PHYSICAL REVIEW E 101, 032116 (2020)

Geometry-induced nonequilibrium phase transition in sandpiles

M. N. Najafi,1,* J. Cheraghalizadeh ,1 M. Lukovic,2,3 and H. J. Herrmann4

1Department of Physics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran2Computational Physics, IfB, ETH Zurich, Stefano-Franscini-Platz 3, CH-8093 Zurich, Switzerland

3Cellulose and Wood Materials, Empa Swiss Federal Laboratories for Materials Science and Technology, CH-8600 Dübendorf, Switzerland4ESPCI, CNRS UMR 7636 - Laboratoire PMMH, F-75005 Paris, France

(Received 25 October 2019; revised manuscript received 26 January 2020; accepted 29 January 2020;published 12 March 2020)

We study the sandpile model on three-dimensional spanning Ising clusters with the temperature T treatedas the control parameter. By analyzing the three-dimensional avalanches and their two-dimensional projections(which show scale-invariant behavior for all temperatures), we uncover two universality classes with differentexponents (an ordinary BTW class, and SOCT =∞), along with a tricritical point (at Tc, the critical temperature ofthe host) between them. The transition between these two criticalities is induced by the transition in the support.The SOCT =∞ universality class is characterized by the exponent of the avalanche size distribution τ T =∞ =1.27 ± 0.03, consistent with the exponent of the size distribution of the Barkhausen avalanches in amorphousferromagnets Durin and Zapperi [Phys. Rev. Lett. 84, 4705 (2000)]. The tricritical point is characterized by itsown critical exponents. In addition to the avalanche exponents, some other quantities like the average height,the spanning avalanche probability (SAP), and the average coordination number of the Ising clusters changesignificantly the behavior at this point, and also exhibit power-law behavior in terms of ε ≡ T −Tc

Tc, defining further

critical exponents. Importantly, the finite-size analysis for the activity (number of topplings) per site shows thescaling behavior with exponents β = 0.19 ± 0.02 and ν = 0.75 ± 0.05. A similar behavior is also seen for theSAP and the average avalanche height. The fractal dimension of the external perimeter of the two-dimensionalprojections of avalanches is shown to be robust against T with the numerical value Df = 1.25 ± 0.01.

DOI: 10.1103/PhysRevE.101.032116

I. INTRODUCTION

In the context of out-of-equilibrium critical phenomena,self-organized critical (SOC) systems have attracted muchattention because of their role in a wide range of systems,from finance [1] and biological [2] to granular matter [3], thebrain [4], and neural networks in general [5]. SOC systemsare characterized by their avalanche dynamics resulting fromslow driving of the system. Vortex avalanche dynamics intype-II superconductors [6], earthquakes [7], solar flares [8],microfracturing processes [9], fluid flow in porous media[10], phase transitionlike behavior of the magnetosphere [11],bursts in filters [12], phase transitions in jammed granularmatter [3], and avalanch dynamics in the rat cortex [13] aresome natural examples for SOC. This large class of naturalsystems inspired theoretical models with the aim of capturingthe dominant internal dynamics that causes the avalanches.

Here we find evidence for a new nonequilibrium uni-versality class that is reached by changing the geometry ofthe underlying graph upon which the model is defined, andanalyze the transition point between SOC models. It mightbe applicable to experiments with spatial flow patterns oftransport in heterogeneous porous media [14], which involvethe toppling of fluid [15]. Another example is the Barkhauseneffect in magnetic systems [16], for which the avalanches have

*morteza.nattagh@gmail.com

been shown to exhibit scaling behavior with an avalanchesize exponent 1.27 ± 0.03 in amorphous ferromagnets (whichconstitutes a disordered medium) [17,18]. From a theoreticalperspective there is also an interest in critical phenomena onrandom geometries [19].

