This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 130.237.122.245

This content was downloaded on 03/06/2014 at 01:37

Please note that terms and conditions apply.

Dynamical percolation transition in the two-dimensional ANNNI model

View the table of contents for this issue, or go to the journal homepage for more

2013 J. Phys.: Condens. Matter 25 136002

(http://iopscience.iop.org/0953-8984/25/13/136002)

Home Search Collections Journals About Contact us My IOPscience

IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 25 (2013) 136002 (7pp) doi:10.1088/0953-8984/25/13/136002

Dynamical percolation transition in thetwo-dimensional ANNNI modelAnjan Kumar Chandra

Theoretical Condensed Matter Physics Division, Saha Institute of Nuclear Physics, 1/AF Bidhannagar,Kolkata-700064, India

E-mail: anjanphys@gmail.com

Received 12 September 2012, in final form 17 December 2012Published 1 March 2013Online at stacks.iop.org/JPhysCM/25/136002

AbstractThe dynamical percolation transition of the two-dimensional axial next nearest-neighbor Isingmodel due to a pulsed magnetic field has been studied by finite size scaling analysis (by MonteCarlo simulation) for various values of frustration parameters, pulse width and temperature(below the corresponding static transition temperature). It has been found that the size of thelargest geometrical cluster shows a transition for a critical field amplitude. Although thetransition points shift, the critical exponents remain invariant for a wide range of frustrationparameters. They are also the same as those obtained for the 2d Ising model. This suggests thatalthough the static phase diagrams of these two models differ significantly in various aspects,the dynamical percolation transitions of these models belong to the same universality class.

(Some figures may appear in colour only in the online journal)

1. Introduction

The response of magnetic systems to time-dependent externalmagnetic fields has been of current interest in statisticalphysics [1–12]. Magnetization reversal dynamics is animportant subject for many technological applications ofmagnetic materials. In the magnetic recording industry, fastmagnetization reversal in storage media is essential for ahigh data transfer rate [13]. These spin systems driven bytime-dependent external magnetic fields have basically gota competition between two time scales: the time periodof the driving field and the relaxation time of the drivensystem, giving rise to interesting non-equilibrium phenomena.Magnetization reversal by a pulsed magnetic field has beenstudied extensively [1–12]. The magnetization reversal wasstudied below the corresponding static transition temperature(T0

c ), where the system remains ordered. A pulse of finitestrength (hp) and finite duration (1t), applied along theopposite direction to the prevalent order, tends to reversethe magnetization. Depending on the temperature (T) andpulse duration (1t), there will be a critical field strength(hc

p) at which the system will be forced to change fromone ordered state (magnetization m0) to another orderedstate (magnetization—m0). Below this critical strength, themagnetization will eventually remain unchanged as the field

pulse is taken off. This transition is called a ‘magnetizationreversal transition’.

A percolation transition is also associated with thismagnetization reversal induced by the external magneticfield. By percolation transition we mean the transition of thelargest ‘geometrical cluster’ (consisting of nearest-neighborparallel spins) size from a large value to zero. In the initiallyordered state, large clusters of a particular spin orientation(‘up’ or ‘down’) exist. However, owing to the pulsed fielddirected along the opposite direction, the spins gradually flip(depending on the magnitude and duration) and for a criticalvalue of the field or pulse duration the pre-existing largeclusters vanish, giving rise to oppositely oriented clusters.Thus, if we measure the largest ‘geometrical cluster’ sizefor a particular orientation at a certain temperature (belowthe critical temperature) and finite duration of field pulse, itshows a transition at a critical field magnitude. Although thethermal percolation transition of the ‘geometrical’ clusters inthe 2d Ising model has been studied extensively [14–17], verylittle is known about the field induced dynamical percolationtransition [18]. It was found numerically (based on MonteCarlo simulation) that for the pure two-dimensional Isingmodel at finite temperature the field induced magnetizationtransition and the percolation transition occur for the samecritical magnetic field. Moreover, the critical exponents were

10953-8984/13/136002+07$33.00 c© 2013 IOP Publishing Ltd Printed in the UK & the USA

J. Phys.: Condens. Matter 25 (2013) 136002 A K Chandra

Figure 1. Schematic phase diagram of the two-dimensionalANNNI model according to recent studies [23, 24].

found (using finite size scaling analysis) to be differentfrom those of the thermal percolation transition, indicatingthe dynamical percolation transition to belong to a separateuniversality class [18]. This transition was also studied forthe 2d Ising model with diagonal next nearest-neighborinteraction for a typical value of the frustration parameter [18]and it was found that the critical exponents remain unchanged.

