- Email: [email protected]
Avalanche properties of the 3d-RFIM
ARTICLE IN PRESS
Physica B 343 (2004) 281–285
Avalanche properties of the 3d-RFIM Eduard Vives*, Francisco-Jose! Pe! rez-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 evolve following an out-of-equilibrium path usually affected by the presence of disorder. We focus here on so-called rate-independent hysteresis, which is observed when the system is quasistatically driven and thermal fluctuations are irrelevant [1]. This phenomenon occurs in a broad set of experimental systems [2] displaying firstorder phase transitions. For instance, this independence on the driving rate is observed in Barkhausen effect experiments in ferromagnets [3] and in acoustic emission detected during martensitic transformations [4]. When driven through the transition all these systems evolve discontinuously and exhibit avalanches. This phenomenology has been successfully described *Corresponding author. Fax: +39-34-934-021-174. E-mail address: [email protected] (E. Vives).
by lattice models at T ¼ 0 with metastable dynamics. The prototype model is the 3d-Random Field Ising Model (3d-RFIM) with metastable dynamics driven by an external field B: Disorder is introduced using random independent local fields and its amount is controlled by the standard deviation s of the distribution of the random fields. Since the model was introduced [5], it has been seen that the response of the system when it is driven exhibits both hysteresis and avalanches. From a macroscopic point of view the behaviour of hysteresis loops is different depending on the amount of disorder. For low disorders, apart from a certain continuous part, the magnetization exhibits a large jump, in which an important fraction of the system is reversed (Fig. 1(a)). On the other hand, for large amounts of disorder (Fig. 1(b)), the large jump disappears and the loop is continuous. It is believed that this change in the shape of the hysteresis loops occurs for a disorder sc defined as
0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2003.08.107
ARTICLE IN PRESS E. Vives, F.-J. P!erez-Reche / Physica B 343 (2004) 281–285
282
m(B)
1 (a)
(b)
0
−1 −5
5 B
0
Fig. 1. Examples of hysteresis cycles corresponding to (a) s ¼ 2:0 and (b) s ¼ 3:0: System size L ¼ 48:
the limiting disorder between sharp ðsosc Þ and smooth ðs > sc Þ hysteresis loops. This behaviour has been understood by assuming the existence of a T ¼ 0 critical point ðsc ; Bc Þ on the metastable phase diagram. The aim of this paper is to study the connection between the macroscopic observations (behaviour of the hysteresis loop) and the microscopic phenomena (avalanches) in the 3dRFIM. We would like to address the question of how the avalanches must behave in order to produce a macroscopic jump in the magnetization. This is done by performing numerical simulations and analyzing the results with finite-size scaling (FSS) techniques. This allows us to propose a scenario for avalanche behaviour in the thermodynamic limit.
2. Model The 3d-RFIM is defined on a cubic lattice of size L L L: At each lattice site, there is a spin variable Si : The Hamiltonian of this set of spins is written as H¼
n:n: X i; j
Si Sj
X i
Si hi B
X
Si ;
ð1Þ
i
where the sum in the first term extends over all the nearest-neighbour (n.n.) pairs, the second term stands for the local interaction with a set of quenched random fields fhi g; and the last term takes into account the interaction with a homogeneous external field B: The random fields fhi g are independent and Gaussian distributed with zero mean and standard deviation s:
The external magnetic field B drives the system through the transition. Standard synchronous relaxation dynamics is used to induce the spins to flip and thus obtain the hysteresis loops [5,6]. The model exhibits hysteresis and avalanches when this type of dynamics is used. The avalanches are characterized by their size s (number of reversed spins). For each half loop ðB ¼ þNNÞ; we analyze the number and size distribution of the avalanches. Once an avalanche has terminated we classify it as being non-spanning, 1dspanning, 2d-spanning, and 3d-spanning depending on whether or not it spans the lattice in the different space directions. We have simulated systems with different sizes (from L ¼ 5 to 48) and, for each size, we have studied different amounts of disorder s: In order to improve the statistics we have also computed averages over many disorder realizations.
