight, the three loops look very simi-lar. In all the cases we start from a metastable state, weincrease or decrease the field by a ∆ H step and then, atconstant field, we recursively minimize the system energyby using a local rule based on single-spin changes. Onlyafter a new metastable state is reached we proceed witha new field change ∆ H .In the extremal selection + extremal update case, wescan the whole system and check which variable S i canchange to a new value with the minimum energy differ-ence ∆ H . The proposed change is accepted if this mini-mum ∆ H is negative. In the random selection + randomupdate case, we randomly choose a spin on the lattice andpropose a random change to a new value. If the proposedchange implies ∆ H < random selection + extremal update dynamics, we -20 -10 0 10 20 H m extremal selection+extremal updaterandom selection+random updaterandom selection+extremal update
16 16,50.020.04
FIG. 1: (Color on line) Hysteresis cycles for three differentdynamics as explained in the text. The parameters of thesimulations are: ∆ H = 0 . L = 16, λ = − σ = 5 and ρ = 2. In the inset: a magnification of a loop region: themagnetization values for the three dynamics coincide only forsome field values. randomly choose a spin on the lattice and check amongthe three possible new values which one represents a min-imum ∆ H . If this minimum value is negative we acceptthe change. From a computational point of view the firstchoice is much more time consuming than the other two,since the effort scales with L .Although the loops are very similar, detailed analysisreveals that the obtained hysteresis loops, as well as thesequence of metastable states, are not identical. Thistells us that the proposed model is not abelian and thatthe final state will depend on the order in which unstablespins will be changed. In order to ensure some robustnessof the results, we are thus forced to choose extremal selec-tion + extremal update dynamics, that is: to propose theoptimal spin change among the whole lattice and amongall the three possible final values at each time step. Thiskind of dynamics is deterministic and thus, by definition,independent of the updating order. We will keep to thisdynamics for the rest of the paper.We now study the effect of changing the value of ∆ H .In Fig. 2 we show three hysteresis cycles obtained for dif-ferent values of the driving rate ∆ H using the extremalselection+extremal update dynamics. A detailed analysisreveals that the differences between the three loops canbe attributed to the fact that driving with a smaller ∆ H allows more metastable intermediate states to be found,but for the same applied field values, in the three real-izations not only the magnetization, but also the finalmicroscopic configurations reached are the same. Theindependence from the field rate is an important prop- FIG. 2: (Color on line) Hysteresis cycles for various val-ues of the applied field rate ∆ H with the extremal selec-tion+extremal update dynamics. The parameters of the sim-ulations are: L = 16, λ = − σ = 4 and ρ = 0. In the inset:a magnification of a portion of the hysteresis cycle. erty from the point of view of the simulations, since itallows us to use a relatively large ∆ H for the study ofthe properties of hysteresis loops. IV. RESULTS
We have performed numerical simulations of systemswith sizes L = 8 , , ,
40 and 60, averaging over10 − realizations of the quenched random fields. Wehave focused our analysis on hysteresis loops behavior,on microstructure formation and on the statistical prop-erties of the avalanches. A. Hysteresis loops
In Figs. 3, 4 and 5, we show some examples of hystere-sis loops simulated with our model in order to illustratethe effect of the different hamiltonian parameters.We consider the cases with λ < λ > λ is, the larger the width of the loop. In the case of verynegative values, the loops start to exhibit a plateau inthe increasing field branch: the retransformation to the0 phase is done in two separate steps. (This effect will bediscussed below). For the second case, increasing lambdatowards positive values increases the tilt of the hysteresisloop so that saturation in the transformed phase can onlybe obtained when the field is very negative. This effect is FIG. 3: (Color on line) Examples of hysteresis loops for dif-ferent values of the parameter λ : (a) λ = − , − , − , − λ = 1 , ,