Here we intend to study the field induced percolationtransition for the 2d axial next nearest-neighbor Ising(ANNNI) model, which is one of the simplest classicalIsing models with a tunable frustration and has been studiedfor a long time (see [19–22] for review). The 2d ANNNImodel is a square lattice Ising model with a nearest-neighborferromagnetic interaction along both the axial directions anda second-neighbor anti-ferromagnetic interaction along oneaxial direction. The Hamiltonian is

H = −J∑x,y

sx,y[sx+1,y + sx,y+1 − κsx+2,y] (1)

where the sites (x, y) run over a square lattice, the spins sx,yhave binary states (±1), J is the nearest-neighbor interactionstrength and κ is a parameter of the model. For positive valuesof κ , the second-neighbor interaction introduces a competitionor frustration. The T–κ phase diagram for this model shows arich behavior. It is easy to prove analytically that [19–22] atzero temperature, the system is in a ferromagnetic state forκ < 0.5, and in antiphase (+ + − − + + − − · · · alongthe x direction and all like spins along the y direction) forκ > 0.5 with a ‘multiphase’ state at κ = 0.5. (The multiphasestate comprises all possible configurations that have nodomain of length 1 along the x direction.) However, the finitetemperature phase diagram is yet to be solved analytically.There had been many predictions from various kinds ofnumerical calculations. Here we present the qualitative phasediagram (figure 1) that has been obtained by extensive MonteCarlo studies [23, 24]. As the previously conjectured floatingphase was not found in [23, 24], it has not been shown infigure 1.

The phase diagram consists of a ferromagnetic phase (forκ < 0.5) and an antiphase (for κ > 0.5) at low temperaturealong with a paramagnetic phase at high temperature (forall κ values). Thus, for both κ < 0.5 and κ > 0.5 there is

a transition temperature depending upon the value of κ . Forκ < 0.5, the study of the percolation transition is done ina similar way as that for the unfrustrated 2d Ising model.However, for κ > 0.5, the cluster cannot be defined as we havefor κ < 0.5, because for κ > 0.5 below the static transitiontemperature we mainly have striped states with like spinsoriented along the y-axis and a periodic modulation alongthe direction of frustration. Here we propose an alternativedefinition of a cluster to study the percolation transition. Bya cluster we mean a set of nearest-neighbor spins which is inthe same state as it was in its initial condition. Starting froman exactly antiphase state (initial cluster size is the systemsize) we achieve a steady state at a particular temperature(due to thermal agitation some of the spins flip and thuspatches of clusters form). Now we apply a field pulse whichvaries spatially along the system such that the field acts on thespin along the direction opposite to that with which it startedthe dynamics i.e. sgn(hp) = −sgn(sx,y(0)), where hp is theapplied field strength at site x, y and sx,y(0) is the initial spinvalue at lattice site x, y. This means that if we start with a‘++−−++−−· · ·’ spin configuration along the x direction,the applied field will be along the downward direction for thefirst two columns and upward along the next two columnsand this pattern continues over the entire system. As soonas we switch on the field pulse (applied along the reversedirection of the initial spin orientation), the spins will tendto orient along the field direction and the spins which are ata state identical to that of its initial state may flip (dependingon the field strength and duration) and thus the cluster sizeseventually decrease giving rise to a percolation transition.If instead of the spatially modulated pulse, we would haveapplied a homogeneous pulse (for κ > 0.5), then the role ofthe applied field would have been different for alternate stripesof the antiphase. Some of the spins would have been favored,while others would have been opposed by the applied externalfield pulse. On the other hand for κ < 0.5, even if we apply ahomogeneous field, all of the spins are equally treated (tendsto orient along the field). Thus, to keep similarity in the natureof the transition, we have applied a spatially modulated fieldfor κ > 0.5.