3. Results. FSS treatment As we have already mentioned, during the simulation we classify the avalanches depending on their spanning properties. For each studied size, we measure the average number of avalanches of each kind that are detected in a hysteresis half loop for different values of s: In this way we directly obtain from the simulation the average numbers of spanning avalanches N1 ; N2 ; and N3 (see Table 1 for the definition of each number). Fig. 2 shows 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 2d-spanning 3d-spanning
N1 ¼ Ly N* 1 ðuL1=n Þ N2 ¼ Ly N* 2 ðuL1=n Þ N3 ¼ N3c þ N3
Critical 3d-spanning Subcritical 3d-spanning
N3c ¼ Ly N* 3c ðuL1=n Þ N3 ¼ N* 3 ðuL1=n Þ
ARTICLE IN PRESS E. Vives, F.-J. P!erez-Reche / Physica B 343 (2004) 281–285
N1 (σ,L)
1.5 1.0 0.5
L=5 L=8 L=10 L=12 L=16 L=24 L=32 L=48
(a)
L−N1(,L) L−N2(,L) Ñ3c(uL1/) Ñ3−(uL1/)
0.8 0.6 (b)
N2 (σ,L)
1.2 1.0
0.0 1.5
0.4
1.0 0.2
0.5
0.0
0.0
−4
1.5 (c) N3 (σ,L)
283
0
2 uL1/
4
6
8
Fig. 3. Scaling collapses characterizing the number of avalanches of each kind. For each set of points there are data corresponding to sizes ranging from L ¼ 5 to 48.
1.0
0.5 1.0
−2
1.5
2.0
2.5 σ
3.0
3.5
4.0
Fig. 2. Number of spanning avalanches in (a) one dimension, (b) two dimensions, and (c) three dimensions. Different sizes are indicated in the legend.
these numbers for different system sizes. As can be seen, both N1 and N2 (Figs. 2(a) and (b)) exhibit a peak at a certain value of s that approaches a fixed value for large systems. This value is identified with the critical amount of disorder ðsc C2:21Þ: On the other hand, the height of the peak increases with L: The number of 3d-spanning avalanches N3 (Fig. 2(c)) also exhibits a peak with a behaviour similar to that observed for N1 and N2 but, in this case, the peak appears to be summed to a function that seems to approach a step function (N3 ¼ 1 for sosc and N3 ¼ 0 for s > sc ) in the thermodynamic limit. Moreover, analyzing these results by the FSS technique [6], we have been able to identify two different types of 3d-spanning avalanches (N3c and N3 ). In Table 1 we indicate the names for these quantities obtained by indirect methods and their FSS behaviour. As regards the number of non-spanning avalanches, they increase when s is larger and, at fixed s; they increase for larger values of L: As we will see, this kind of avalanche will not be of major importance for the proper understanding of the magnetization jump, but they are important for the complete under-
standing of the hysteresis loops. Fig. 3 shows the scaling collapses obtained for the number of spanning avalanches of each kind. The scaling dependence is indicated in Table 1. In these fits we have used sc ¼ 2:21 and a second-order approximation u ¼ ðs sc Þ=sc 0:2½ðs sc Þ=sc 2 for the scaling variable u: In this way, the critical exponents obtained are n ¼ 1:270:1 and y ¼ 0:1070:02: These values of sc and n are slightly different from previously reported values [7]. This is due to the different scaling variable that we have used. For the purposes of this paper it is also interesting to compute the average number of spins belonging to each type of avalanche in a half loop (we will refer to this quantity as the volume filled by each type of avalanche). For a given kind a; this quantity is obtained by multiplying the number of avalanches Na by their mean size /sSa : The mean size of spanning avalanches is larger than that for non-spanning. We will restrict the study of the filled volume to the spanning avalanches because only spanning avalanches can contribute to the discontinuity in the hysteresis loop. In Table 2 we summarize the scaling relations for the filled volume and the corresponding scaling collapses are shown in Fig. 4. In the scaling relations introduced in Table 2 we indicate the fractal dimension d3 of the subcritical 3dspanning avalanche and the fractal dimension df of the 1d-, 2d-, and critical 3d-spanning avalanches.
