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Physica B 343 (2004) 281–285

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doi:10.1016

Avalanche properties of the 3d-RFIM

Eduard Vives*, Francisco-Jos!e P!erez-Reche

Departament d’Estructura i Constituents de la Mat"eria. Universitat de Barcelona. Diagonal 647, Facultat de F!ısica, 08028 Barcelona,

Catalonia, Spain

Abstract

The behaviour of the hysteresis loop in the 3d-Gaussian Random-Field Ising Model at T ¼ 0 has been studied by

numerical simulations. We have reviewed the relation between the macroscopic behaviour of the system (magnetization

hysteresis loops) and the microscopic observations (avalanches). Conclusions are drawn on a scenario for the behaviour

of the system in the thermodynamic limit and give, from a microscopic point of view, the reason for the existence of a

macroscopic jump (magnetization discontinuity) in the hysteresis loop for disorder values that are lower than the

critical amount of disorder.

r 2003 Elsevier B.V. All rights reserved.

Keywords: Hysteresis; Avalanches; Magnetization discontinuity

1. Introduction

Hysteresis occur in driven systems which evolvefollowing an out-of-equilibrium path usuallyaffected by the presence of disorder. We focushere on so-called rate-independent hysteresis,which is observed when the system is quasi-statically driven and thermal fluctuations areirrelevant [1]. This phenomenon occurs in a broadset of experimental systems [2] displaying first-order phase transitions. For instance, this inde-pendence on the driving rate is observed inBarkhausen effect experiments in ferromagnets[3] and in acoustic emission detected duringmartensitic transformations [4]. When driventhrough the transition all these systems evolvediscontinuously and exhibit avalanches. Thisphenomenology has been successfully described

onding author. Fax: +39-34-934-021-174.

ddress: eduard@ecm.ub.es (E. Vives).

- see front matter r 2003 Elsevier B.V. All rights reserve

/j.physb.2003.08.107

by lattice models at T ¼ 0 with metastabledynamics. The prototype model is the 3d-RandomField Ising Model (3d-RFIM) with metastabledynamics driven by an external field B: Disorder isintroduced using random independent local fieldsand its amount is controlled by the standarddeviation s of the distribution of the randomfields. Since the model was introduced [5], it hasbeen seen that the response of the system when it isdriven exhibits both hysteresis and avalanches.From a macroscopic point of view the beha-

viour of hysteresis loops is different depending onthe amount of disorder. For low disorders, apartfrom a certain continuous part, the magnetizationexhibits a large jump, in which an importantfraction of the system is reversed (Fig. 1(a)). Onthe other hand, for large amounts of disorder(Fig. 1(b)), the large jump disappears and the loopis continuous.It is believed that this change in the shape of the

hysteresis loops occurs for a disorder sc defined as

d.

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−5 0

−1

0

1

B5

m(B

)

(a) (b)

Fig. 1. Examples of hysteresis cycles corresponding to (a) s ¼2:0 and (b) s ¼ 3:0: System size L ¼ 48:

E. Vives, F.-J. P!erez-Reche / Physica B 343 (2004) 281–285282

the limiting disorder between sharp ðsoscÞ andsmooth ðs > scÞ hysteresis loops. This behaviourhas been understood by assuming the existence ofa T ¼ 0 critical point ðsc;BcÞ on the metastablephase diagram. The aim of this paper is to studythe connection between the macroscopic observa-tions (behaviour of the hysteresis loop) and themicroscopic phenomena (avalanches) in the 3d-RFIM. We would like to address the question ofhow the avalanches must behave in order toproduce a macroscopic jump in the magnetization.This is done by performing numerical simulationsand analyzing the results with finite-size scaling(FSS) techniques. This allows us to propose ascenario for avalanche behaviour in the thermo-dynamic limit.

Table 1

Notation of the average number of spanning avalanches in a

half loop. We have indicated the scaling relation for the

numbers to which only one type of avalanche contributes. We

have classified the quantities into two groups depending on

whether they are measured during the simulation (above) or

(below) they are calculated later from the data directly obtained

using the methods described in Ref. [6]

Avalanche kind Dependence

1d-spanning N1 ¼ Ly *N1ðuL1=nÞ2d-spanning N2 ¼ Ly *N2ðuL1=nÞ3d-spanning N3 ¼ N3c þ N3�

Critical 3d-spanning N3c ¼ Ly *N3cðuL1=nÞSubcritical 3d-spanning N3� ¼ *N3�ðuL1=nÞ

2. Model

The 3d-RFIM is defined on a cubic lattice of sizeL � L � L: At each lattice site, there is a spinvariable Si: The Hamiltonian of this set of spins iswritten as

H ¼ �Xn:n:

i; j

SiSj �X

i

Sihi � BX

i

Si; ð1Þ

where the sum in the first term extends over all thenearest-neighbour (n.n.) pairs, the second termstands for the local interaction with a set ofquenched random fields fhig; and the last termtakes into account the interaction with a homo-geneous external field B: The random fields fhigare independent and Gaussian distributed withzero mean and standard deviation s:

