Cluster formulation for frustrated spin models

Physica A 192 (1993) 167-174 North-Holland

Cluster formulation for frustrated spin models a

•

• a b

V. C a t a u d e l l a , A. Conlgho ' , L. de Arcangelis b and F. di L i b e r t o ~ aDipartimento di Scienze Fisiche, Universit& di Napoli, Mostra d'Oltremare Pad. 19, 1-80125 Naples, Italy bLPMMH, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France

Received 7 A u g u s t 1992

A q-state frustrated Potts model is introduced which generalizes the K a s t e l e y n - F o r t u i n formalism to frustrated systems. For q = 2 the Ising spin is recovered. For q = 1 it gives the frustrated percolation model, which combines frustration and connectivity features and might be relevant to systems like gels of glasses. The solution on a decorated lattice shows that a line of critical temperatures To(q) appears when frustration is introduced. Tp(q) is the percolation t e m p e r a t u r e where the clusters used in the Swendsen and Wang dynamics diverge. T h e critical behaviour at Tp(q) is found to be the same as the ferromagnetic q/2 state Potts model, implying the universality class of the ferromagnetic 1/2 state Potts model for frustrated percolation.

The cluster dynamics algorithm of Swendsen and Wang [1] (SW) for ferromagnetic Potts models [2] has opened a new avenue for accelerated simulations in spin systems, stimulating a large amount of research [3,4] in this direction. The extremely high efficiency of the SW algorithm is based on a cluster formulation [5,6], in which the size of the clusters representing thermal fluctuations diverges at the critical temperature of the q-state Potts model. For q = 1 these clusters describe the standard random percolation problem [7]. For frustrated systems such as Ising spin glasses [8] the natural extension of the cluster formulation and its connection to the thermodynamic properties [9] showed on the basis of numerical simulations [10,11] that these clusters diverge at a temperature Tp much higher than the spin glass critical temperature Tsg. It is the formation of an infinite cluster at such high temperature that prevents the SW algorithm to be efficient in the spin glass case. In this paper we introduce a formalism to study connectivity properties in frustrated systems. In particular we investigate the nature of the percolation transition at Tp, its universality class and its effect on the free energy. To do so, we introduce a q-state Hamiltonian model which allows the generalization of the cluster formulation of the ferromagnetic q-state Potts model to flus0378-4371/93/$06.00 (~) 1993 - Elsevier Science Publishers B.V. All rights reserved

168

V. Cataudella et al. / Cluster formulation for frustrated spin models

trated systems with competing interactions. As a particular case, for q = 2 the Ising spin glass is recovered, whereas for q = 1 we obtain the frustrated percolation model [9] which describes percolation properties in frustrated systems. We will argue that for frustrated systems there is a line of percolation transitions depending on q. For the Ising spin glass case (q = 2), the transition corresponds to random percolation and, due to a vanishing amplitude in the free energy, it does not induce any singularity in the thermodynamic quantities. For the frustrated percolation problem ( q = 1), the transition is in the same universality class as the ferromagnetic s = 1/2 state Potts model. In general, for any q the transition is in the same universality class as for the s = q/2 state Potts model. We introduce the following Hamiltonian: --H = Z

(q)

[Jao-ic~j(~-ijSiSj -~ 1) -

2J],

(1)

where at each site there is an Ising variable S i =---1 and an s-state Potts variable o-i = 1 . . . . , s. The e# = -+ 1 are quenched variables and specify the sign of the interaction of the Ising variables. The sum is extended to all nearest neighbour pairs of sites of the lattice. For each pair (ij } the interaction energy H q = [ . . . ] in eq. (1) can be zero, if % = % and e q S f l j = l , or H q = - 2 J otherwise. Note that if eq = 1 (ferromagnetic case), eq. (1) is equivalent to a q-state ferromagnetic Potts model with q = 2s. For a random distribution of Eq the Hamiltonian (1) is a q-state model with competing interactions, which for s = 1 gives the Ising spin glass. For general s, a plaquette is frustrated when it is impossible for all spins to satisfy at the same time their mutual interactions so that eqSflj = 1. Note that the Potts variables are never frustrated. We introduce now a cluster formulation for the model Hamiltonian (1). For a fixed realization of {eq} and a fixed configuration of spin and Potts variables {Si, %} we introduce a bond between a pair (i j) only if the interaction energy is satisfied (fig. 1), each bond being present with probability given by

p = 1- exp(-2J/kT),

(2)

where T is the temperature. Two sites belong to the same cluster if they are connected by at least one chain of bonds. For a given configuration of variables {Si, %} and bonds C, compatible with the variable configuration, the weight W({Si, ~'}, C) is given by

W({Si, cri}, C) = plCf(1 - p)fBf e x p ( - H / k T ) ,

(3)

