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Spin glasses and frustrated percolation: a renormalization group approach
PHYSlCA/!', ELSEVIER
Physica A 237 (1997) 353 362
Spin glasses and frustrated percolation: a renormalization group approach Umberto Pezzella*, Antonio Coniglio INFM and INFN Sezione di Napoli, Dipartimento di Scienze Fisiche, Universita' di Napoli. 1-80125 Napoli, Italy Received II October 1996
Abstract
We develop a Hamiltonian formalism for the frustrated percolation model. This is a cluster model which has recently received attention for its relevance to glassy systems and granular materials. The exact solution of the model on the frustrated hierarchical lattice of Mc Kay et al. exhibits a percolation transition with critical exponents of the ferromagnetic ½-state Potts model and a spin glass transition characterized by chaotic renormalization group trajectories. Finally, we discuss the model in light of recent experiments. PA CS: 05.50
One of the features of many complex systems is the appearance of many minima in the free energy, which allows the system to exist in one of the many different states for a given set of external parameters. The corresponding rough landscape in configurational space induces a complex dynamical behavior, characterized by relaxation phenomena occurring on many time scales. At low temperatures, the system may become trapped in one of the minima and exhibit non-ergodic behavior. One mechanism leading to a landscape with many minima is based on the concept of frustration. In spin glasses [1] (SG), frustration occurs when the ferromagnetic and antiferromagnetic interactions are distributed in such a way that not all the spins can satisfy all the interactions simultaneously. In glass-forming systems [2] without underlying crystalline order, frustration arises because the molecules are intrinsically unable to fon-n close-packed configurations at low temperature or high density, while for systems with underlying crystalline order, frustration arises when the local arrangement of molecules kinetically prevents all the molecules from reaching the crystalline state. Examples of * Corresponding author. 0378-4371/97/$17.00 Copyright (~ 1997 Published by Elsevier Science B.V. All rights reserved Pll S03 7 8 - 4 3 7 1 ( 9 6 ) 0 0 4 3 4 - 7
354
u. Pezzella, A. Coniglio/Physica A 237 (1997) 353-362
other systems where frustration is present are proteins, liquid crystals, granular media and many others. In order to understand the complex behavior of frustrated systems, recently, a model has been introduced in both site and bond version [3], which contains as essential ingredient only geometrical frustration. Since percolation plays an important role, this model has been called frustrated percolation (FP). To define site FP, let us consider a regular lattice and randomly assign to each edge of the lattice a quenched interaction eij = +1 with equal probability. A loop is considered frustrated if the product of the signs along the loop is - 1 . We now randomly introduce particles on the vertices of this lattice, with the constraint [4] that frustrated loops cannot be fully occupied Fig. l(a). Frustration therefore induces defects or holes in the system, since a frustrated loop must contain at least one hole. The dramatic effect of frustration on particle motion can be seen by allowing the particles to diffuse such that no frustrated loops become completely occupied. At low particle densities, motion is not inhibited by frustration because of the abundance of holes. However, at high densities, a given particle can diffuse through the system only by a large scale, cooperative rearrangement of many particles. The site version, which has been proposed as the model for glass-fomaing liquids, exhibits a complex dynamical behavior characterized [5] by a two-step relaxation function, typical of glass-forming systems. Recently, the site version has also been applied to granular material [6] to explain the complex properties of such systems. In the bond version [7], we assign to each edge of the lattice a random bond, each with probability p under the constraint that a frustrated loop cannot be fully occupied Fig. l(b). That is, to a given bond configuration C, we assign zero weight W ( C ) = 0 if the configuration C contains at least one fully occupied frustrated loop and a weight given by W ( C ) = plCl(1 - p)lAI
(1)
(a)
I
Q,,-
0
0
II ~
'
(b)
I !
,,,~..,.~. I I ~
II ~
Ii
II
Fig. I. A configuration of (a) sites and (b) bonds in the frustrated percolation model. Straight lines and wavy lines correspond,respectively,to ci/= +1, and cii - -1. In (a) particles arc represented by filled circles. In (b) present bonds are represented by thick lines.