The external magnetic field B drives the systemthrough the transition. Standard synchronousrelaxation dynamics is used to induce the spinsto flip and thus obtain the hysteresis loops [5,6].The model exhibits hysteresis and avalancheswhen this type of dynamics is used. The avalanchesare characterized by their size s (number ofreversed spins). For each half loop ðB ¼ þN-�NÞ; we analyze the number and size distributionof the avalanches. Once an avalanche has termi-nated we classify it as being non-spanning, 1d-spanning, 2d-spanning, and 3d-spanning depend-ing on whether or not it spans the lattice in thedifferent space directions. We have simulatedsystems with different sizes (from L ¼ 5 to 48)and, for each size, we have studied differentamounts of disorder s: In order to improve thestatistics we have also computed averages overmany disorder realizations.

3. Results. FSS treatment

As we have already mentioned, during thesimulation we classify the avalanches dependingon their spanning properties. For each studied size,we measure the average number of avalanches ofeach kind that are detected in a hysteresis half loopfor different values of s: In this way we directlyobtain from the simulation the average numbers ofspanning avalanches N1; N2; and N3 (see Table 1for the definition of each number). Fig. 2 shows

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0.0

0.5

1.0

1.5

0.0

0.5

1.0

1.5

1.0 1.5 2.0 2.5 3.0 3.5 4.00.5

1.0

1.5

σ

N1

(σ,L

)

(a)

(b)

(c)

N2

(σ,L

)N

3 (σ

,L)

L=5L=8L=10L=12L=16L=24L=32L=48

Fig. 2. Number of spanning avalanches in (a) one dimension,

(b) two dimensions, and (c) three dimensions. Different sizes are

indicated in the legend.

−4 −2 0 4 8uL1/�2 6

L−�N1(�,L)L−�N2(�,L)Ñ3c(uL1/�)Ñ3−(uL1/�)

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Fig. 3. Scaling collapses characterizing the number of ava-

lanches of each kind. For each set of points there are data

corresponding to sizes ranging from L ¼ 5 to 48.

E. Vives, F.-J. P!erez-Reche / Physica B 343 (2004) 281–285 283

these numbers for different system sizes. As can beseen, both N1 and N2 (Figs. 2(a) and (b)) exhibit apeak at a certain value of s that approaches a fixedvalue for large systems. This value is identifiedwith the critical amount of disorder ðscC2:21Þ: Onthe other hand, the height of the peak increaseswith L: The number of 3d-spanning avalanches N3

(Fig. 2(c)) also exhibits a peak with a behavioursimilar to that observed for N1 and N2 but, in thiscase, the peak appears to be summed to a functionthat seems to approach a step function (N3 ¼ 1 forsosc and N3 ¼ 0 for s > sc) in the thermody-namic limit. Moreover, analyzing these results bythe FSS technique [6], we have been able toidentify two different types of 3d-spanning ava-lanches (N3c and N3�). In Table 1 we indicate thenames for these quantities obtained by indirectmethods and their FSS behaviour. As regards thenumber of non-spanning avalanches, they increasewhen s is larger and, at fixed s; they increase forlarger values of L: As we will see, this kind ofavalanche will not be of major importance for theproper understanding of the magnetization jump,but they are important for the complete under-

standing of the hysteresis loops. Fig. 3 shows thescaling collapses obtained for the number ofspanning avalanches of each kind. The scalingdependence is indicated in Table 1. In these fits wehave used sc ¼ 2:21 and a second-order approx-imation u ¼ ðs� scÞ=sc � 0:2½ðs� scÞ=sc2 for thescaling variable u: In this way, the criticalexponents obtained are n ¼ 1:270:1 and y ¼0:1070:02: These values of sc and n are slightlydifferent from previously reported values [7]. Thisis due to the different scaling variable that we haveused.For the purposes of this paper it is also

interesting to compute the average number ofspins belonging to each type of avalanche in a halfloop (we will refer to this quantity as the volumefilled by each type of avalanche). For a given kinda; this quantity is obtained by multiplying thenumber of avalanches Na by their mean size /sSa:The mean size of spanning avalanches is largerthan that for non-spanning. We will restrict thestudy of the filled volume to the spanningavalanches because only spanning avalanches cancontribute to the discontinuity in the hysteresisloop. In Table 2 we summarize the scalingrelations for the filled volume and the correspond-ing scaling collapses are shown in Fig. 4. In thescaling relations introduced in Table 2 we indicatethe fractal dimension d3� of the subcritical 3d-spanning avalanche and the fractal dimension df ofthe 1d-, 2d-, and critical 3d-spanning avalanches.

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−4 −2 0 4 8

0.60.00

0.02

0.04

Θ1(uL1/�

)

Θ2(uL1/�)

Θ3c

(uL1/�)Θ

3−(uL1/�)

1.2

1.0

0.8

0.6

0.4

0.2

0.0

2 6uL1/�

log(

Θ3−

)

log(|u|L1/�)0.4 0.8

Fig. 4. Scaling collapses corresponding to the filled volume of

the spanning avalanches. The inset reveals the power-law

behaviour Y3�ðuL1=nÞ ¼ ðjujL1=nÞb3� with b3� ¼ 0:024: As inFig. 3, there are data corresponding to sizes ranging from L ¼ 5

to 48.