169

V. Cataudella et al. / Cluster formulation for frustrated spin models

L2 a

b

Fig. 1. (a) Example of a configuration of up and down Ising spins, represented respectively by the

circles and the squares, and Potts variables gi = 1, 2. Straight and wavy lines indicate respectively ferromagnetic and antiferromagneticinteractions between pairs of Ising spins. The Potts variables always interact ferromagnetically. (b) Clusters obtained from the configuration given in (a) by putting bonds (heavy lines) between spins satisfyingthe interaction with probabilityp given in eq. (2). where Icl is the number of bonds in the configuration C and [B I the number of bonds absent between pairs of spins for which the interaction is satisfied ( H , = o). From the Hamiltonian (i), exp(-H/kT)= [exp(-2J/kT)] IDl, where IDI is the number of pairs for which H~j = - 2 Z Using (3), the weight can be written as

W({S,, o-i}, C) = plCl(1 - p)lal

l~

(ij)~C F

6s,sj

l-I

{mn)~C A

(1-6sins)

I-I s~,,~i , (4)

(ij)EC

where [A[ = IBI + [DI is the total number of absent bonds, C F and C A the subsets of C made respectively of ferromagnetic (Eij= 1) and antiferromagnetic (eij= - 1 ) bonds. The products of the Kroenecker delta 6sisj and 6~i~ take into account the fact that the weight (3) is zero whenever the bonds in the configuration C do not satisfy the rule of being inserted between pairs {ij) satisfying the interaction (H/j = 0). As a consequence those sets of bonds C which contain a frustrated loop always have zero weight since there is never a configuration of spins consistent with such bonds. The partition function can formally be written in two ways. By taking first the sum in (3) over all allowed bond configurations C, we have the usual expression Z = E{s~,~J e x p ( - / 3 H ) . Alternatively, from (4) by taking first the sum over the variable configurations {Si, o-i} we have Z =~*

plCl(l - p)lA](2s)U(C)

(5)

where (2s) N(c) is the number of spin configurations compatible with the bond configuration C and N(C) is the number of clusters in C. The star means that

V. Cataudella et al. / Cluster formulation for frustrated spin models

170

the sum must be taken only over those configurations C that do not contain frustration. Eq. (5) generalizes the cluster formulation of Kasteleyn and Fortuin for the q = 2s state Ports model to frustrated systems. In particular it gives for q = 2 the Ising spin glass and for q = 1 frustrated percolation [9]. In this model, for any realization of interactions {%} a bond configuration C which does not contain frustration has a weight given by W(C) = plCl(1 - p)lAI, while it is zero for all configurations which do contain frustration. The frustrated percolation model plays the same role in the spin glass problem as random percolation in an Ising model. It may apply also to systems like glasses and gels where both percolation and frustration concepts are present. For instance, a system made of three-functional monomers exhibits a sol-gel transition when a macroscopic molecule is formed. If we model the system on a lattice of coordination number z > 3, the monomers can only react with 3 out of z nearest neighbours, resulting in frustration. Finally, in the limit s---~0 we obtain the frustrated tree percolation, which allows the same configuration as the random tree percolation since the trees do not contain frustration. An advantage of the Hamiltonian (1) introduced here is that these problems can be studied with techniques which make use of a Hamiltonian formulation such as the renormalization group. Following ref. [9] it is straightforward to prove the following relations for each fixed realization of {%}: (Si)

= Pi~ --) Pi,~ ,

(6)

(7)

( SiSj) = Pqll - Piix ,

where Pir (P~,) is the probability that the spin at site i is up (down) and belongs to the infinite cluster, and Pqll (Pijk() is the probability that the spins at site i and j are parallel (antiparallel) and belong to the same cluster. The angular brackets stand for an average over all configurations of spins and Potts variables {S~, cri} and bonds C with weight given by (4). Similarly ((s6~s I - 1)) = (s - 1)Pi, ( ( S t ~ i 1 -- 1)(S(~ j I -- 1 ) )

(8) l ) p qfin + (s - .,2 inf

= (S -- ~

1) Pij

'

(9)

where Pi =Pi¢ + Pi~ is the probability that site i is in the infinite cluster and p qf i n . ,( p qinfx ) is the probability that i and j are in the same finite (infinite) cluster.

V. CataudeUa et al. / Cluster formulation for frustrated spin models

171

T h e above relations are obtained for a fixed realization of interactions {Eq}. Relations ( 6 ) - ( 9 ) are still valid if we average over all bond configurations. In particular, from (6) we can construct the E d w a r d s - A n d e r s o n order parameter and relate it to the connectivity properties: qEA

/

1

.

(S,)2

N

.

(Pi'f - Pi$) 2,

(10)

where 2¢ is the n u m b e r of sites and the bar stands for the average over the realization of interactions. Moreover, from (8) the density of sites in the infinite cluster can be obtained: inf P

1 - N(s - 1)

~i

((S~°il

1 ~/ . E.

