Percolation transition in two-dimensional ±J Ising spin glasses

PHYSICA ELSEVIER Physica A 246 (1997) 18-26

Percolation transition in two-dimensional +J Ising spin glasses

Hitoshi Imaoka a'*, Hideo Ikeda b, Yasuhiro Kasai b

aFundamental Research Laboratories, NEC Corporation, 34 Miyukigaoka, Tsukuba lbaragi 305, Japan bDepartment of Applied Physics, Osaka University, Suita 565, Japan

Received 3 October 1997

Abstract

The percolation properties of geometrical clusters are investigated for the Ising spin glasses in the square and triangular lattices with the asymmetric weights of ferromagnetic and antifer- romagnetic bonds. By Monte Carlo simulation, we obtain the phase diagram of the percolation transition temperature as a function of the weight of ferromagnetic bonds. At each transition temperature, we estimate the critical exponents v and y which agree well with those exponents belonging to the universality class of the random bond percolation except for the pure ferromag- netic case.

PACS: 05.50.+q; 75.10.Nr; 64.60.Ak; 75.40.Mg Keywords: Ising percolation; Spin glass; Monte Carlo simulations

1. Introduction

In the last decade, the Ising spin glass has been studied as the central problem of the complex system [1]. Here, we investigate the percolation of the _~J model, one of the Ising spin glasses, in the square and triangular lattices [2-8].

The study of the percolation problem in the Ising model was started by Kasteleyn and Fortuin [2,3] for the ferromagnet. They transferred the partition function of the Ising ferromagnet to the generating function of the percolation problem and proposed an idea of percolation cluster with the Ising correlation. Coniglio and Klein [4] also rep- resented the percolation cluster in a simple way as a set of the nearest neighbor parallel spins connected by the bonds present, each bond being present with probability p =

1 - e x p ( - Z l / k B T ) where J , kB and T denote the interaction, the Boltzmann constant and the temperature, respectively. The cluster defined in this rule has a remarkable property that the connected probability of clusters between two sites agrees precisely

* Corresponding author.

0378-4371/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved PH S0378-4371 (97)0035 1-8

H. Imaoka et al. IPhysica A 246 (1997) 18-26 19

with the two-spin correlation function. From this property, it was derived that the per- colation transition temperature Tp becomes exactly the same as the Curie temperature

To. In the percolation cluster, a present bond means that two spins at both ends of the bond are aligned due to the interaction effect, and an absent bond means that the interaction does not work on between those spins due to the thermal noise. Namely, the percolation representation of the Ising system is superior to the spin representation in the point that the thermal effect can be distinguished from the interaction effect.

For the Ising spin glasses containing the fully frustrated model, the percolation rep- resentation was obtained by Kasai and Okiji [5], and Coniglio et al. [6]. They showed that the percolation cluster can be defined in a similar way as that of the ferromagnet. Here, each bond is present with the probability p for a satisfied bond, which means that the parallel bond is for the ferromagnetic interaction and antiparallel one for the an- tiferromagnetic interaction. For several models, the percolation transition temperatures Tp were estimated numerically using the Monte Carlo [9-11]. However, the values of Tp did not agree with any transition temperatures we had known. The magnitude order of transition temperatures was Tg < Tp < /'PUre(= To) for the spin glasses and Tc < Tp < Tf ure for the fully frustrated models. Here, Tg, T p~re and TG denote the

freezing temperature, the Curie temperature of the pure ferromagnet and the Griffiths temperature [12] of the spin glass proposed by Randeria [13], respectively. The con- crete examples of this relation are as follows: for the ~ model (c = 0.5) in the square lattice Tg(= 0 [14]) < Tp(-= 1.80(5) [9]) < TPure(= 2.269), for the fully frustrated model in the square lattice T~ = (0 [15]) < Tp(= 1.69(5) [9]) < Tf ure and for the antiferromagnet (c = 0, fully-frustrated) in the triangular lattice To(= 0 [16]) < Tp (= 2.62 [10]) < TPU~e(= 3.641). Here, the temperature is scaled by J/k~ and c denotes

the weight of ferromagnetic bonds. Coniglio et al. [6] derived the relation of T q <<, Tp from the fact that the percolation cluster does not reflect the net spin correlation in the frustrated model. Then, what does the percolation transition in the spin glass mean? One of answers for this question is considered in the "damage spreading" dynamics [10,17-24].