In this paper we study the percolation transition for awide range of frustration parameters (0.0 < κ < 1.0) andfind how the critical field strength varies with the frustrationparameter for a particular pulse duration. We have used MonteCarlo simulation to obtain the numerical results and a fullyperiodic boundary condition throughout our entire study. Wehave calculated the critical exponents numerically by finitesize scaling analysis to determine whether the exponents areaffected by the frustration parameter. In section 2, we discussthe model studied. In section 3, we present the results obtainedfor the finite size scaling analysis of the percolation orderparameter and in section 4 the fractal dimension and thescaling analysis of the Binder cumulant. In the final sectionwe discuss some consequences related to our study. The mainobservation of this work is that the critical exponents areindependent of the frustration parameter and also the sameas those of the normal 2d Ising model.

2

J. Phys.: Condens. Matter 25 (2013) 136002 A K Chandra

Figure 2. Variations of the largest cluster size (Pmax) with the fieldpulse amplitude (hp) in the case of the 2d ANNNI model whereκ = 0.3,T = 0.2 and 1t = 4.

2. The model

The model studied here is the 2d ANNNI model under atime-dependent external magnetic field pulse, described bythe Hamiltonian

H = −J∑x,y

sx,y[sx+1,y + sx,y+1 − κsx+2,y] − h(t)∑x,y

sx,y

(2)

where the sites (x, y) run over a square lattice, the spinssx,y are ±1, J is the nearest-neighbor interaction strength(we keep J = 1.0 here on) and κ is the competition orfrustration parameter. The static critical temperature (T0

c )depends on the value of κ (figure 1). For κ = 0,T0

c = 2/ ln(1+√

2) ≈ 2.269 . . .. For κ < 0.5, the field pulse applied isspatially uniform (directed opposite to that of the initial spinconfiguration we start with) having a time dependence:

h(t) =

{−hp, t0 ≤ t ≤ t0 +1t

0 otherwise(3)

where 1t is the duration of the field pulse applied (in termsof the number of Monte Carlo sweeps). For κ > 0.5, the fieldpulse applied is such that the field direction is opposite to thatof the starting configuration (antiphase order) of the spins, butthe time dependence is the same as that of equation (3). Inour simulation we ensure that at the time t0 at which the pulseis applied, the system reaches its equilibrium configuration atthat temperature T (<Tc).

The spins are selected randomly and are flipped followingthe normal Metropolis algorithm. The energy difference (1E)to flip the spin is calculated and flipped with probabilitymin{1, exp(−1E/T)}. At t = t0, the system achieves a steadystate at the corresponding temperature (in the absence of anyfield). As soon as the field pulse is switched on, a large numberof spins which were intact in their initial state, flip during thepulse duration. Thus, the percolation order parameter Pmax =

SL/L2 (where SL is the size of the largest cluster and L is thelinear size of the system) decreases during this duration and

Figure 3. Plot of hcp against the frustration parameter κ . (We have

omitted the study close to κ = 0.5, because in this region thetransition temperatures are too low.)

for a particular combination of T, κ and 1t, for a particularvalue of hc

p, the system undergoes a percolation transition(figure 2).

It is found that for a particular temperature for κ < 0.5,as the value of κ is increased, the transition field hc

p decreases,but for κ > 0.5, hc

p increases with κ (figure 3). For κ < 0.5,as κ is increased, the increasing frustration itself destabilizesthe order and thus the pulse amplitude needed for transitiondecreases. However, for κ > 0.5, the situation is different.The increasing frustration parameter stabilizes the antiphasemore and more, thus requiring a higher pulse amplitude forthe percolation transition (figure 3). Typical domain structuresfor a 100 × 100 lattice both for κ < 0.5 (κ = 0.2) andκ > 0.5 (κ = 0.8) at hp � hc

p, hp = hcp and hp � hc

p (T =0.2,1t = 4) are shown in figures 4 and 5 respectively. It canbe seen from both figures that for hp � hc

p and hp � hcp,

ordered structure prevails. Only the spin orientation changes.This change (transition) occurs at hp = hc

p.

3. Finite size scaling analysis of the percolationorder parameter

In this section we discuss the percolation transition behavior.For a particular combination of κ,T and 1t, we studythe variation of Pmax with hp. The percolation transition ischaracterized by several exponents which also determine theuniversality class. The order parameter i.e., the relative size ofthe largest cluster, varies as

Pmax ∼ (hcp − hp)

β . (4)

The correlation length diverges near the percolationtransition point as

ξ ∼ (hcp − hp)

−ν, (5)

where hcp is the critical field amplitude. The other exponents

can be calculated from various scaling relations [25].However, owing to finite size effects it is difficult to

3

J. Phys.: Condens. Matter 25 (2013) 136002 A K Chandra

Figure 4. Typical domain structures of a 100 × 100 lattice for hp � hcp, hp = hc

p and hp � hcp for κ = 0.2,1t = 4 and T = 0.2. The shaded

regions represent the domains corresponding to up spins.