-- 1)) = ~

1(1)

Note that there are two order parameters, qEA and pinf. They define two critical temperatures, respectively T c and Tp, above which they are zero. Since Pi = Pie + Pg~ from (10) and (11) it follows that qEA ~< pinf ,

(12)

which implies T c ~< Tp. The two temperatures coincide when there is no frustration. For instance, in the ferromagnetic case Pi~ = 0 (in the phase of positive magnetization), from (6), (8), (10) and (11) it then follows that T c = Tp. For s = 1 and for a random distribution of the interactions {eq}, T c is the spin glass critical temperature. Numerically in 3D it is found [12] that Tc --~ 1.2, whereas the percolation transition Tp = 3.95 [10]. In 2D, T c = 0 and Tp -----1.80 [11]. For the fully frustrated model [4,11] T c = 0 and Tp --~ 1.69 [11]. Is there a singularity in the free energy at the percolation transition To? If we look at the partition function (5), one is led to conclude that there is a singularity at Tp due to the singularity in the cluster number N(C). To have an insight on the frustration at Tp, we consider a decorated lattice, where each unit cell is made of four frustrated plaquettes (fig. 2). After decimation we obtain an equivalent model in which two nearest neighbour Ising spins interact at the same time with two opposite interactions + J1 and - J ~ . The Hamiltonian for the decimated model is given by (1) where for each pair (ij) one has to sum over the ferromagnetic and antiferromagnetic interactions H = Z (q)

+

(13)

where H F = J,a~,~)(SiS~ + 1) - 2 J 1 and H A = J,6~,~j(-SiS j + 1) - J,. The percolation transition Tp, where the order parameter (11) goes to zero, is easily

V. Cataudella et al. / Cluster formulation for frustrated spin models

172

a

Fig. 2. (a) Unit cell of a frustrated model where straight and wavy lines indicate ferromagnetic and antiferromagnetic interactions between Ising spins. Potts variables interact ferromagnetically according to eq. (1). (b) Unit cell obtained from (a) after decimating on the sites of coordination number two (black sites). The ferromagnetic interactions +J1 have the same strength as the antiferromagnetic interactions J1. calculated if we consider that the H a m i l t o n i a n (13) can be written as -H

= 2J l E

(fio.yi - 2 ) .

(14)

(ij)

This is the H a m i l t o n i a n of the f e r r o m a g n e t i c s-state Potts model. T h e r e f o r e for e a c h s the o r d e r p a r a m e t e r (11) goes to zero at a t e m p e r a t u r e Tp(S) with a critical e x p o n e n t of the s-state Potts model. N o t e that the o t h e r o r d e r p a r a m e t e r (10) is always zero since the Ising variables do not a p p e a r in (14). F r o m (14) the critical b e h a v i o u r of the free e n e r g y is given by In Z = A ( s ) [ T

-

Tp(s)]

2-~(s)

,

(15)

w h e r e a(s) is the specific heat critical e x p o n e n t of the s-state Potts m o d e l and A ( s ) is an amplitude that vanishes as s ~ 1, since in this limit the H a m i l t o n i a n (14) r e d u c e s to a constant. L e t us discuss the role of frustration in the m o d e l and the c o n s e q u e n c e s on the S W cluster dynamics. If in the m o d e l of fig. 2a all the interactions were positive (no frustration), after decimation we would have o b t a i n e d an effective H a m i l t o n i a n given by (1) with % = 1 . This H a m i l t o n i a n , as noticed, is equivalent to a q-state f e r r o m a g n e t i c Potts m o d e l with q -- 2s, leading to a singularity in the free e n e r g y of the q-state Potts model. T h e presence of frustration, as we have s h o w n , has the effect of inducing a new critical b e h a v i o u r at Tp(S), drastically changing the universality class from the q to the s = q / 2 state Potts m o d e l . F o r the Ising case q = 2 the e x p o n e n t s are those of the s = 1 state Potts m o d e l , n a m e l y r a n d o m percolation. O f course this percolation transition does