The method of "damage spreading" traces the time evolution of two configurations {a ~ } and {a B} in which only a small number of spins have different orientations initially and both systems evolve by using the same dynamics and the same sequence of random numbers. Then, an order parameter, called "damage", is defined as the Hamming distance between two systems

N 1

D(t) = ~-~ ~ [o~/(t) - o~/(t)[ , i=1

(l)

where N denotes the number of spins in each system. When D(t) is not zero after a sufficiently long time at a given temperature, the corresponding phase is called a chaotic one. In the spin glasses, this phase appears below the dynamical transition temperature Td by the damage spreading. For many systems, the values of Td were investigated numerically and in most systems, the close agreements between Td and Tp

20 I-L Imaoka et al. IPhysica A 246 (1997) 18-26

were certified [10,17]. The agreement problem was also investigated theoretically by Coniglio et al. [18] and Chikyu [19,20]. In Ref. [19], Chikyu stated that the percolating phase between Tg and Tp can be compared to the liquid phase, because the spin configurations translate mainly between the macroscopic degenerating states with the strong topological constraint. For the same phase, Campbell [17] also described it as one complex valley in the phase space, where "complex" means that the phase space is maze-like so that the two configurations take an extremely long time to collide under the damage spreading procedure. However, a clear picture in the percolating phase is not still obtained. In this paper, we estimate the percolation transition temperatures and some exponents in the + J model on the square and triangular lattices for various weights of ferromagnetic bonds c using the Monte Carlo method.

In Section 2 we introduce the formulation of the percolation problem in the spin glass, in Section 3 we show the numerical result by simulation and in Section 4 we summarize it.

2. Formulation of percolation problem in Ising spin glass

In this section, we formulate the percolation problem in the i J model with the asym- metric weights of ferromagnetic and antiferromagnetic bonds [5,6]. The Hamiltonian is defined by

H = - - J ) , (2)

where Ising spin si takes a value + I. The nearest-neighbor interaction Jij is randomly distributed independently at each bond with the probability

P(Jij) = c¢~(Jij - J ) + (1 - c)•(gij + J ) , (3)

where J > 0. Then, the partition function is given by

Z = Z e-#H' (4) {s,}

where fl =- 1/kBT. According to Kasteleyn-Fortuin transformation [2,3], the partition fimction is expanded to

Z = ~ pb(CuF)(1 -- p)B--b(CvF)2N(CvF), ( 5 )

{CuF}

where B denotes the number of interactions in the whole lattice, and b(CuF) and N(CuF) denote the number of present bonds and the number of clusters in the unfrus- trated bond configuration CUE, respectively.

We introduce an exact relation between the connected probability of the percolation cluster and the two-spin correlation function. It is written by

(sisj> = @+> - <V~j > , (6)

H. Imaoka et al . /Physica A 246 (1997) 18-26 21

where (-) means thermal average, and ~+(7/~) is 1 or 0 depending whether or not spins

si and sj belong to the same cluster in parallel (antiparallel) state. In the ferromagnet,

it is noted that 7~ is always zero. An order parameter of the percolation problem is defined as the percolating prob-

ability. Here, the term "percolating" indicates that one or plural number of clusters extend from one side of the lattice to another. Generally, below Tp, one large cluster is percolating and above Tp, all clusters are finite. The percolating probability is given

by

--]

(7)

On the other hand, the magnetization per site is expressed by using the cluster formu-

lation

M : <~+)- <7~) . (8)

From the above two definitions, we have a relation p ~ ~>M at any temperature. Hence, we obtain Tp >~ Tc for any c. In the pure ferromagnet c = l, since 7~ is zero, the two order parameters coincide and Tp agrees exactly with Tc. The mean cluster size

corresponds to the susceptibility in the spin system. It is defined by

Z 2 (9) S : S2ns - S o o ,

s

where n~. and so~ denote the average number per site of clusters with s spins and the size of the infinite cluster, respectively. At the percolation transition temperature, the mean cluster size diverges. In the next section, we estimate the percolation transition temperatures and some exponents by calculating the mean cluster size using the Monte

Carlo method.