U. Pezzella, A. Coniglio/Physica A 237 (1997) 3 5 3 - 3 6 2
355
if the bond configuration C does not contain a fully occupied jrustrated loop. Here, [C 1 is the number of present bonds and [A[ the number of absent bonds. The FP interpolates between standard percolation, in which all configurations with and without loops are allowed, and Tree Percolation in which only loopless configurations are allowed. However, as we will show now, the model is much richer due to interfering effects caused by frustration. In fact, the bond FP is related to the i J [sing spin-glass model, just as the random bond percolation is related to the Ising model. We have studied the static properties of the bond FP model via a renormalization group calculation on a hierarchical lattice, using a Hamiltonian formalism which will be introduced in the next section. We find two transitions: a percolation transition with critical exponents of the ferromagnetic ½-state Potts model [8] (as opposed to standard percolation which is described by the ferromagnetic l-state Potts model), and in the percolating phase a second-transition in the same universality class of the Ising spinglass transition. Each critical point is characterized by a diverging length, associated with the quantities Pij = P+ + P~, ,q, i =
-
~2)
p;.
Here p+ ( p g ) is the probability that (1) sites i and j are connected by at least one path of bonds, and (2) the phase qii defined as the product over all the signs ~:..... along the path connecting i and j is +1 ( - 1 ) [9] (Fig. 2). The length ~p associated with the pair connectedness function ~ diverges at the percolation critical point, while the length ~ associated with O~i diverges at the second critical point (the bar represents the average over all possible interaction configurations {~:ij}). Since Pij gives the probability that i and j belong to the same cluster of sites connected by bonds, ~p corresponds to the linear dimension of these clusters. What is the physical meaning of the second diverging length? The second transition, like the quantum percolation transition [10], occurs at a bond density higher than the usual percolation transition, due to the interference of paths with different phases. At high density the number of allowed configurations is extremely reduced and most of lhe a)
b~
Fig. 2. Examples of clusters in frustrated percolation. Straight lines and wavy lines correspond, respectively. to ~:ij = +1 and ~:iJ --1. Thick lines are present bonds. The phase *lij = +1 in (a) and (b) and ~lii I in (c). -
356
U. Pezzella, A. ConiolioIPhysica A 237 (1997) 353-362
configurations connecting a given pair of sites i and j will have a common path. We call this path a "quasi-frozen" path, since in a dynamical sequence of configurations exploring the allowed phase space, this path will be present most of the time. In conclusion, the major contribution to 9i) is due to those configurations in which i and j are connected by quasi-frozen bonds [11] and 19~ijl roughly coincides with the probability that i and j belong to the same quasi-frozen cluster, and the length associated with 9 2 roughly represents the linear dimension of these clusters. Hamiltonian formalism. Consider the following hamiltonian: qJ - H = ~ - ~_~[6~a,(eijS,.Sj + 1) - 2], @)
(4)
where each site contains both an Ising variable S/ = + 1 and an s-state Potts variable ai = 1. . . . . s. As before, the e0 = + l are randomly distributed quenched variables. Note that for s = 1 (6~,,j = 1), the Hamiltonian (4) coincides with the spin glass model. It has been shown that, for a given set of interactions {~S}, the partition function for this Hamiltonian can be expressed, using the cluster formalism of Fortuin-Kasteleyn [12] and Coniglio-Klein [13], in terms of the bond FP configurations:
= X
= Z
{s,o,}
(s)
c
where Wq(C) = 0 for a bond configuration C which contains at least one fully occupied frustrated loop and Wq(C) = plCl(1 - p)lAlq N(C)
(6)
for a bond configuration C which does not contain a fully occupied frustrated loop. Here q = 2s, p = 1 e -qlU, N ( C ) is the number of clusters in the bond configuration C and [C[ and IAI are the numbers respectively of present and absent bonds. Moreover, following the Fortuin and Kasteleyn [12] (FK) formalism for the ferromagnetic Ising model, it is possible to show [14] that the pair correlation functions of the spin variables and Potts variables are related to the connectivity quantities (2) and (3). More precisely, for each fixed realization of {e,ij} -
Oij ~ (SiSj} = Vii-{+)- P{i{ ).
(7)
The angular brackets stand for an average over all configurations of spins and Potts variables {Si, at} weighted with the Boltzmann factor e -I~H, where H is given by (4). p{+) (/ ( p l f ) ) is the probability that (1) sites i and j are connected by at least one path of bonds, and (2) the phase ~lij along the path connecting i and j is +1 ( - 1 ) . Now the bond configurations are weighted according to the weights Wq(C) defined above. Similarly ((saael - 1)(sra,!