II. THE MODEL

We implement the dynamics of the Bak-Tang-Wiesenfeldmodel (BTW) [20], also known as the Abelian sandpilemodel, on a diluted cubic lattice. This lattice comprises sitesthat are either active (through which sand grains can pass)or inactive (completely impermeable to sand grains), whichare labeled by the quenched variable s (called spin) that is+1 for the active case, and −1 for the inactive one. Weuse the Ising model at finite temperatures (T ) to obtain thespin configuration, which is expressed by the HamiltonianH = −J

∑<i, j> sis j , where si is the spin on site i, J > 0 the

ferromagnetic coupling constant, and 〈i, j〉 means that i andj are nearest neighbors. The three-dimensional (3D) Isingmodel undergoes a magnetic phase transition at T = Tc ≈4.51 for the cubic lattice [26]. Since there is at least onespanning spin cluster at any temperature (two sites belongto the same cluster if they are nearest neighbors and havethe same spin) of the 3D Ising model on the cubic lattice,no percolation transition takes place at Tc. This can be un-derstood by noting that the critical site percolation thresholdfor the cubic lattice is around 0.32 < 0.5 (= occupancy

2470-0045/2020/101(3)/032116(8) 032116-1 ©2020 American Physical Society

M. N. NAJAFI et al. PHYSICAL REVIEW E 101, 032116 (2020)

101 102 103

|ε|L1/νM

10−1

100

101

ML(ε

)Lβ

M/ν

M

βM = 0.34 ± 0.03

L = 64L = 100L = 150

4.2 4.4 4.6T

0.0

0.2

0.4

0.6

ML

1 2

0.2

0.4

0.6

| ln(T − Tc)||ln

M|βM = 0.34 ± 0.02

4.2

(b)(a)

4.3 4.4 4.5 4.6 4.7T

0.0

0.5

1.0

1.5

2.0

[Z(T

)−

Z∞

]Lβ

Z/ν Z

×101

L = 64L = 100L = 150L = 200

101 102

|ε|L1/νZ

100

101

βZ = 0.39 ± 0.04

−2.0 −1.5 −1.0 −0.5

−0.2

0.0

0.2

0.4 βZ = 0.39 ± 0.04

ln(T − Tc)ln(Z

−Z

∞)

FIG. 1. (a) The data collapse of the average magnetization ML (ε) in terms of ε ≡ T −TcTc

with exponents βM = 0.34 ± 0.02 and νM =0.63 ± 0.03. The bare graph (ML in terms of T ) is shown in the lower inset. (Top inset) | log M| in terms of log ε with the exponent βM .(b) The average coordination number Z (T ) in terms of T . It is seen that at T = Tc, Z changes behavior in a power-law form with the exponentsβZ = 0.39 ± 0.04, νZ = 0.75 ± 0.05. (Top inset) The log-log plot of LβZ /νZ (Z − Z∞) in terms of L1/νZ |ε|. (Lower inset) The log(Z − Z∞) interms of log ε.

probability for T → ∞ of the Ising model). After construct-ing an Ising configuration at a given temperature using MonteCarlo, we implement the BTW dynamics on top of the span-ning (majority) spin cluster (SSC), i.e., a cluster compris-ing spins with the same orientation connecting two oppositeboundaries of the lattice. Free boundary conditions are im-posed in all directions. In the BTW dynamics, we consideron each site i a height hi (the number of sand grains) takinginitially randomly (independently and uncorrelated) with thesame probability one integer from {1, ..., Zi}, in which Zi isthe number of active neighbors of the ith site. Then we adda sand grain at a random site i, so that hi → hi + 1. If thissite becomes unstable (hi > Zi), then a toppling process starts,during which h j → h j − �i, j , where �i, j = −1 if i and j areneighbors, �i, j = Zi if i = j, and is zero otherwise. After asite topples, it may cause some neighbors to become unstableand topple, and so on, continuing until no site is unstableanymore. Then another random site is chosen and so on. Theaverage height grows with time, until it reaches a stationarystate after which the number of grains that leave the systemthrough the boundary is statistically equal to the number ofadded ones. The dynamics can be implemented with eithersequential or parallel updating. Criticality of the 3D systemsis also manifest in two-dimensional (2D) observables, whichenables us to apply 2D techniques like critical loop ensembletheory [21–24]. Here we consider 3D avalanches, as well astheir 2D projections on the horizontal plane.