ARTICLE IN PRESS E. Vives, F.-J. P!erez-Reche / Physica B 343 (2004) 281–285
284
Table 2 Scaling relations for the volume filled by the spanning avalanches Avalanche kind
Scaling relation
1d 2d 3c 3
N1 /sS1 ¼ Lyþdf Y1 ðuL1=n Þ N2 /sS2 ¼ Lyþdf Y2 ðuL1=n Þ N3c /sS3c ¼ Lyþdf Y3c ðuL1=n Þ N3 /sS3 ¼ Ld3 Y3 ðuL1=n Þ
1.2 log(Θ3−)
0.04
1.0 0.8 0.6 0.4
Θ1(uL1/) Θ2(uL1/) Θ3c(uL1/) Θ3−(uL1/)
0.02
0.00
0.4
0.6
0.8
log(|u|L1/)
0.2 0.0 −4
−2
0
2 uL1/
4
6
8
Fig. 4. Scaling collapses corresponding to the filled volume of the spanning avalanches. The inset reveals the power-law behaviour Y3 ðuL1=n Þ ¼ ðjujL1=n Þb3 with b3 ¼ 0:024: As in Fig. 3, there are data corresponding to sizes ranging from L ¼ 5 to 48.
*
*
that in the thermodynamic limit these avalanches do not exist for sosc : For s ¼ sc : The four kinds of spanning avalanches exist together with the non-spanning avalanches. There is, on average, approximately 0:8 subcritical 3d-spanning avalanches and an infinite number of the other kinds (since y > 0). For s > sc only non-spanning avalanches exist since the scaling relation for the other kinds decays exponentially for u > 0:
These observations allow us to conclude that the large jump observed in the hysteresis loop for sosc may be associated with the existence of one subcritical 3d-spanning avalanche. Besides, we have shown above that this subcritical 3d-spanning avalanche is characterized by a fractal dimension d3 ¼ 2:98: On the other hand, for an avalanche to contribute a macroscopic jump to the magnetization, it is necessary that this avalanche fills a finite fraction of the system volume. The question is therefore how is it possible that a fractal object fills a finite fraction of the system? From Table 2, the fraction f3 filled by the subcritical 3d-spanning avalanche can be written as f3 ðs; LÞ ¼
From the collapses we obtain d3 ¼ 2:9870:02 and df ¼ 2:7870:05:
4. Discussion and conclusions From the scaling collapses in Fig. 3 and the scaling equations given in Table 1 it is possible to deduce the behaviour of the number for each type of avalanche in the thermodynamic limit ðL-NÞ: Apart from the non-spanning avalanches that exist for all s > 0; we find: *
For sosc ; only one spanning avalanche coexists with the non-spanning avalanches. This spanning avalanche is of the 3 kind (subcritical 3d-spanning) because N* 3 goes exponentially to 1 for uo0: The corresponding scaling relations for the other kinds decay exponentially to zero for uo0: This implies
N3 /sS3 ¼ Ld3 3 Y3 ðuL1=n Þ: L3
ð2Þ
On the other hand, as shown in the inset of Fig. 4, for sosc ðuo0Þ; the scaling function Y3 shows an asymptotic power-law behaviour Y3 ðuL1=n Þ ¼ ðjujL1=n Þb3 with b3 ¼ 0:02470:012: Given this value, within error bars, one can say that the following hyperscaling relation holds: b3 ¼ nð3 d3 Þ:
ð3Þ
From this relation we conclude that, in the thermodynamic limit, the volume filled by the subcritical 3d-spanning avalanche is given by N3 /sS3 ¼ L3 jujb3 : This indicates that, below sc ; this object does not behave as fractal at length scales comparable to the system size (large length scales). In this limit, from Eq. (2) one obtains that the fraction filled by the subcritical 3d-spanning avalanche is finite and can be written as f3 ðsÞ ¼ jujb3 ;
sosc :
ð4Þ
ARTICLE IN PRESS E. Vives, F.-J. P!erez-Reche / Physica B 343 (2004) 281–285
Note that for this to occur, it is crucial that the hyperscaling relation (3) is satisfied. As we have demonstrated for sosc ; apart from the subcritical 3d-spanning avalanche, only non-spanning avalanches exist. This implies that the fraction of sites not belonging to the subcritical 3d-spanning avalanche are filled by the non-spanning avalanches. On the other hand, at the critical amount of disorder (u ¼ 0), the volume filled by the subcritical 3d-spanning avalanche is N3 /sS3 ¼ Ld3 Y3 ð0Þ and, as a consequence, in the thermodynamic limit f3 ¼ 0: This indicates that the subcritical 3d-spanning avalanche is fractal to all length scales for s ¼ sc only. In fact, as occurs with the infinite cluster in the problem of percolation [8], the subcritical 3dspanning avalanche looks homogeneous (nonfractal) on length scales larger than the correlation length. It looks fractal for smaller length scales. In our case, at s ¼ sc the correlation length diverges and, therefore, in the thermodynamic limit the subcritical 3d-spanning avalanche looks fractal. For sosc the correlation length is finite and, therefore, for large length scales the subcritical 3dspanning avalanche looks homogeneous and fills a finite fraction of the system.