-
1)} = (s - 1)p{j in + (s
-
112 i,¢ , ;'is,
(8)
where ,_}j o fin (pi~ f ) is the probability that i and j are in the same finite (infinite) cluster.
U. Pezzella, A. Coni.qliolPhysica A 237 (1997) 353
362
357
By taking the limit q ~ 2 (s ~ l) in (5), we obtain the FK cluster formulation of the partition function of the spin glass mode [7], while in the limit q ~ 1 (s ~ ½) Eq. (5) gives the partition function of the bond FP model (1) [15]. In the ferromagnetic case (~:i,j = 1 ) Eq. (5) reproduces the FK cluster formulation of the partition function of the q-state Potts model. In this case Pij(-) = 0 and Pij(+) = P~/ = p.lji,; ,j + P,:/i,;! • In this formulation it is clear how the FP model is closely related to the SG model. The only difference is that the bond configurations are weighted by (6) with q 1 for FP and q 2 for SG. The Hamiltonian formalism (4) allows to study FP using a renormalization group approach. We, in fact, apply an exact renormalization group to the Hamiltonian (4) on the hierarchical lattice introduced by Mc Kay et ah [16] (Fig. 3). The renormalization group generates a new term 26~;~/ in the Hamiltonian (4). So the Hamiltonian which keeps the same form under renormalization apart from an unessential constant [17] is the following: --#
= ~-~ (J (eijsisj + I)4-).)6,7~<. (,,/)
(9)
For 2 -+ vc the Potts variables are all identical and the Hamiltonian (4) reduces to SG model, while for J = 0 it reduces to the s-state Potts model. After decimating the internal sites, we have obtained [18] renormalization group equations in the variables x = e -qJ,'kBT and y = e->/'~Brwhich exhibit for any s > 0, in the plane 0~
-
o
o
(c) o
o
N~ o
Fig. 3. Construction of hierarchical lattice (c); made of N8 ~ units (a) and N c ~ units (b) in parallel. Straight lines and wiggly lines represent, respectively, ferromagnetic and antiferromagnetic interactions, n is the number of antiferromagnetic interactions in (a) and ml + m2 is the number of ferromagnetic interactions in (b). We have chosen NB -- 31, N c = 1, ml -- m2 -- 7.
U. Pezzella, A. Coniglio/Physica A 237 (1997) 353-362
358
Xp , ~ :
,
,
,
,
,
,i
\
(sG)
~(s=fl k
\ \ \ \
\ \ "..
i\
I
0
,
,
,'..~Y
Xs9
(a)
1
×
Xp
Xc
//
(p)
\,~,. ,yFs : in)
(sG) \ \ \ \ \ \
0
(b)
I
,
,
T ×sg ×
Fig. 4. Fixed points (y = 0, x = XSG) and (x = 1, y = yp(S)) and chaotic band (bold lines on the y = 0 axis). Also are shown schematically the two critical lines (continuous) and the renormalization group trajectories (dashed) for (a) spin glass (s = I) and (b) frustrated percolation (s = ½). In (c) is shown for s = ½ the chaotic behavior of the renormalization group trajectories. Starting anyway in the spin-glass phase they end up in the chaotic band.