III. RESULTS

Before analyzing the avalanches, it is worth discussing themagnetic phase transition of the 3D Ising model at T = Tc

which is estimated to be 0.451(3) consistent with the previousstudies [26]. The finite-size scaling (FSS) analysis of themagnetization per spin (M) is presented in Fig. 1 showing that

it fulfills the relation ML(ε) = L− βMνM GM (εL

1νM ) where βM =

0.34 ± 0.01, νM = 0.63 ± 0.03, and GM is a universal func-tion, consistent with previous studies; see Ref. [25]. There-fore a similar behavior change takes place for the fractionalnumber of sites in the majority-spin cluster [= 1

2 (1 + M(T ))].Although the Ising model does not undergo a percolationtransition at Tc, the average coordination number Z of themajority spin cluster dramatically changes behavior at Tc.From Fig. 1(b) we observe that Z − Z∞ decays as a powerlaw with ε, where Z∞ is the average coordination number inthe T → ∞ limit. It similarly fulfills the FSS hypothesis, i.e.,

4.2 4.3 4.4 4.5 4.6 4.7 4.8T

2.9

3.0

3.1

3.2

3.3

3.4

3.5

3.6

h(T

,L)

Tc

L = 200

L = 150

L = 100

L = 64

4.2 4.4 4.6 4.80.0

0.5

1.0

1.5×10−3

Var

[h]

T

1.25 1.50 1.75 2.00

0.6

0.8

σh = 0.44 ± 0.03

| ln(Tc − T )|

|ln( h

−h

T)|

FIG. 2. (a) The average height h(T, L) in terms of T for variouslattice sizes L. It exhibits power-law behavior around Tc as shown inthe lower inset: |h(T ) − h(Tc )| ∝ |T − Tc|σh with an exponent σh =0.44 ± 0.03 for L = 200. (Top inset) The height fluctuation Var[h] ≡〈h2〉 − 〈h〉2

showing a peak at Tc.

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4.3 4.4 4.5 4.6 4.7T

0

1

2

3

4

5

6

7

ζ(T

)

×10−2

L = 64L = 100L = 150L = 200

4.4 4.6 4.8

0.0

0.5

1.0

×104

χ(T

)

T

100 101 102

10−1

|ε|L1/ν

ζ L( T

)Lβ/ν β = 0.19 ± 0.02

FIG. 3. The order parameter ζ (T ) in terms of T for variouslattice sizes. (Top inset) Data collapse of ζ with exponents β =0.19 ± 0.02 and ν = 0.75 ± 0.05. (Lower inset) χ (T ) ≡ ∂ζ/∂T interms of T , showing a peak at Tc.

Z (T ) − Z∞ = L− βZνZ GZ (εL

1νZ ) with βZ = 0.39 ± 0.04, νZ =

1.0 ± 0.05, where GZ is a universal function. These FSS be-haviors are the fingerprint of the second-order phase transitionat T = Tc.

Our model undergoes a phase transition at T = Tc sepa-rating two different SOC phases, which is induced by thechange of the geometry of the support. As the temperatureincreases, the average height h(T ) undergoes a substantialchange as shown in Fig. 2. Close and below Tc it exhibits apower-law decay h(T ) − h(Tc) ∝ |T − Tc|σh with an exponentσh = 0.44 ± 0.03, whereas above this temperature we observea gentle slow-varying function of T . All graphs (for varioussystem sizes L) cross each other right at T = Tc, at which apronounced peak arises for the variance of h, i.e., Var[h] ≡〈h2〉 − 〈h〉2

.