285
Acknowledgements This work has received financial support from CICyT (Project No MAT2001-3251) and CIRIT (Project 2001SGR00066). F.J.P. acknowledges financial support from DGICyT (Spain).
References [1] G. Bertotti, Hysteresis in Magnetism, Academic Press, San Diego, 1998. [2] J.P. Sethna, K.A. Dahmen, C.R. Myers, Nature 410 (2001) 242. [3] J.S. Urbach, R.C. Madison, J.T. Market, Phys. Rev. Lett. 75 (1995) 4694. [4] F.J. P!erez-Reche, E. Vives, Ll. Man˜osa, A. Planes, Phys. Rev. Lett. 87 (2001) 195701. [5] J.P. Sethna, K. Dahmen, S. Kartha, J.A. Krumhansl, B.W. Roberts, J.D. Shore, Phys. Rev. Lett. 70 (1993) 3347. [6] F.J. P!erez-Reche, E. Vives, Phys. Rev. B 67 (2003) 134421. [7] O. Perkovi!c, K.A. Dahmen, J.P. Sethna, Phys. Rev. B 59 (1999) 6106. [8] J. Stauffer, A. Aharony, Introduction to Percolation Theory, 2nd Edition, Taylor & Francis, London, 1994.
Physica B 343 (2004) 281–285
Avalanche properties of the 3d-RFIM Eduard Vives*, Francisco-Jose! Pe! rez-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 evolve following an out-of-equilibrium path usually affected by the presence of disorder. We focus here on so-called rate-independent hysteresis, which is observed when the system is quasistatically driven and thermal fluctuations are irrelevant [1]. This phenomenon occurs in a broad set of experimental systems [2] displaying firstorder phase transitions. For instance, this independence on the driving rate is observed in Barkhausen effect experiments in ferromagnets [3] and in acoustic emission detected during martensitic transformations [4]. When driven through the transition all these systems evolve discontinuously and exhibit avalanches. This phenomenology has been successfully described *Corresponding author. Fax: +39-34-934-021-174. E-mail address: [email protected] (E. Vives).
by lattice models at T ¼ 0 with metastable dynamics. The prototype model is the 3d-Random Field Ising Model (3d-RFIM) with metastable dynamics driven by an external field B: Disorder is introduced using random independent local fields and its amount is controlled by the standard deviation s of the distribution of the random fields. Since the model was introduced [5], it has been seen that the response of the system when it is driven exhibits both hysteresis and avalanches. From a macroscopic point of view the behaviour of hysteresis loops is different depending on the amount of disorder. For low disorders, apart from a certain continuous part, the magnetization exhibits a large jump, in which an important fraction of the system is reversed (Fig. 1(a)). On the other hand, for large amounts of disorder (Fig. 1(b)), the large jump disappears and the loop is continuous. It is believed that this change in the shape of the hysteresis loops occurs for a disorder sc defined as
0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2003.08.107
ARTICLE IN PRESS E. Vives, F.-J. P!erez-Reche / Physica B 343 (2004) 281–285
282
m(B)
1 (a)
(b)
0
−1 −5
5 B
0
Fig. 1. Examples of hysteresis cycles corresponding to (a) s ¼ 2:0 and (b) s ¼ 3:0: System size L ¼ 48:
the limiting disorder between sharp ðsosc Þ and smooth ðs > sc Þ hysteresis loops. This behaviour has been understood by assuming the existence of a T ¼ 0 critical point ðsc ; Bc Þ on the metastable phase diagram. The aim of this paper is to study the connection between the macroscopic observations (behaviour of the hysteresis loop) and the microscopic phenomena (avalanches) in the 3dRFIM. We would like to address the question of how the avalanches must behave in order to produce a macroscopic jump in the magnetization. This is done by performing numerical simulations and analyzing the results with finite-size scaling (FSS) techniques. This allows us to propose a scenario for avalanche behaviour in the thermodynamic limit.