10. In all the cases, L = 16, ∆ H = 0 . σ = 3 and ρ = 0 due to the competition between the Potts and the dipolarterms. Many domains in the final stages of the transfor-mation are frustrated and can only be transformed by avery negative H , as occurs in ferromagnets that containa small percentage of antiferromagnetic bonds.In Figs. 4 and 5 we show the effect of the two dis-order parameters σ (Fig. 4, in the low (a) and high (b) ρ regimes) and ρ (Fig. 5, in the low (a) and high (b) σ regimes). In all the cases it can be seen that increasingthe amount of disorder increases the tilt and decreasesthe width of the loop. Moreover, as expected, for low val-ues of the amount of disorder ( σ or ρ ) the loops exhibitsharp discontinuous (ferromagnetic-like) behavior. Thisfeature is in agreement with recent results on the stan-dard RFIM, concerning the observation that the transi-tion from sharp to smooth loops can be induced by dif-ferent kinds of disorder parameters: not only the randomfield variance σ but also random anisotropy [18], the va-cancy concentration [19], etc.. Our model shows that thecorrelation with the random fields of intensity ρ can alsoact in a similar way.Let us now discuss the plateau observed in Fig. 3(a)in the increasing field branch. As shown in the examplesin Fig. 6, this plateau occurs at smaller magnetizationswhen the system size is increased. This suggest that itmay be due to the stabilization of “slab” domains thatcross the whole system from one face to the other and -10 0 10 H m σ =2 σ =3 σ =4 σ =5 σ =6 -20 -10 0 10 20 30 H m σ =2 σ =5 σ =10 (a)(b) FIG. 4: (Color on line) Examples of hysteresis loops for var-ious values of the disorder parameter σ : (a) for σ = 4 , , ρ = 0; and (b) for σ = 2 , ,
10 with ρ = 5. In all thecases λ = − L = 16 and ∆ H = 0 . that due to the periodic boundary conditions behave asinfinitely large. Such slabs become less and less frequentby increasing the system size. This suggestion has beenconfirmed by analyzing sequences of configurations. Anexample will be discussed in Sec. IV B.In order to perform a quantitative analysis of the hys-teresis loops it is important to measure some of theirproperties. One of the most studied hysteretic featuresis loop area. In fact, it represents the amount of en-ergy dissipated during a cycle and thus it is an importantquantity to be controlled both from the theoretical andmaterials application point of view. In Fig. 7 we show theloop area, averaged over several disorder configurations,as a function of the parameter λ , for various values of σ .As can be seen, the area shows a much more impor-tant dependence on λ for negative than for positive λ .This behavior can also be seen by studying the coerciv-ity (amplitude) of the hysteresis cycles at m = 0 .
5, whichdisplays a very similar dependence on λ .In Figs. 3, 4 and 5 we can see that the hysteresis cy-cles obtained with our model are asymmetric, i.e. thedecreasing field branch (transformation) cannot be re-lated by an inversion operation to the increasing fieldbranch (retransformation). This is an interesting prop-erty since, experimentally, materials displaying a transi- -10 0 10 H m ρ =1 ρ =2 ρ =4 -20 0 20 40 H m ρ =1 ρ =5 ρ =10 (a)(b) FIG. 5: (Color on line) Examples of hysteresis cycles for vari-ous values of the disorder parameter ρ : (a) for ρ = 1 , , σ = 1 .
5; and (b) for ρ = 1 , ,
10 with σ = 4. In all the cases, λ = − L = 16, and ∆ H = 0 . tion to a multivariant phase show such behavior, whichcannot be reproduced with the RFIM (see section I). Inour model, this feature descends from the intrinsic dif-ference of the physical processes occurring in the twobranches (transition from the 0 state to the three vari-ant phase in the first branch, and the opposite process inthe second branch). In order to study this feature morequantitatively, we define an asymmetry factor A as: A = ( dM/dH ) − ( dM/dH ) ( dM/dH ) + ( dM/dH ) , (4)where ( dM/dH ) and ( dM/dH ) are the derivatives ofthe hysteresis cycle at the coercive fields (defined as thetwo fields at a height m = 0 .
5) in the two branches.As can be seen in Fig. 8, A is greater than zero for neg-ative values of λ , while for λ > A is essentially 0, irrespective of the value of σ . We havecut the curve with σ = 3 in Fig. 8 at the value λ = − λ hysteresis curves begin to show the plateaux explainedabove, and thus our definition of asymmetry loses sense.The effect of disorder on hysteresis loop properties isillustrated in Fig. 9, where we show the width W of theloops at m = 0 . ρ , for different values of σ . As mentioned in FIG. 6: (Color on line) Hysteresis loop for various values ofthe system size L = 16 , , , ,
60. The model parametersare: λ = − σ = 3, ρ = 0 and ∆ H = 0 . -10 -5 0 5 10 λ a r ea σ =3 σ =4 σ =5 σ =6 FIG. 7: (Color on line) Hysteresis loop area as a function ofthe parameters λ , for σ = 3 , , ,