4.2 4.4 4.6 4.8 5.0 5.2T

2

3

4

5

6

SA

P(T

,L)

×10−3

L = 64L = 100L = 150L = 200

102 2 × 102

2

4

6

L−1

L−0.65

SA

P

L

×10−3

T = 4.15T = Tc

T = 4.80

0.6 0.8 1.0

3.1

3.2

| ln(Tc − T )||ln(S

AP

−S

AP

)| σS = 0.33 ± 0.01

FIG. 4. SAP as a function of T and system size L, undergoing anabrupt change at Tc. The upper inset is the finite-size dependence ofSAP for T = 4.15 < Tc, T = Tc, and T = 4.80 > Tc from which wesee that they all go to zero, for L → ∞ as power laws. (Lower inset)The power-law dependence of SAP on T for T < Tc close to Tc forL = 200, defining the exponent σS = 0.33 ± 0.01.

4.2 4.3 4.4 4.5 4.6 4.7 4.8T

1.9

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

τ r3(T

,L)

L = 64L = 100L = 150L = 200

0.01 0.02 0.032.0

2.4

2.8

τT

cr 3

L−1

4.2 4.4 4.6

1.98

1.88T < Tc T > Tc

FIG. 5. The critical exponent of the distribution function of gyra-tion radius τr3 in terms of T . (Upper inset) The finite-size dependence(linear in terms of L−1) of τr3 at Tc, revealing that τ Tc

r3(L → ∞) =

1.86 ± 0.03. (Lower inset) The same finite-size extrapolation of τr3

for all temperatures. The exponent undergoes a clear jump at Tc.

Contrary to fixed energy sandpiles, in which h acts as atuning parameter [27], in our model the system organizesitself in a critical state, experiencing additional dominantfluctuations (for, e.g., h, see Fig. 2) at the transition point. Tocharacterize more precisely the two SOC phases, we analyzethe average number of topplings per site (toppling density)in avalanches. We may define as the order parameter ζ (T ) ≡fperc − f (T ), where f (T ) = m(T )

N (T ) , m(T ) being the number oftopplings, N (T ) the number of sites in the SSC, and fperc ≡f (T = ∞). Figure 3 reveals that ζ (T ) is approximately zero(is a weak function of T ) for T > Tc, and starts to growcontinuously in a power-law fashion when T is decreasedbelow Tc signaling a phase transition at T = Tc, at whichχ (T ) ≡ ∂ζ

∂T shows a distinct peak. It is notable that ζ is notprecisely zero at T > Tc, and has some fluctuations aroundzero. However, these fluctuations diminish as the temperatureraises. The FSS relation for ζ is ζL(T ) = L−β/νGζ (εL1/ν ) (see

10−1

r3L−νr3

103

104

105

106

107

108

109

p(r 3

)Lβ

r3

L = 32L = 64L = 100L = 150L = 200

101

r3

101

103

105

p(r 3

)

FIG. 6. Data collapse for the distribution function of r3 at T = Tc

giving βr3 = 1.86 ± 0.03, νr3 = 1.00 ± 0.03, and τr3 = 1.86 ± 0.03.(Inset) Distribution of r3 before collapse. The dashed line shows a fitwith the slope τr3 = 1.83 ± 0.04.

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M. N. NAJAFI et al. PHYSICAL REVIEW E 101, 032116 (2020)

1 2 3 4

〈ln(r)〉2

3

4

5

6

7

8

9

〈ln(l

)〉

T = TcL = 32L = 64L = 100L = 150L = 200

4.2 4.4 4.6 4.8

1.24

1.26

γlr

T

FIG. 7. The ensemble average of ln l in terms of ln r for theexternal perimeter of avalanches at T = Tc. The slope is the fractaldimension Df ≡ γlr = 1.25 ± 0.01. (Inset) The temperature depen-dence of Df .

upper inset of Fig. 3) in which ε ≡ T −TcTc

, Gζ (x) is a scalingfunction with Gζ (x)|x→∞ → xβ , and β = 0.19 ± 0.02 andν = 0.75 ± 0.05 are the resulting critical exponents. The caseT → ∞ corresponds to a site percolation cluster with occupa-tion probability p = 1