Figure 5. Typical domain structures of a 100 × 100 lattice for hp � hcp, hp = hc

p and hp � hcp for κ = 0.8,1t = 4 and T = 0.2. The shaded

regions represent the domains corresponding to up spins.

4

J. Phys.: Condens. Matter 25 (2013) 136002 A K Chandra

Figure 6. Inset: the scaled largest cluster size (PmaxLβ/ν) has beenplotted with hp for six different system sizes(L = 100, 200, 300, 400, 800 and 1000) for a typical value ofκ < 0.5. The crossing point determines the critical field amplitude(hc

p). Main: the value of hcp and β/ν have been used to make the

plots for the scaled variables PmaxLβ/ν and (hcp − hp)L1/ν collapse

by tuning the value of 1/ν for all of the different L values.

determine these exponents precisely. We have determinedthese exponents from the finite size scaling relations. Pmaxfollows the scaling form

Pmax = L−β/νF[L1/ν

(hc

p − hp

)], (6)

where F is a suitable scaling function. This scaling functionshows that the largest cluster grows with system size andapplied field. It also indicates that at the critical field thescaled largest cluster has a constant value. By tuning β/ν, allof the PmaxLβ/ν–hp curves can be made to cross at a singlepoint and this point determines the critical field amplitude(hc

p). Now, once β/ν and hcp has been estimated, by tuning

1/ν, one can collapse all of the PmaxLβ/ν–(hcp − hp)L1/ν

curves on one another, and thus ν is determined. We haveshown both these plots for κ = 0.2 (κ < 0.5) and κ = 0.7(κ > 0.5) in figures 6 and 7, respectively. For both the curveswe get the same values for β/ν = 0.20 ± 0.05 and 1/ν =0.85±0.05 and more importantly this value is the same as thatwe obtained for the normal 2d Ising model found earlier [18],which indicates that the dynamical percolation transition ofthe 2d ANNNI model also belongs to the same universalityclass as that of the normal 2d Ising model. Only the criticalfield amplitude changes (as shown in figure 3). We haverepeated the same exercise for some other combination of κ,Tand 1t, but the exponents remain unchanged. This suggeststhat the magnitude of frustration does not have any drasticeffect on the dynamical percolation phenomenon except thequantitative shift of the transition point.

To have a picture of the effect of various parameters onthe critical field amplitude we present two graphs, one forκ < 0.5 (figure 8) and the other for κ > 0.5 (figure 9), eachshowing the variation of hc

p with 1t for different sets of κand T . It is quite clear from figure 8 that, for any combinationof κ and T, hc

p decreases with 1t. Also for any combination

Figure 7. Inset: the scaled largest cluster size (PmaxLβ/ν) has beenplotted with hp for six different system sizes(L = 100, 200, 300, 400, 800 and 1000) for a typical value ofκ > 0.5. The crossing point determines the critical field amplitude(hc

p). Main: the value of hcp and β/ν have been used to make the

plots for the scaled variables PmaxLβ/ν and (hcp − hp)L1/ν collapse

by tuning the value of 1/ν for all of the different L values.

Figure 8. Plot of hcp against 1t for three different combinations of

(κ,T) for the regime κ < 0.5.

of κ and 1t, hcp decreases with T , because thermal excitation

always helps in percolation transition and thus reducing thenecessary external field amplitude. However, the frustrationparameter plays opposite roles in the two halves of the κregime. For a fixed pair of T and 1t, when we are belowκ < 0.5, with an increase in κ we need less field to reach thetransition point because increasing the frustration parameteritself reduces the parallel alignment between the adjacentspins and thus facilitates the transition. However, above κ =0.5, the increase in the frustration parameter supports theantiphase structure (with which we begin the dynamics forκ > 0.5), and thus we have to apply more field amplitude toflip the spins to the opposite direction. Thus, the frustrationparameter plays opposite roles in cluster formation in the tworegimes.