V. Cataudella et al. / Cluster formulation for frustrated spin models

173

not affect the free energy due to the vanishing amplitude. For all the other values of q the singularity is present in the free energy. In particular we find that the frustrated percolation model is in the class of universality of the s = 1/2 state Potts model, while, as expected, the frustrated tree percolation problem corresponds to the s = 0 state Potts model as the standard tree percolation. The decorated model introduced so far exhibits only the percolation transition at Tp, where the clusters, defined before, which are used in the SW dynamics, diverge. It is not difficult to modify the model to exhibit also the second transition at Tc, by replacing the frustrated plaquette with the plaquette of fig. 3. For simplicity we consider here only the Ising Hamiltonian (s = 1), - - H = J E ( i j ) e 6 S i S j. After decimating the internal sites, the spins interact ferromagnetically via J2 = J1 + J and antiferromagnetically via - J 1 , with an effective ferromagnetic interaction Jeff = -/2 - J1 = J between the undecimated spins. Therefore we find beside the percolation temperature Tp, also an Ising transition at the critical temperature T c < Tp. If we had to employ the SW cluster dynamics on the decorated model of fig. 3, the dynamics would become highly inefficient below the percolation transition Tp due to the presence of the infinite cluster. This problem could be avoided by considering the system of undecimated spins with an effective ferromagnetic interaction J~,. The SW cluster dynamics applied to this system with a bond probability p = 1 exp(-2J~ff/kT) restores then the usual efficiency of the pure Ising model. We expect that the above picture with percolation exponents in the connectivity properties of the clusters and without singularity in the free energy at Tp is valid in general for Ising spin glasses and other frustrated systems. In fact, numerical simulations in 2D and 3D spin glasses and 2D fully frustrated models show at Tp random percolation exponents in the percolation quantities [10,11]. Moreover, since in the fully frustrated model it is known analytically that there are no singularities for T > 0, we can exclude the presence of a singularity in the free energy at T o .

@

Jeff

-J

a

©

©

1

b

c

Fig. 3. (a) Frustrated plaquette of the unit cell as in fig. 2a. (b) Effective interactions - J l =

½kT ln{2 exp(-2J/kT)/[1 + exp(-4J/kT)]) and J2 = J1 + J after decimating on the sites of coordination n u m b e r two. (c) Effective ferromagnetic interaction Jeff = J2 - Jl = J-

V. Cataudella et al. / Cluster formulation for frustrated spin models

174

T o show that there is no singularity in the free energy for spin glasses also, we have calculated numerically the specific heat and its derivative with respect to the t e m p e r a t u r e for a 3D Ising spin glass on a cubic lattice of size L = 32 o v e r a range of t e m p e r a t u r e s about Tp ( 3 . 9 < T < 4 . 2 ) . We have found a s m o o t h curve with no signal of singular behaviour. The Monte Carlo simulations have been p e r f o r m e d o v e r 10 6 MC steps after having thermalized the system for 105 M C steps. T h e results have been averaged over 12 configurations of interactions. Similar results have also been obtained for 2D Ising spin glasses (L = 64) and the 2D fully frustrated model (L = 64) for comparison. F o r these two models we have also studied the time-depending e n e r g y - e n e r g y correlation function defined as

F(t) = ( E ( t ' ) E(t' + t)) - ( E ( t ' ) ) 2 (E(t,)2) _ (E(t,))2

(16)

for t e m p e r a t u r e s around Tp. In this case too, we do not find any indication of an a n o m a l o u s increase of the relaxation time r a t Tp. In conclusion, we have introduced a 2s-state model (1) to study connectivity properties in frustrated systems. While s = 1 corresponds to the standard spin glass model, other values of s give new models which may be relevant to other physical systems where both frustration and connectivity properties are relevant. Since not much information is available for these models, further investigation would be of much interest. The formalism introduced here, which allows to study cluster properties in Ising spin glasses, might also be useful to help to develop an efficient cluster dynamics for frustrated systems. This w o r k was partially supported by the Ministero dell'Universith e della Ricerca Scientifica e Tecnologica ( M U R S T ) and the Consiglio Nazionale delle Ricerche (CNR).

References [1] R.H. Swendsen and J.S. Wang, Phys. Rev. Lett. 58 (1987) 86. J.S. Wang and R.H. Swendsen, Physica A 167 (1991) 565. [2] Wu, Rev. Mod. Phys. 54 (1982) 235. [3] F. Niedermayer, Phys. Rev. Lett. 61 (1988) 2026. U. Wolff, Phys. Rev. Lett. 62 (1989) 361. [4] D. Kandel, R. Ben-av and E. Domany, Phys. Rev. Lett. 65 (1990) 941. [5] C.M. Fortuin and P.W. Kasteleyn, Physica 57 (1972) 536. [6] A. Coniglio and W. Klein, J. Phys. A 12 (1980) 2775. [7] D. Stauffer, Introduction to Percolation Theory (Taylor and Francis, London, 1985). [8] K. Binder and A.P. Young, Rev. Mod. Phys. 58 (1986) 801. [9] A. Coniglio, F. di Liberto, G. Monroy and F. Peruggi, Phys. Rev. B 44 (1991) 12605. [10] L. de Arcangelis, A. Coniglio and F. Peruggi, Europhys. Lett. 14 (1991) 515. [11] V. Cataudella, Physica A 183 (1992) 249. [12] A.T. Ogielski, Phys. Rev. B 37 (1985) 7384.