2 . For T > Tc we will therefore call thisphase SOCp= 1

2. The behavior in the T < Tc region, however,

is dominated by T = 0, i.e., the regular lattice, and thereforewe call this phase BTW. The transition between these phasesis driven by the connectivity inside the cluster. It becomesclear if we note that Z changes character at Tc (Fig. 1(b)). Themost direct effect of the strong variation of the local coordina-tion number at Tc is the change in the range of avalanches.Let us consider the spanning avalanche probability (SAP),which is the ratio of the number of spanning avalanches (theavalanches that connect two opposite boundaries) to the totalnumber of avalanches, i.e., the probability that an avalanchepercolates. This function is a convex monotonic function ofT in the T < Tc phase and a slow-varying function at high

10−1 100

|ε|L1/νh

100

4 × 10−1

6 × 10−1

(h−

h∞

)Lβ

h/ν h

βh = 0.20 ± 0.04L = 64L = 100L = 150L = 200

−0.5 0.0 0.5log(T − Tc)

0.0

0.2

log(h

−h∞

) βh = 0.20 ± 0.01

FIG. 9. The data collapse for hL (T ) − hL (∞) in terms of ε. Theexponents are βh = 0.2 ± 0.02 and νh = 2.0 ± 0.1.

temperatures (T > Tc) (Fig. 4). It shows a sharp peak atT = Tc. We have found that SAP extrapolates to zero forall temperatures, faster for T < Tc than for T > Tc. In theupper inset of Fig. 4 we show this function for three cases:T = 4.15 < Tc, T = Tc, and T = 4.80 > Tc which reveal apower-law decay for all the cases. The increase in SAP whendiluting the system is to be expected, since reducing the activechannels through which the sand grains can pass increases therange of the corresponding avalanches [28]. However, this isnot true in the vicinity of Tc in which case SAP decreases.This is reminiscent of the subdiffusive dynamics of randomwalkers (sand grains here) on the critical percolation cluster[29]. Also the fact that the paths along which the avalanchetopples become more tortuous with T (and there are lesspaths) has a competing effect, and can result in the decreaseof the spanning range. Also since the coordination number isreduced, the local firings are less powerful for increasing T . Inthe vicinity of Tc, SAP exhibits a power-law behavior in termsof T , although it always exhibits power-law behavior in termsof L. The corresponding exponent σS is shown in the lowerinset of Fig. 4 which is defined by |SAP(T ) − SAP(Tc)| ∼

FIG. 8. (a) A three-dimensional (3D) BTW sample, and (b) its two-dimensional (2D) projection on the X -Y plane (Z = 0). (c) Externalperimeter of 2D shadow.

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1.0 1.5 2.0 2.5 3.0 3.5 4.0

r

0

2

4

6

8

10

12

P( r

) T = 4.15T = 4.35T = 4.47T = 4.50T = 4.51T = 4.54T = 4.70

(a)

4.2 4.3 4.4 4.5 4.6 4.7 4.8T

2.62

2.64

2.66

2.68

2.70

2.72

2.74

γr 2

,m2(T

,L)

L = 64L = 100L = 150L = 200

0.01 0.02 0.03

2.65

2.70

2.75

γT

cr 2

,m2

L−1

γTr ,m (∞) = 2.76(2)

(b)

4.2 4.3 4.4 4.5 4.6 4.7 4.8T

1.9

2.0

2.1

2.2

2.3

τ r( T

,L)

L = 64L = 100L = 150L = 200

0.01 0.02 0.03

2.0

2.2

τT

L−1

τT (∞) = 1.77(3)

1.25 1.50 1.75 2.00

0.8

1.0

| log(Tc − T )|

|log(

τ−

τT

)| στ = 0.52(5)

(c)

4.2 4.3 4.4 4.5 4.6 4.7 4.8T

1.28

1.30

1.32

1.34

1.36

1.38

1.40

1.42

τ m3(T

,L)

L = 64L = 100L = 150L = 200

0.01 0.02 0.03

1.3

1.4

τT

cm

3

L−1

τTm (∞) = 1.26(2)

1.25 1.50 1.75 2.00

1.4

1.6

| log(Tc − T )||log(

τ m−

τT m

)| στ = 0.48(3)

(d)

4.2 4.3 4.4 4.5 4.6 4.7 4.8T

1.35

1.40

1.45

1.50

1.55

τ m2(T

,L)