5

J. Phys.: Condens. Matter 25 (2013) 136002 A K Chandra

Figure 9. Plot of hcp against 1t for three different combinations of

(κ,T) for the regime κ > 0.5.

Figure 10. Variation of the largest cluster size with the linearsystem size for two different values of κ = 0.2 and 0.7 at the criticalfield for T = 0.2 and 1t = 4. The log–log plot gives the fractaldimension to be D = 1.82± 0.01 for both the values of κ .

4. Fractal dimension and Binder cumulant

At the transition point, the percolation clusters becomeself-similar. The largest cluster depends on the linear systemsize and it varies as SL ∼ LD, where SL is the largestcluster and D is the fractal dimension. The fractal dimensionmeasured (figure 10) is found to be independent of the value ofκ and the value is D = 1.82 ± 0.01, which is again the sameas that we found in the case of the normal 2d Ising model.This fractal dimension is known to be related to the spatialdimension (d) as D = d − β/ν. This relation is also satisfiedfor our estimated values of β/ν = 0.20± 0.05 (mentioned insection 3) and D for all values of κ .

For further verification of the critical points and theexponents, we have also studied the Binder cumulant of theorder parameter, defined as [26],

U = 1−〈P4

max〉

3〈P2max〉

2 , (7)

Figure 11. Main: the Binder cumulant has been plotted with hp forsix different system sizes (L = 200, 300, 400, 800 and 1000) for atypical value of κ < 0.5. The crossing point determines the criticalfield amplitude (hc

p). Inset: the value of hcp has been used to make the

plots for U and (hcp − hp)L1/ν collapse by tuning the value of 1/ν for

all of the different L values.

Figure 12. Main: the Binder cumulant has been plotted with hp forsix different system sizes (L = 200, 300, 400, 800 and 1000) for atypical value of κ > 0.5. The crossing point determines the criticalfield amplitude (hc

p). Inset: the obtained value of hcp has been used to

make the plots for U and (hcp − hp)L1/ν collapse by tuning the value

of 1/ν for all of the different L values.

where Pmax is the percolation order parameter and the angularbrackets denote the ensemble average. The crossing point ofthe curves U–hp for different system sizes gives the criticalfield amplitude (hc

p). The Binder cumulant has also been usedto determine the correlation length exponent as it follows thescaling form

U = U((hcp − hp)L

1/ν), (8)

where U is a suitable scaling function. Having obtained hcp

from the crossing point, if we plot U with the scaled fieldamplitude (hc

p − hp)L1/ν , for a suitable value of ν we willobtain a data collapse and that value of ν determines thecorrelation length exponent. We have studied these resultsboth for κ < 0.5 (figure 11) and κ > 0.5 (figure 12).

6

J. Phys.: Condens. Matter 25 (2013) 136002 A K Chandra

For all values of κ (only two sets of κ,T are presentedin figures 11 and 12), the value of 1/ν remains the same(0.85± 0.05) and also equal to that we have already obtainedfrom the finite size scaling of the percolation order parameter.The value of the Binder cumulant at the crossing point forvarious system sizes is also identical (0.62 ± 0.01). The onlyobvious quantitative change is seen for the transition point.

5. Discussions

Spatially modulated phases of metallic alloys can beinvestigated in terms of the ANNNI model and consideringthe application of these alloys in the magnetization reversalphenomenon, we have studied the magnetic field pulseinduced percolation phenomenon in the 2d ANNNI model.We have focused on the role of the frustration parameter inaffecting the percolation transition of the 2d ANNNI model.By applying a magnetic field pulse of finite duration (1t)for a certain frustration parameter (κ) and temperature (T)(below the corresponding static transition temperature), wehave determined the critical field strength (hc

p) for which thetransition occurs. Although we have found that the frustrationparameter highly enriches the static phase diagram of the 2dANNNI model (various novel phases and transitions appeardue to κ), quite surprisingly it does not have any largeimpact on the dynamical percolation transition apart fromshifting the transition points. We have restricted our study tothe regime 0.0 < κ < 1.0. From finite size scaling analysiswe have showed that both the percolation order parameterand the Binder cumulant lead to the same values of criticalexponents as we have found for the normal 2d Ising model(thus, it belongs to the same universality class as that of thedynamical percolation transition of the 2d Ising model). Thevalues of exponents determined are β/ν = 0.20 ± 0.05 and1/ν = 0.85 ± 0.05. The fractal dimension D = 1.82 ± 0.01also confirms the relations between the exponents.