Table 2

Scaling relations for the volume filled by the spanning

avalanches

Avalanche kind Scaling relation

1d N1/sS1 ¼ LyþdfY1ðuL1=nÞ2d N2/sS2 ¼ LyþdfY2ðuL1=nÞ3c N3c/sS3c ¼ LyþdfY3cðuL1=nÞ3� N3�/sS3� ¼ Ld3�Y3�ðuL1=nÞ

E. Vives, F.-J. P!erez-Reche / Physica B 343 (2004) 281–285284

From the collapses we obtain d3� ¼ 2:9870:02and df ¼ 2:7870:05:

4. Discussion and conclusions

From the scaling collapses in Fig. 3 and thescaling equations given in Table 1 it is possible todeduce the behaviour of the number for each typeof avalanche in the thermodynamic limit ðL-NÞ:Apart from the non-spanning avalanches that existfor all s > 0; we find:

* For sosc; only one spanning avalanche coex-ists with the non-spanning avalanches. Thisspanning avalanche is of the 3� kind (sub-critical 3d-spanning) because *N3� goes expo-nentially to 1 for uo0: The correspondingscaling relations for the other kinds decayexponentially to zero for uo0: This implies

that in the thermodynamic limit these ava-lanches do not exist for sosc:

* For s ¼ sc: The four kinds of spanningavalanches exist together with the non-spanningavalanches. There is, on average, approximately0:8 subcritical 3d-spanning avalanches and aninfinite number of the other kinds (since y > 0).

* For s > sc only non-spanning avalanches existsince the scaling relation for the other kindsdecays exponentially for u > 0:

These observations allow us to conclude that thelarge jump observed in the hysteresis loop forsosc may be associated with the existence of onesubcritical 3d-spanning avalanche. Besides, wehave shown above that this subcritical 3d-span-ning avalanche is characterized by a fractaldimension d3� ¼ 2:98: On the other hand, for anavalanche to contribute a macroscopic jump to themagnetization, it is necessary that this avalanchefills a finite fraction of the system volume. Thequestion is therefore how is it possible that afractal object fills a finite fraction of the system?From Table 2, the fraction f3� filled by the

subcritical 3d-spanning avalanche can be writtenas

f3�ðs;LÞ ¼N3�/sS3�

L3¼ Ld3��3Y3�ðuL1=nÞ: ð2Þ

On the other hand, as shown in the inset of Fig. 4,for sosc ðuo0Þ; the scaling function Y3� showsan asymptotic power-law behaviour Y3�ðuL1=nÞ ¼ðjujL1=nÞb3� with b3� ¼ 0:02470:012: Given thisvalue, within error bars, one can say that thefollowing hyperscaling relation holds:

b3� ¼ nð3� d3�Þ: ð3Þ

From this relation we conclude that, in thethermodynamic limit, the volume filled by thesubcritical 3d-spanning avalanche is given byN3�/sS3� ¼ L3jujb3� : This indicates that, belowsc; this object does not behave as fractal at lengthscales comparable to the system size (large lengthscales). In this limit, from Eq. (2) one obtains thatthe fraction filled by the subcritical 3d-spanningavalanche is finite and can be written as

f3�ðsÞ ¼ jujb3� ; sosc: ð4Þ

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E. Vives, F.-J. P!erez-Reche / Physica B 343 (2004) 281–285 285

Note that for this to occur, it is crucial that thehyperscaling relation (3) is satisfied.As we have demonstrated for sosc; apart from the

subcritical 3d-spanning avalanche, only non-spanningavalanches exist. This implies that the fraction of sitesnot belonging to the subcritical 3d-spanning avalancheare filled by the non-spanning avalanches.On the other hand, at the critical amount of

disorder (u ¼ 0), the volume filled by the sub-critical 3d-spanning avalanche is N3�/sS3� ¼Ld3�Y3�ð0Þ and, as a consequence, in the thermo-dynamic limit f3� ¼ 0: This indicates that thesubcritical 3d-spanning avalanche is fractal to alllength scales for s ¼ sc only.In fact, as occurs with the infinite cluster in the

problem of percolation [8], the subcritical 3d-spanning avalanche looks homogeneous (non-fractal) on length scales larger than the correlationlength. It looks fractal for smaller length scales. Inour case, at s ¼ sc the correlation length divergesand, therefore, in the thermodynamic limit thesubcritical 3d-spanning avalanche looks fractal.For sosc the correlation length is finite and,therefore, for large length scales the subcritical 3d-spanning avalanche looks homogeneous and fills afinite fraction of the system.

Acknowledgements

This work has received financial support fromCICyT (Project No MAT2001-3251) andCIRIT (Project 2001SGR00066). F.J.P. ack-nowledges financial support from DGICyT(Spain).

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