L = 64L = 100L = 150L = 200

0.01 0.02 0.03

1.4

1.5

1.6

1.7

τT

cm

2

L−1

τTm (∞) = 1.30(2)

1.25 1.50 1.75 2.001.2

1.3

1.4

1.5

| log(Tc − T )||log(

τ−

τT

)| στ = 0.36(3)

(e)

4.2 4.3 4.4 4.5 4.6 4.7 4.8T

1.9

2.0

2.1

2.2

2.3

2.4

2.5

τ l(T

,L)

L = 64L = 100L = 150L = 200

0.01 0.02 0.03

2.0

2.5

3.0

τT

c

l

L−1

τTl (∞) = 1.62(2)

1.25 1.50 1.75 2.00

1.0

1.1

1.2

| log(Tc − T )|

|log(τ

l−

τT l

)| στ = 0.31(5)

(f)

FIG. 10. (a) The log-log plot of the distribution function of the gyration radius for L = 200. The temperature dependence of (b) γr2,m2 ,(c) τr , (d) τm3 , (e) τm2 , and (f) τl . The corresponding exponents, along with finite-size analysis are shown in the insets.

|T − Tc|σS for T < Tc. It is shown that σS = 0.33 ± 0.03 forL = 200.

The two phases (ordinary BTW, and SOC 12) and the transi-

tion point (SOCTc ) can also be characterized in terms of geo-metrical quantities. For 3D avalanches, we have m3 (the mass),r3 (the gyration radius), and m2 (the number of sites in thesurface of 3D avalanches); for 2D projections of avalanches,we have r (the gyration radius) and l (the total length of the

external perimeter). For these quantities we study two types ofexponents: the exponent of the distribution function P(x), i.e.,τx in P(x) ∝ x−τx , and γxy in y ∝ xγyx (x, y = m3, r3, m2, r, andl). At the transition point all of these exponents show a sharpchange (Fig. 10 in Appendix). For example, τr3 in the lowerinset of Fig. 5 abruptly changes its value from BTW to SOC 1

2

at T = Tc, its value being completely different for T < Tc andT > Tc. It is steplike in the limit L → ∞ which is obtained

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M. N. NAJAFI et al. PHYSICAL REVIEW E 101, 032116 (2020)

by linear extrapolation in terms of 1/L. We observed thatfor all temperatures T < Tc, τr3 extrapolates to 1.94 ± 0.04,and for T > Tc it is 1.76 ± 0.04. At T = Tc, this exponent is1.86 ± 0.03 which is different from both values.

To be more precise we used also the data collapse tech-nique. Figure 6 shows that the distribution function for r3 (andall other quantities that are analyzed here) fulfills a finite-sizescaling relation, i.e., p(x) = L−βx G(xL−νx ) in which βx andνx are critical exponents and τx = βx/νx that can be used asa check for consistency. Also note that the cutoff of power-law behavior for the gyration radius (which is commonlyconsidered as the correlation length in sandpile models) aswell as the cutoff for the other observables do not practicallychange with T (see Fig. 10(a) in Appendix for details). Theexponents for r3 and m3 for two universality classes (T = 0and T = ∞) and the transition point T = Tc, along with theexponents of m2, r, l at T = Tc are gathered in Table I (for acomplete set of exponents see Tables I and II in Appendix).We have observed that for T < Tc all the exponents are withinthe error bars equal to the exponents in the T = 0 system,whereas for T > Tc they are consistent with T = ∞. Mostof the exponents are different for two universality classes,and also some exponents for the transition point T = Tc

are different from both of them. We note that the size andmass exponent (τ T =∞

m3= 1.27 ± 0.03) in SOC 1

2is consistent

with the exponent of size distribution of the Barkhausenavalanches in amorphous ferromagnets [17,18], which hasbecome important due to its considerable connections withdisordered systems and nonequilibrium critical phenomena.Although some authors relate it to the depinning transition[30,31], and some others propose a critical point tuned bythe disorder in the framework of disordered spin models [32],the exact nature of the critical behavior is still debated. Ourobservation (the similarity between the dynamics of flexiblemagnetic domain walls in random media, i.e., Barkhausenavalanches and the avalanches of our model) adds the BTW-like avalanches on a diluted lattice to the list of possibilities.This has root in the fact that the maximum pinning forceper unit vortex length f max