6. The system parametersare: L = 16, ∆ H = 0 .
05 and ρ = 0. Each point represents anaverage over 800 realizations of the disorder. Solid lines area guide to the eye. Error bars are not visible on the scale ofthe picture. the qualitative description above, both ρ and σ decreasethe width of the loops. -10 -5 0 5 10 λ A σ =3 σ =4 σ =5 σ =6 FIG. 8: (Color on line) Hysteresis loop asymmetry as a func-tion of the parameters λ , for σ = 3 , , ,
6. The system pa-rameters are: : L = 16, ∆ H = 0 .
05 and ρ = 0. Each pointrepresents an average over 800 realizations of the disorder.Solid lines are a guide to the eye. ρ W σ =2 σ =4 σ =6 FIG. 9: (Color on line) Hysteresis loop width as a function ofthe parameters ρ , for σ = 2 , ,
6. The system parameters are: L = 16, ∆ H = 0 .
05 and λ = −
3. Each point represents anaverage over 800 realizations of the disorder. Solid lines are aguide to the eye.
B. Microstructures
As we have already mentioned (see section II), whenthe dipolar term is large enough compared to the Pottsterm, the transformed domains lose their spherical sym-metry and start to show a non-trivial microstructure.The microstructure of a system is defined as the arrange-ment of the variants of the product phase.
FIG. 10: (Color on line) Saturation configuration for systemparameters: L = 32, ∆ H = 0 . λ = − σ = 20 and ρ = 20. Different colors correspond to different spin variants(see the legend). In Fig. 10 we can see an example of these three-dimensional microstructures. We represent the views ofthe yz , xz and xy surfaces, when the sample has reachedsaturation (fully transformed state). In this case ( λ < ~S i = (0 , , z direction both in the xz and zy planes.This effect can be quantitatively measured as will be ex-plained below.For λ >
0, we observe the formation of oblate domains,as shown in Fig. 11, developing in the plane perpendicu-lar to the spin direction. This effect generates a sort of“chessboard” correlation as can be seen, for instance inthe yz plane by observing the red and green domains. FIG. 11: (Color on line) Saturation configuration for systemparameters: L = 32, ∆ H = 0 . λ = 3, σ = 1 and ρ = 1.Different colors correspond to different spin variants (see thelegend). In order to quantify the shape of the domains in suchmicrostructures, we have calculated the average linearsize h D x i , h D y i and h D z i , of the domains of the threevariants x , y and z , along the three spatial directions ˆ x ,ˆ y and ˆ z , at the saturation configuration. For instance, theaverage size matrix corresponding to Fig. 10 and Fig. 11,are shown in Tables I and II. In the first case (corre-sponding to the prolate domains), the diagonal elementsof the matrix are sensibly larger than the others, con-firming the growth tendency of domains along the orien-tation of each variant. As is quite obvious, for symmetryreasons, the diagonal elements can be averaged givingwhat we will call the average linear size in the paralleldirection h D k i and the off-diagonal elements can also beaveraged giving the average linear size in the perpendic-ular direction h D ⊥ i . For the case of Fig. 10 ( λ <
0) weget h D k i = 4 . ± .
20 and h D ⊥ i = 2 . ± .
05. Forthe case of Fig. 11 ( λ > h D k i = 1 . ± .
005 and h D ⊥ i = 1 . ± . h D x i h D y i h D z i ˆ x ± ± ± y ± ± ± z ± ± ± h D x i h D y i h D z i ˆ x ± ± ± y ± ± ± z ± ± ± For the sake of completeness, as a final microstruc-ture example, in Fig. 12 we show a system configurationwith high disorder and λ = 0. As expected, no domainasymmetry arises and every spin just aligns with its lo-cal random field. The values of h D k i = 1 . ± .
14 and h D ⊥ i = 1 . ± .