Although we have not presented the details, in thiscontext we have also investigated the transition behavior ofthe magnetization (for κ < 0.5) and sub-lattice magnetization(for κ > 0.5). The sub-lattice magnetization is defined asx-direction; the sub-lattice magnetization is defined by ms =

1L

∑ L4−1x=0 (m4x+1 + m4x+2 − m4x+3 − m4x+4) with, mx =

1L

∑Ly=1sx,y, where L is the length of the system. The transition

points were found to be the same as those of the percolationtransition (for a certain set of values of κ,T and 1t). Thestudy of finite size scaling of the order parameters also givesthe same critical exponents (β/ν and 1/ν). Only the value ofthe crossing point of the Binder cumulant for magnetizationshows non-universal behavior for different temperatures. Thisvalue (U?

= 0.62 ± 0.01) is robust in the case of percolationtransition.

This study may be useful in the application ofmagnetization reversal of spatially modulated magnetic

materials where the only change necessary is the value ofthe magnitude of the critical field, keeping the nature of thetransition intact. Although we have limited our regime ofinvestigation to below κ = 1.0, one can also explore thistransition beyond κ = 1.0 or other modulated structures.

Acknowledgments

The author acknowledges many fruitful discussions with andsuggestions of Professors Bikas K Chakrabarti, ProfessorSubinay Dasgupta and Mr Soumyajyoti Biswas. The authoracknowledges the financial support from DST (India) underthe SERC Fast Track Scheme for Young Scientists Sanc. No.SR/FTP/PS-090/2010(G).

References

[1] Chakrabarti B K and Acharyya M 1999 Rev. Mod. Phys.71 847

[2] Acharyya M and Chakrabarti B K 1995 Phys. Rev. B 52 6550[3] Acharyya M, Bhattacharjee J K and Chakrabarti B K 1997

Phys. Rev. E 55 2392[4] Misra A and Chakrabarti B K 1997 Physica A 246 510[5] Misra A and Chakrabarti B K 1998 Phys. Rev. E 58 4277[6] Stinchcombe R B, Misra A and Chakrabarti B K 1999 Phys.

Rev. E 59 R4717[7] Misra A and Chakrabarti B K 2000 J. Phys. A: Math. Gen.

33 4249[8] Misra A and Chakrabarti B K 2002 Comput. Phys. Commun.

147 120[9] Chatterjee A and Chakrabarti B K 2004 Phase Transit. 77 581

[10] Chatterjee A and Chakrabarti B K 2003 Phys. Rev. E67 046113

[11] Korniss G, White C J, Rikvold P A and Novotny M A 2001Phys. Rev. E 63 016120

[12] Korniss G, Rikvold P A and Novotny M A 2002 Phys. Rev. E66 056127

[13] Pizzini S, Vogel J, Bonfim M and Fontaine A 2003 Top. Appl.Phys. 87 155

[14] Muller-Krumbhaar H 1974 Phys. Lett. A 48 459[15] Stoll E, Binder K and Schneider E 1972 Phys. Rev. B 6 2777[16] Muller-Krumbhaar H and Stoll E P 1976 J. Chem. Phys.

65 4294[17] Binder K and Stauffer D 1972 J. Stat. Phys. 6 49[18] Biswas S, Kundu A and Chandra A K 2011 Phys. Rev. E

83 021109[19] Selke W 1988 Phys. Rep. 170 213[20] Selke W 1992 Phase Transitions and Critical Phenomena

vol 15, ed C Domb and J L Lebowitz (New York:Academic)

[21] Yeomans J M 1988 Solid State Physics vol 41,ed H Ehrenrich and D Turnbull (London: Academic)

[22] Liebmann R 1986 Statistical Mechanics of Periodic FrustratedIsing Systems (Berlin: Springer)

[23] Shirahata T and Nakamura T 2001 Phys. Rev. B 65 024402[24] Chandra A K and Dasgupta S 2007 J. Phys. A: Math. Theor.

40 6251[25] Stauffer D and Aharony A 1994 Introduction to Percolation

Theory 2nd edn (London: Taylor and Francis)[26] Binder K and Heermann D 1988 Monte Carlo Simulations in

Statistical Physics: An Introduction (Berlin: Springer)

7