2. Model The 3d-RFIM is defined on a cubic lattice of size L L L: At each lattice site, there is a spin variable Si : The Hamiltonian of this set of spins is written as H¼
n:n: X i; j
Si Sj
X i
Si hi B
X
Si ;
ð1Þ
i
where the sum in the first term extends over all the nearest-neighbour (n.n.) pairs, the second term stands for the local interaction with a set of quenched random fields fhi g; and the last term takes into account the interaction with a homogeneous external field B: The random fields fhi g are independent and Gaussian distributed with zero mean and standard deviation s:
The external magnetic field B drives the system through the transition. Standard synchronous relaxation dynamics is used to induce the spins to flip and thus obtain the hysteresis loops [5,6]. The model exhibits hysteresis and avalanches when this type of dynamics is used. The avalanches are characterized by their size s (number of reversed spins). For each half loop ðB ¼ þNNÞ; we analyze the number and size distribution of the avalanches. Once an avalanche has terminated we classify it as being non-spanning, 1dspanning, 2d-spanning, and 3d-spanning depending on whether or not it spans the lattice in the different space directions. We have simulated systems with different sizes (from L ¼ 5 to 48) and, for each size, we have studied different amounts of disorder s: In order to improve the statistics we have also computed averages over many disorder realizations.
3. Results. FSS treatment As we have already mentioned, during the simulation we classify the avalanches depending on their spanning properties. For each studied size, we measure the average number of avalanches of each kind that are detected in a hysteresis half loop for different values of s: In this way we directly obtain from the simulation the average numbers of spanning avalanches N1 ; N2 ; and N3 (see Table 1 for the definition of each number). Fig. 2 shows 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 2d-spanning 3d-spanning
N1 ¼ Ly N* 1 ðuL1=n Þ N2 ¼ Ly N* 2 ðuL1=n Þ N3 ¼ N3c þ N3
Critical 3d-spanning Subcritical 3d-spanning
N3c ¼ Ly N* 3c ðuL1=n Þ N3 ¼ N* 3 ðuL1=n Þ
ARTICLE IN PRESS E. Vives, F.-J. P!erez-Reche / Physica B 343 (2004) 281–285
N1 (σ,L)
1.5 1.0 0.5
L=5 L=8 L=10 L=12 L=16 L=24 L=32 L=48
(a)
L−N1(,L) L−N2(,L) Ñ3c(uL1/) Ñ3−(uL1/)
0.8 0.6 (b)
N2 (σ,L)
1.2 1.0
0.0 1.5
0.4
1.0 0.2
0.5
0.0
0.0
−4
1.5 (c) N3 (σ,L)
283
0
2 uL1/
4
6
8
Fig. 3. Scaling collapses characterizing the number of avalanches of each kind. For each set of points there are data corresponding to sizes ranging from L ¼ 5 to 48.
1.0
0.5 1.0
−2
1.5
2.0
2.5 σ
3.0
3.5
4.0
Fig. 2. Number of spanning avalanches in (a) one dimension, (b) two dimensions, and (c) three dimensions. Different sizes are indicated in the legend.