20 are equal to within statistical errors. h D x i h D y i h D z i ˆ x ± ± ± y ± ± ± z ± ± ± The formation of microstructures such as those inFig. 10 and 11 is quite clearly affected by the dynamics
FIG. 12: (Color on line) Saturation configuration for systemparameters: L = 32, ∆ H = 0 . λ = 0, σ = 40 and ρ = 40.Different colors correspond to different spin variants (see thelegend). due to the effect of kinetic constraints. In fact, when adomain of one variant starts to grow, it necessarily blocksthe growth of neighboring domains of other variants and vice versa . Thus the first variant to locally break sym-metry will facilitate the nucleation of large domains, thusrestricting the dimensions of the other variants. This ef-fect could be seen, for instance, by analyzing the decreas-ing field branch in Fig. 13: the formation of the domainsof type x and y that cross the system block the growthof domains of the z variant.Moreover, with the help of the microstructure repre-sentation, we can analyze in more detail the bump for-mation due to finite size effects, discussed in section IV A(see Fig. 13): in fact, if the system size is finite there isthe formation of domains spanning a whole system side,such as the y domains in microstructures labeled with4 and 5 in Fig. 13. As already pointed out, these kindof slab domains are actually infinite due to the periodicboundary conditions and are thus very stable. As wecan see in the figure, they keep existing up to high driv-ing field, giving rise to the existence of the plateau anddisappear when the field overcomes a certain threshold.As we already mentioned in section II, the truncationof the dipolar term does not allow elastic effects to be re-produced, which would lead to more realistic microstruc-ture. In particular, it would be interesting to controlthe tendency for the different variants to exhibit a pre-ferred habit plane, as would be the case of a real cubic totetragonal transition. We expect that including a next-nearest-neighbor dipolar interaction will allow for suchan interesting property to occur. The model presentedhere is, therefore, a promising starting point for modelingmaterials with a phase transition from a single variant toa multivariant phase. C. Avalanches
Another phenomenon that may be analyzed with ourmodel is the avalanche dynamics. The hysteresis loopsin athermal first-order phase transitions consist of a se-quence of jumps between metastable states. Such discon-tinuities are, in general, microscopic. However, for cer-tain values of the disorder one or more may become com-parable to the system size and then correspond to the ob-served macroscopic discontinuities in the ferromagneticloops. In the magnetic case, microscopic avalanches canbe detected experimentally by appropriate coils. Theycorrespond to the so-called Barkhausen noise [12]. Instructural transitions, avalanches can also be detectedtypically by acoustic emission techniques [20]. Knowl-edge of the distribution of the number of avalanches alongthe transition, as well as their size and duration, is animportant piece of information in order to characterizeathermal FOPT.Good discrimination of individual avalanches in thesimulations can only be performed in the limit of ∆ H →
0. This will require a true adiabatic simulation algorithmwhich is not available at present, as opposed to the caseof the standard RFIM [14]. In our case, after a small (butfinite) change ∆ H some spins can be updated. We willconsider all of them as being part of a single avalanche.This is an approximation and, therefore, we should keepin mind the fact that we are slightly overestimating thesize of the observed avalanches due to some unavoidableoverlaps. In the experiments recording avalanches thesame problem occurs [21, 22].With this definition we can study, for instance, theaverage number of avalanches h N av i as a function of theexternal field H . As an example, in Fig. 14(a) we presentthe behavior of this number for the case λ = − σ = 4, ρ = 0, compared with the behavior of the average hys-teresis loop in Fig. 14 (b). h N av i presents two peaksin correspondence with the two coercive fields and goesto zero (as expected) at the m = 0 and m = 1 satura-tions. This kind of information is very relevant for theunderstanding of the measurements of acoustic emissionin structural transitions using the pulse-counting tech-nique [23].More interesting information can be obtained by mea-suring the avalanche sizes S and computing the inte-grated probability distribution P ( S ), by analyzing allthe avalanches in a single branch of the loop (the twobranches must be analyzed separately since they are notsymmetric). As a naive approximation, in our case onecan define in our case the size S of the avalanche asthe order parameter variation ∆ M associated with anavalanche (i.e. when the field is varied by ∆ H .) Never-theless, this definition imported from the standard RFIMshould be carefully adapted to our multivariant FOPT.Inside an avalanche, in fact, one can distinguish betweendifferent kinds of processes taking place, depending ontheir effect on the order parameter variation. Let us focuson the decreasing field branch starting from the m = 0 FIG. 13: (Color on line) Parameters: L = 32, λ = − σ = 1 .
5, ∆ H = 0 .
05 and ρ = 0. The configuration snapshots are takenfor (1) m = 0 .
03; (2) m = 0 .
98; (3) m = 1; (4) m = 0 .
78; (5) m = 0 .