3. Numerical results

We simulate the i J models in the square and triangular lattices of linear size L = 30,60, 100, 140 and 180. The boundary condition is helical in the vertical direction and free in the horizontal direction. More precisely, the sites are labeled from 1 to L 2, such that the nearest neighbors of site i are i+ 1 and i+L in the square lattice and izi: l, i+L and i+(L-1 ) in the triangular lattice, and the interactions are missing if the site number is outside the range 1 to L z. The update of the spins is performed by the heat-bath dynamics and the cluster multiple labeling technique is used [25]. The thermalization time chosen is 2500 Monte Carlo sweeps and after that, the thermodynamic quantities

are averaged over 7500 Monte Carlo sweeps. We first simulate the square lattice. We calculate the mean cluster size S, where the

size of the infinite cluster Soo in Eq. (9) is replaced with the size of the maximum

22 H. Imaoka et al./Physica A 246 (1997) 18-26

113

500 S

400

300

Z00

100

0

O L=30

O L=60

A L=100

X L~I 4O

1.41.51.61.71.81.9 2 2.1 2.22.32.4

T

Fig. 1. The mean cluster size S as a function of the temperature T in the spin glass with c = 0.5 in the square lattice. The curve is only a guide for the eye.

0.08-

0.07-

0.06-

0.05

E~ 0.04-

0.03-

0.02-

0.01

X

O

z~

©

6, [] <>

O L=30

0 L=60

/x. L=100

X L=140

[ ] L=IB0

x ~

0 u') 0 u? 0 u'~ o u~ o u'~ o

i i

T'

Fig. 2. The finite-size scal ing plot o f the mean cluster sizes for the data o f Fig. 1. The quant i ty S r = S L -~/v

as a funct ion o f T ~ = (T - Tp)L ]/v for Tp = 1.80, v --- 1.31 and 7 = 2.31.

cluster, since we treat the finite-size system. For example, we show the result for the

system with c --- 0.5 in Fig. 1. The obtained mean cluster sizes have a well-pronounced

peak about T ---- 1.8-1.9 in the largest size o f the L = 180. In order to estimate the

percolation transition temperature precisely, we use a finite-size scaling method. In Fig.

2, we show the finite-size scaling plot for the data o f Fig. 1. From the data-collapse

analysis, the percolation transition temperature Tp is estimated to be 1.80 + 0.02, which

is the same value obtained by Cataudella [9].

For various weights o f ferromagnetic bonds, we estimate the percolation transition

temperatures (Fig. 3) and critical exponents v and 7 (Fig. 4). The numerical results

H. Imaoka et al./ Physica A 246 (1997) 18-26 23

3

2:1 . . . . . . . . . . . . . . .

o.5 0:6 o:7 o:8 0.9

Fig. 3. The percolation transition temperatures for various weights of ferromagnetic bonds c in the square lattice. The values of Tp are plotted by the full circles. Below Tp, the corresponding phase is a percolating phase (P-phase). Here, FM, PM and RAS denote the ferromagnetic phase, the paramagnetic phase and the random antiphase state [26], respectively. These boundary lines are drawn following Ref. [27]. The temperatures on the dotted line is the Griffiths temperature.

2 ¸

r- e

o r l x ¢~1

o o o 0 L)

- - - X - - - × - - - X - - - -N- ~<-

0 o.s 016 017 0 .8 '019

c

Fig. 4. The critical exponents v (x ) and y (o) in the square lattice. The dotted and thin lines denote the 43 of the random bond percolation, respectively. exact values of v = 34- and 7 =

are shown in Table 1. It is found that the percolat ion transi t ion temperature decreases

accord ingly as the weight o f the fer romagnet ic bonds c decreases, namely , the ef-

fect o f the frustration increases. The critical exponents agree a lmost wi th those o f the

random bond percolat ion (v = 4 and ~ = ~8) for c ~ 1 and agree with those o f

the two-d imens iona l Is ing fe r romagnet (v -- 1 and 7 = ¼) at c = 1. In Table 1, we

s imulate twice with different ini t ia l -bond configurat ions for each c. This is because

the the rmodynamics proper ty o f the physical quantit ies m a y depend on the bond

24 H. Imaoka et al. IPhysica A 246 (1997) 18-26

Table 1 The numerical results of the percolation transition temperatures Tp and the critical exponents v and 7 in the square lattice

c Te v

1 2 1 2 1 2

0.5 1.80(2) 1.79(3) 1.31 1.35 2.31 2.36 0.6 1.79(3) 1.80(3) 1.32 1.33 2.30 2.33 0.7 1.81 (2) 1.82(3) 1.31 1.35 2.25 2.36 0.8 1.83(2) 1.83(3) 1.34 1.31 2.33 2.29 0.9 1.92(3) 1.88(2) 1.33 1.32 2.35 2.30 0.95 1.99(4) 2.02(3) 1.32 1.31 2.34 2.32 1.0 2.26(2) 2.26(3) 0.99 1.01 1.74 1.76

4

3-

© o o © o 0 o o o 0

t - t~

o" -X- -X- K- X-)~ -)~ -×- -x- XX

0 0 01z o14 o18

C

Fig. 5. The percolation transition temperatures Tp (e) and the critical exponents v (x) and 7 (o) in the 4 and 43 of the random triangular lattice. The dotted and thin lines denote the exact values of v = ~ 7 =

bond percolation, respectively.

configurat ion sensit ively, especial ly in the spin glass. However, these two trials show

almost the same values for any c.