these numbers for different system sizes. As can be seen, both N1 and N2 (Figs. 2(a) and (b)) exhibit a peak at a certain value of s that approaches a fixed value for large systems. This value is identified with the critical amount of disorder ðsc C2:21Þ: On the other hand, the height of the peak increases with L: The number of 3d-spanning avalanches N3 (Fig. 2(c)) also exhibits a peak with a behaviour similar to that observed for N1 and N2 but, in this case, the peak appears to be summed to a function that seems to approach a step function (N3 ¼ 1 for sosc and N3 ¼ 0 for s > sc ) in the thermodynamic limit. Moreover, analyzing these results by the FSS technique [6], we have been able to identify two different types of 3d-spanning avalanches (N3c and N3 ). In Table 1 we indicate the names for these quantities obtained by indirect methods and their FSS behaviour. As regards the number of non-spanning avalanches, they increase when s is larger and, at fixed s; they increase for larger values of L: As we will see, this kind of avalanche will not be of major importance for the proper understanding of the magnetization jump, but they are important for the complete under-
standing of the hysteresis loops. Fig. 3 shows the scaling collapses obtained for the number of spanning avalanches of each kind. The scaling dependence is indicated in Table 1. In these fits we have used sc ¼ 2:21 and a second-order approximation u ¼ ðs sc Þ=sc 0:2½ðs sc Þ=sc 2 for the scaling variable u: In this way, the critical exponents obtained are n ¼ 1:270:1 and y ¼ 0:1070:02: These values of sc and n are slightly different from previously reported values [7]. This is due to the different scaling variable that we have used. For the purposes of this paper it is also interesting to compute the average number of spins belonging to each type of avalanche in a half loop (we will refer to this quantity as the volume filled by each type of avalanche). For a given kind a; this quantity is obtained by multiplying the number of avalanches Na by their mean size /sSa : The mean size of spanning avalanches is larger than that for non-spanning. We will restrict the study of the filled volume to the spanning avalanches because only spanning avalanches can contribute to the discontinuity in the hysteresis loop. In Table 2 we summarize the scaling relations for the filled volume and the corresponding scaling collapses are shown in Fig. 4. In the scaling relations introduced in Table 2 we indicate the fractal dimension d3 of the subcritical 3dspanning avalanche and the fractal dimension df of the 1d-, 2d-, and critical 3d-spanning avalanches.
ARTICLE IN PRESS E. Vives, F.-J. P!erez-Reche / Physica B 343 (2004) 281–285
284
Table 2 Scaling relations for the volume filled by the spanning avalanches Avalanche kind
Scaling relation
1d 2d 3c 3
N1 /sS1 ¼ Lyþdf Y1 ðuL1=n Þ N2 /sS2 ¼ Lyþdf Y2 ðuL1=n Þ N3c /sS3c ¼ Lyþdf Y3c ðuL1=n Þ N3 /sS3 ¼ Ld3 Y3 ðuL1=n Þ
1.2 log(Θ3−)
0.04
1.0 0.8 0.6 0.4
Θ1(uL1/) Θ2(uL1/) Θ3c(uL1/) Θ3−(uL1/)
0.02
0.00
0.4
0.6
0.8
log(|u|L1/)
0.2 0.0 −4
−2
0
2 uL1/
4
6
8
Fig. 4. Scaling collapses corresponding to the filled volume of the spanning avalanches. The inset reveals the power-law behaviour Y3 ðuL1=n Þ ¼ ðjujL1=n Þb3 with b3 ¼ 0:024: As in Fig. 3, there are data corresponding to sizes ranging from L ¼ 5 to 48.