28 and (6) m = 0 . phase up to the m = 1 saturated configuration: there areseveral microscopic possibilities for a spin jump. A spincould jump from the 0 state to one of the three variants x , y and z thus giving rise to a positive contribution to themagnetization change ∆ M ; it could jump from the x , y or z states to 0, causing a negative contribution ∆ M < − and 0. Instead of only measuring the totalsize ∆ M of an avalanche, for each of them we will mea-sure the three quantities n + , n − and n , which are thenumber of spin updates of each kind. Moreover, since aswe have seen in section IV A, the hysteresis loops are notsymmetric, we should separately analyze the avalanchesduring the decreasing field branch and during the in-creasing field branch. This gives, therefore, 6 possibledistributions: P down + ( n ), P down − ( n ), P down ( n ) for the de-creasing field branch and P up + ( n ), P up − ( n ), P up ( n ) for theincreasing field branch. It should be mentioned that inthe increasing field branch the total number of events ofthe + and 0 kind are much smaller than the number of − events, typically by 2-3 orders of magnitude. For thedecreasing branch the − events are rare, but 0 eventsare frequent, since an important number of transitionswithin variants may occur during the avalanches in thelater stages of the transition. Of course such an optimiza-tion between variants cannot occur in the reverse processduring the increasing field branch.In Figs. 15 and 16 we show some examples of the prob-ability distributions for varying values of the two disorderparameters on log-log plots, respectively σ and ρ . Actu-ally, we show only the P down + ( n ) and the P up − ( n ) distri-butions which account for the majority of the events. InFig. 15(b) it is possible to notice another finite size effect:the slab domains of size corresponding to multiples of thesystem size L present the tendency to disappear abruptly(and thus the P − up ( n ) presents some peaks in correspon-dence of values multiple of L ), as we have already pointedout in section IV B.An interesting result for the case of ρ = 0 (seeFig. 15(a)) is that the distribution P down + ( n ) shows aexponentially damped behavior but seems to become apower-law for a certain critical value of σ (which will0 H < N a v > H M (a) (b) FIG. 14: (a) Number of avalanches N av (arbitrary units) asa function of the external field. N av is computed in each∆ H interval over 600 loops, and (b): The related averagedhysteresis cycle. Simulation parameters: L = 16, ∆ H = 0 . λ = − σ = 4, ρ = 0. be located around σ ≃ . ± .
03 [24], but this fea-ture could be due to finite size effect. Only a detailedfinite-size scaling analysis and the study of the geometryof the avalanches [25] (out of the scope of this paper)will reveal if the observed power law conforms to univer-sal behavior or not. Interestingly it seems that for theincreasing field branch (see Fig. 15(b)), the distributionis always exponentially damped, at least for the studiedrange of values of σ . Therefore, the critical point for theincreasing field branch, if it exists, would be located at adifferent (smaller) value of σ . V. SUMMARY
The analysis of microstructure formation in ferroic sys-tems undergoing a first-order phase transitions is an in-teresting issue both from a purely theoretical and an ap-plicative point of view. Microstructures arise since theproduct phase, arising from the balance of many ener-getic terms, may show energetically equivalent variants.Despite the interest in this issue, the models that havebeen used up to now for the study of the interplay be-tween disorder and athermal evolution (for example, theRandom Field Ising model [11]) are not suitable for theanalysis of microstructure formation, due to the equiva-lence of the variants of the product phase.In the present work, we have introduced a modifica-tion of the Random Field Potts model, which consists ofadding a dipolar term truncated to the nearest-neighborapproximation, which represents a promising step to- × × × n × -6 × -4 × -2 P + do w n ( n ) σ =3 σ =4 σ =6 σ =8 × × × n × -6 × -4 × -2 P - up ( n ) σ =3 σ =4 σ =6 σ =8 (a)(b) FIG. 15: (Color on line) Avalanche size probability distribu-tions P down + ( n ) (a) and P up − ( n ) (b) corresponding to the twobranches of the hysteresis loop, averaged over 600 disorder re-alizations and for four values of σ as indicated by the legend.Simulation parameters are : L = 16, ∆ H = 0 . λ = − ρ = 0. wards the analysis of athermal transitions from a degen-erate to a multivariant phase.In our simulations we have chosen extremal updatingin order to preserve the independence of the trajectoryfrom the applied field rate. We have studied the depen-dence of the hysteresis loop shape on the hamiltonianparameters values. From a quantitative point of view,this has been performed by measuring the the loop area,its asymmetry and width. Our loops display a large areaand asymmetry regime for very negative values of the’dipolar’ term parameter λ , associated with the forma-tion of microstructures with prolate domains, orientedalong three equivalent spatial directions. On the otherhand, for λ > × × × n × -6 × -4 × -2 × P + do w n ( n ) ρ =1 ρ =4 ρ =8 × × × n × -6 × -4 × -2 × P - up ( n ) ρ =1 ρ =4 ρ =8 (a)(b) FIG. 16: (Color online) Avalanche size probability distribu-tions P down + ( n ) and P up − ( n ) corresponding to the two branchesof the hysteresis loop, averaged over 600 disorder realizationsand for three valued of ρ . Simulation parameters are: L = 16,∆ H = 0 . λ = − σ = 8. Acknowledgments
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