p (the defects being realized by 3Dpercolation) play the role of the threshold for sandpiles abovewhich the particles (vortices) move isotropically toward theneighboring defect site to be pinned, that is reminiscent of theBTW model considered here. Note also that τr3 (T = 0) wasconjectured to be 2 in Ref. [33], and also τm3 (T = 0) = 4

3 ,which are consistent with the values reported in Table I.

An interesting observation is that the fractal dimension ofthe avalanche projections (D f defined by 〈ln l〉 = D f 〈ln r〉)is 1.25 ± 0.01, consistent with the ordinary 2D BTW model(Fig. 7), and within error bars robust against a change in T . Asimilar phenomenon has been observed in a yet unpublishedwork in which the fractal dimension of the shadows of clouds(which show some strong similarities with SOC systems) is1.25 and is quite robust against the environmental conditions,e.g., temperature.

IV. CONCLUSION

Summarizing, we have found a geometry-induced phasetransition between two different SOC universality classes atthe critical temperature of the Ising model. While at low tem-

TABLE I. The critical exponents β, ν, τ , and σ of r3, m3, r, l , andm2 for two universality classes (T = 0 and T = ∞) and the transitionpoint T = Tc. β and ν have been calculated using the data collapsemethod, and τ is the linear extrapolation for L → ∞.

Quantity r3 m3 r l m2

β(T = 0) 1.95(5) 3.71(5) 2.05(5) 2.15(5) 3.85(5)β(T = Tc ) 1.86(3) 3.6(1) 1.83(3) 1.96(2) 3.70(5)β(T = ∞) 1.70(5) 3.5(1) 1.80(5) 1.95(3) 3.70(5)ν(T = 0) 1.00(3) 2.74(5) 0.98(3) 1.23(3) 2.75(5)ν(T = Tc ) 1.00(3) 2.80(5) 1.00(3) 1.20(2) 2.75(5)ν(T = ∞) 0.96(4) 2.75(5) 1.05(5) 1.20(2) 2.73(5)τ (T = 0) 1.94(4) 1.32(4) 2.00(5) 1.79(3) 1.38(3)τ (T = Tc ) 1.86(3) 1.28(2) 1.77(3) 1.62(2) 1.30(2)τ (T = ∞) 1.76(4) 1.27(3) 1.75(3) 1.60(3) 1.30(3)σ 0.38(4) 0.48(3) 0.52(5) 0.31(5) 0.36(3)

peratures the model has ordinary BTW exponents τ = 1.34 ±0.04 for the avalanche size distribution, at Tc and above τ =1.27 ± 0.03. This exponent has been experimentally observedfor the size distribution of the Barkhausen avalanches inamorphous ferromagnets [17,18], proposing that SOC 1

2is a

candidate for the universal aspects of Barkhausen noise inmagnetic materials.

APPENDIX: OTHER STATISTICAL OBSERVABLES

As stated in the paper, we simulate the 3D as well as 2DBTW avalanches on top of the Ising clusters. A sample ofthe present model (BTW on the cubic Ising lattice) is shownin Fig. 8(a), whose projection is shown in Fig. 8(b). Alsothe external perimeter of this sample is shown in Fig. 8(c).An important quantity that changes at this transition point isthe average number of toppings per (active) site (the orderparameter). The behavior change of the other statistical ob-servables is also interesting at this point. In Fig. 2 the averageheight h is shown in terms of the host temperature T , whosechanges in behavior is evident at T = Tc, with a sharp peakat this temperature. Also h(T ) − h(Tc) ∝ |T − Tc|σh , showingpower-law behavior around Tc. The data collapse result for his shown in Fig. 9.