*
*
that in the thermodynamic limit these avalanches do not exist for sosc : For s ¼ sc : The four kinds of spanning avalanches exist together with the non-spanning avalanches. There is, on average, approximately 0:8 subcritical 3d-spanning avalanches and an infinite number of the other kinds (since y > 0). For s > sc only non-spanning avalanches exist since the scaling relation for the other kinds decays exponentially for u > 0:
These observations allow us to conclude that the large jump observed in the hysteresis loop for sosc may be associated with the existence of one subcritical 3d-spanning avalanche. Besides, we have shown above that this subcritical 3d-spanning avalanche is characterized by a fractal dimension d3 ¼ 2:98: On the other hand, for an avalanche to contribute a macroscopic jump to the magnetization, it is necessary that this avalanche fills a finite fraction of the system volume. The question is therefore how is it possible that a fractal object fills a finite fraction of the system? From Table 2, the fraction f3 filled by the subcritical 3d-spanning avalanche can be written as f3 ðs; LÞ ¼
From the collapses we obtain d3 ¼ 2:9870:02 and df ¼ 2:7870:05:
4. Discussion and conclusions From the scaling collapses in Fig. 3 and the scaling equations given in Table 1 it is possible to deduce the behaviour of the number for each type of avalanche in the thermodynamic limit ðL-NÞ: Apart from the non-spanning avalanches that exist for all s > 0; we find: *
For sosc ; only one spanning avalanche coexists with the non-spanning avalanches. This spanning avalanche is of the 3 kind (subcritical 3d-spanning) because N* 3 goes exponentially to 1 for uo0: The corresponding scaling relations for the other kinds decay exponentially to zero for uo0: This implies
N3 /sS3 ¼ Ld3 3 Y3 ðuL1=n Þ: L3
ð2Þ
On the other hand, as shown in the inset of Fig. 4, for sosc ðuo0Þ; the scaling function Y3 shows an asymptotic power-law behaviour Y3 ðuL1=n Þ ¼ ðjujL1=n Þb3 with b3 ¼ 0:02470:012: Given this value, within error bars, one can say that the following hyperscaling relation holds: b3 ¼ nð3 d3 Þ:
ð3Þ
From this relation we conclude that, in the thermodynamic limit, the volume filled by the subcritical 3d-spanning avalanche is given by N3 /sS3 ¼ L3 jujb3 : This indicates that, below sc ; this object does not behave as fractal at length scales comparable to the system size (large length scales). In this limit, from Eq. (2) one obtains that the fraction filled by the subcritical 3d-spanning avalanche is finite and can be written as f3 ðsÞ ¼ jujb3 ;
sosc :
ð4Þ
ARTICLE IN PRESS E. Vives, F.-J. P!erez-Reche / Physica B 343 (2004) 281–285
Note that for this to occur, it is crucial that the hyperscaling relation (3) is satisfied. As we have demonstrated for sosc ; apart from the subcritical 3d-spanning avalanche, only non-spanning avalanches exist. This implies that the fraction of sites not belonging to the subcritical 3d-spanning avalanche are filled by the non-spanning avalanches. On the other hand, at the critical amount of disorder (u ¼ 0), the volume filled by the subcritical 3d-spanning avalanche is N3 /sS3 ¼ Ld3 Y3 ð0Þ and, as a consequence, in the thermodynamic limit f3 ¼ 0: This indicates that the subcritical 3d-spanning avalanche is fractal to all length scales for s ¼ sc only. In fact, as occurs with the infinite cluster in the problem of percolation [8], the subcritical 3dspanning avalanche looks homogeneous (nonfractal) on length scales larger than the correlation length. It looks fractal for smaller length scales. In our case, at s ¼ sc the correlation length diverges and, therefore, in the thermodynamic limit the subcritical 3d-spanning avalanche looks fractal. For sosc the correlation length is finite and, therefore, for large length scales the subcritical 3dspanning avalanche looks homogeneous and fills a finite fraction of the system.
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Acknowledgements This work has received financial support from CICyT (Project No MAT2001-3251) and CIRIT (Project 2001SGR00066). F.J.P. acknowledges financial support from DGICyT (Spain).
References [1] G. Bertotti, Hysteresis in Magnetism, Academic Press, San Diego, 1998. [2] J.P. Sethna, K.A. Dahmen, C.R. Myers, Nature 410 (2001) 242. [3] J.S. Urbach, R.C. Madison, J.T. Market, Phys. Rev. Lett. 75 (1995) 4694. [4] F.J. P!erez-Reche, E. Vives, Ll. Man˜osa, A. Planes, Phys. Rev. Lett. 87 (2001) 195701. [5] J.P. Sethna, K. Dahmen, S. Kartha, J.A. Krumhansl, B.W. Roberts, J.D. Shore, Phys. Rev. Lett. 70 (1993) 3347. [6] F.J. P!erez-Reche, E. Vives, Phys. Rev. B 67 (2003) 134421. [7] O. Perkovi!c, K.A. Dahmen, J.P. Sethna, Phys. Rev. B 59 (1999) 6106. [8] J. Stauffer, A. Aharony, Introduction to Percolation Theory, 2nd Edition, Taylor & Francis, London, 1994.