We simulate the t r iangular lattice similarly. The results are shown in Fig. 5. In this

lattice, the pure ant i ferromagnet (c -- 0) is the fully frustrated system in contrast to

the square lattice. However, the results are a lmost the same with those o f the square

lattice. Namely , the percolat ion transi t ion temperature decreases accordingly as the

ferromagnetic bond weight c decreases, and the critical exponents agree almost with

those o f the r andom bond percolat ion except for c = 1. We summarize the numerica l

results o f Tp, v and 7 for the t r iangular lattice in Table 2. Here, the values o f Tp = 2.56

at c ----- 0 is a little lower than 2.62 as estimated by Zhang and Yang [10].

H. Imaoka et al./Physica A 246 (1997) 18-26 25

Table 2 The numerical results of the percolation transition temperatures Tp and the critical exponents v and 7' in the triangular lattice

c Tp v 7

1 2 1 2 1 2

0.0 2.56(2) 2.56(2) 1.33 1.33 2.30 2.33 0,1 2.64(2) 2.64(2) 1.34 1.33 2.35 2.31 0.2 2.69(2) 2.69(2) 1.33 1.32 2.31 2.32 0.3 2.72(2) 2.72(2) 1.33 1.32 2.30 2.30 0.4 2.73(4) 2.73(4) 1.34 1.34 2.35 2.33 0.5 2.74(2) 2.73(2) 1.33 1.33 2.30 2.33 0.6 2.74(3) 2.73(4) 1.32 1.33 2.28 2.33 0.7 2.75(3) 2.76(3) 1.30 1.31 2.30 2.32 0.8 2.83(3 ) 2.83(3 ) 1.31 1.32 2.31 2.31 0.9 3.06(2) 3.04(2) 1.26 1.30 2.19 2.30 0.95 3.25(3) 3.24(3) 1.30 1.33 2.28 2.31 1.0 3.63(3) 3.63(3) 1.02 0.98 1.76 ! .73

4. S u m m a r y and discuss ion

We have investigated the percolation transition for the i J model on the two-

dimensional lattice with asymmetric weight o f ferromagnetic bonds. Using the Monte

Carlo method, we have estimated the percolation transition temperatures Tp with the exponents v and ~ which belong to the universality class of the random bond percola- tion except for the pure ferromagnet. The percolation transition temperatures are nearly

equal to but lower than the Curie temperature corresponding to the pure ferromagnet

and each of them decreases accordingly as the effect o f the frustration increases re- gardless of the lattice structure, It is clear that the frustration acts as an obstacle to

form the spin cluster by introducing "the unsatisfied bonds". At the ground states of

the frustrated lattice, however, the spin cluster covers all the spins in spite of "the un- satisfied bonds". Namely, the frustration assists the spin cluster to percolate effectively. The percolation transition temperature dependence on c is considered to be a result of these competing effects.

Finally, we discuss the relation between the percolation transition and the phase

space in the spin glass. In the phase space, spin configuration is indicated by a point and its time evolution is drawn by a trajectory of the points. When the system is be- low the percolation transition temperature, the spin configuration migrates around the ground states with the macroscopic degeneracy. In other words, the trajectory of the spin configuration moves only in the restricted phase space around the ground states. When the temperature exceeds the percolation transition temperature, the macroscopi- cally percolating cluster vanishes suddenly. This means that the topological constraint of the spin structure is removed at that moment and, in the phase space, the trajectory of the spin configurations spreads to the whole phase space. However, this transition is not observed in the ordinary thermodynamic quantities. (the free energy and the

26 H. Imaoka et al./ Physica A 246 (1997) 18-26

susceptibility, etc.). Cataudella et al. [28,29] explained the mechanism of non- observability by extending the frustrated Ising model to the Potts model with the in- ternal degrees of freedom q. Their conclusion is that in the Potts model q ~ 2, the real transition occurs in the free energy, but in the Ising case (q = 2) the transition disappears, because the amplitude of the singular term is zero on account of the degree of the internal freedom. It is expected that the significance of the percolation transition is more clarified in the study of the dynamic process like damage spreading [10,19,20].

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