We saw that the observables change their behavior at thetransition point. An important question then arises concerningthe extent of the power-law behaviors in the phase transi-tions. For conventional phase transitions, the extent of thepower-law region reduces as we get away from the criticalpoint. For example, for the Ising model in the vicinity of

TABLE II. The fractal dimensions γm3r3 and γlr for two univer-sality classes, and T = Tc. The last row is the σ exponent for the twofractal dimensions.

Quantity γm3r3 Df ≡ γlr

T = 0 2.96(3) 1.25(1)T = Tc 2.88(3) 1.25(1)T = ∞ 2.84(3) 1.25(1)σ γm3r3 Df ≡ γlr

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10−1

rL−νr

103

104

105

106

107

108

109

p(r)

Lβ

r

βr = 1.83(3) νr = 1.0(3)L = 32L = 64L = 100L = 150L = 200

101 102

r

101

103

105

p(r)

τr(∞) = 1.80(4)

10−1 100

lL−νl

103

104

105

106

107

108

109

p(l)

Lβ

l

βl = 1.96(2) νl = 1.20(2) L = 32L = 64L = 100L = 150L = 200

102 103

l

101

103

105

p(l)

τl(∞) = 1.63(3)

(b)(a)

FIG. 11. Data collapse for distribution function of (a) r and (b) l at T = Tc.

the critical temperature, this extent reduces with a powerof |T − Tc| for all observables. In Fig. 10(a) we show thelog-log plot of the distribution function of the gyration radiusfor L = 200. We have observed that rcut (the cutoff value ofr at which the distribution function falls off rapidly) doesnot run considerably with T , in contrast to the conventionalsecond-order phase transitions. For example, for two extremes(T = 0 and T = ∞) we know that the power-law behaviorssurvive already known critical exponents.

We have also two type of exponents: the exponent of thedistribution function, i.e., τx in P(x) ∝ x−τx (in which x ischosen from the above list of statistical quantities), and γxy

in y ∝ xγyx .These quantities (γr2,m2 , τr3 , τr, τm3 , τm2 , and τl ) are plot-

ted against T in Fig. 10 for various lattice sizes. An ap-parent change of behavior is seen at T = Tc, below whichthe exponents are concave (decreasing) functions of T ,and above which the exponents become (asymptotically)constant. These functions are not analytic at T = Tc, i.e.,have discontinuity in the first derivative. This discontinu-ity becomes more significant for larger lattice sizes. Al-though the exponents are steplike in the thermodynamiclimit L → ∞ (see Fig. 4 in the paper), for finite sizesthe exponents show power-law dependence with secondaryexponents. These behaviors (concavity for T > Tc, changeof behavior at Tc) have been observed for all other ex-

ponents defined above, and power-law behavior around Tc

for finite sizes. In each graph both the power-law behav-ior [τx(T ) − τx(Tc) ∼ (Tc − T )σx ] of the exponent and thefinite-size dependence of the exponent at the critical pointare shown. The relevant critical exponents (obtained fromthe power-law behaviors around T = Tc) are reported in thegraphs.

The other thing that we have analyzed are the criticalexponents at T = Tc, which helps to recognize the universalityclass of the mentioned phase transition. It is done in Fig. 11in which, using the data collapse technique we extracted therelevant exponents. The analysis has been done for r3, r,l , revealing that τr3 (Tc)L→∞ = 1.83(4), τr (Tc)L→∞ = 1.80(4)and τl (Tc)L→∞ = 1.63(3). The exponents for each class (T =0 and T = ∞), and also for T = Tc are presented in Table I.It is notable that for T < Tc all exponents are more or lessconsistent with the T = 0 universality class, whereas for T >

Tc they are consistent with the T = ∞ phase. Some exponentsof the transition point T = Tc are different from the ones in thetwo universality classes.

Also the fractal dimension of the 2D projections is shownto be 1.25 ± 0.01 consistent with the ordinary 2D BTW model(Fig. 7). As stated in the paper, we see that this exponent isrobust against temperature T . The fractal dimensions (γm3r3

and γlr ≡ D f ) can be seen in Table II for T = 0, T = Tc, andT = ∞.

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