LL Pezzella, A. ConigliolPhysica A 237 (1997) 353 362
359
0.8
0.6
0.4
0.2
0
(c)
0
02
0.4
0.6
08
Xg
x
Fig. 4 (continued)
the line y = 0 below the SG transition xsc we find a chaotic band as found by Mc Kay et al. [16] Starting anywhere in the SG phase, the renormalization group trajectories end up in the chaotic band with chaotic behavior. (Fig. 4(c)) The fixed point structure leads to two critical lines (Fig. 4(a) and (b)), one governed by the critical behavior of the Ising spin glass fixed point separating the spin-glass phase (SG) from the percolating phase (P); the other by the ferromagnetic s-state Potts model separating the (P) phase from the nonpercolating phase (NP). The intersections of the critical lines with y = 1 give the two critical points Xp(S) and x~(s) with
x~(s = 1) = xsc. The fact that the percolation transition of FP is described by a ferromagnetic ½-state Potts model, is consistent with the observation that such model interpolates between random percolation s = 1 and tree percolation s - 0, of the ferromagnetic s-state Potts model. In general, the RG analysis applied to the Hamiltonian (8) and therefore to the model (6) with q = 2s predicts a disordered high-temperature phase above Tp(q), a percolating phase between Tp(q) and Tc(q) and a spin-glass phase below T,.(q). At T~(q) the transition is for all q in the same universality class of the Ising SG transition. The critical behavior at Tp(q) depends on q and corresponds to the same universality class of the ferromagnetic s = q/2-state Potts model. For the spin-glass model (q = 2) the percolation temperature Tp(2) corresponds to the divergence of the clusters introduced by Coniglio and Klein [13] for the Ising model. However, in the lsing model the
360
U. Pezzella, A. Coniglio/Physica A 237 (1997) 353 362
percolation temperature coincides with the Ising critical temperature. In spin glasses instead, we find the percolation temperature Tp(2) larger than the SG temperature TsG = Tc(2). This result is consistent with numerical simulations [19], and explains why the cluster dynamics introduced by Swendsen and Wang [20] is not efficient at low temperature since an infinite cluster is present already above the spin glass transition Tsc. For the spin glass, numerically it is found Tsc ~ 1.1 [21] whereas the percolation transition Tp(2) ~- 3.95 [19]. In 2-d Tsc = 0 and Tp(2) ~_ 1.80 [22] and for the 2-d fully frustrated model Tc = 0 and Tp ~_ 1.69 [22]. If we look at the partition function (5), one is lead to conclude that there is a singularity at Tp(q) for any q (including q = 2) due to the singularity in the cluster number N ( C ) , as shown by the renormalization group calculation included q = 2. This consideration would predict a singularity in the spin-glass free energy at Tp(2) well above the spin-glass transition. However, this singularity has never been observed. To understand this puzzle better, we have applied the renormalization group calculation to the spin glass Hamiltonian obtained from (4) putting 6~,~j = 1. In this case we find the spin glass singularity at Tsc and no singularity at Tp(2). This result is interpreted by postulating the following critical behavior of the free energy of the frustrated Potts model (4) near Tp(q) : Fq(T) - Fq(Tp) ,-~ A ( q ) ( T - Tp(q)) 2-~(q),
where A(q) is an amplitude which vanishes as q approaches 2, and c~(q) is the specific heat critical exponent of the feromagnetic q-state Potts model [14]. This result is also consistent with relation (8) which shows that in the limit s = 1 (q = 2) the right-hand side vanishes. However, it is interesting to note that Tp in the 3-D [19,23] SG, in the 2-D fully frustrated model [24] and in the site-frustrated percolation model [5] numerically was found very close to the temperature below which the autocorrelation functions decay with stretched exponentials. This seems to suggest that the percolation transition may coincide with the onset of the stretched exponential behavior [25]. Since FP seems to well-describe the glass transitions in glass forming liquids [3], the model suggests that the presence of a percolation-type transition may be a general feature below which frustration effect starts to be manifested. This transition may be responsible for various precursor phenomena, such as the onset of stretched exponential behavior, breakdown of Stokes Einstein relation, presence of spatial heterogeneity. The presence of a percolation transition well above the glass transition has recently been discovered by Tomida and Egami [26] in a molecular dynamics simulation of monoatomic liquids. It is also interesting to note that Kivelson et al. [27] showed that the viscosity of 15 glass-forming liquids could be collapsed on one single curve, by assuming only one characteristic temperature well above the glass transition. Finally, we note that FP may also turn out to be a useful model for complex systems such as gels and rubber, where both connectivity properties and frustration properties are important. Large polymer molecules, in fact, cannot easily reach a closed packed configuration.
U. Pezzella. A. Conialio/Physica A 237 (19971 353-362
361
Adam et al. [28] have recently reported a dynamical study of a polymer cluster solution near the gelation threshold. In this experiment, they first obtained a cluster distribution by chemical gelation, then by dilution they changed the polymer concentration C. They found two critical concentrations: C* the concentration when the polymer chains start to interpenetrate and a second freezing concentration C/. Between the two critical concentrations they found the stretched exponential relaxation, and for large concentration, inside the glassy phase, power-law relaxation behavior. The presence of the two critical concentrations and the similarity with the spin-glass phenomenology is strongly suggestive of a description in terms of the FP model. We would like to thank A. De Candia, M. Nicodemi and S. Glotzer for interesting discussions. This work has been supported by CNR. References [1] K. Binder and A.P. Young, Rev. Mod. Phys. 58 (1986) 801; M. Mezard, G. Parisi and M.A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 19871. [2] For a review see for example M.D. Ediger, C.A. Angell and S.R. Nagel, J. Phys. Chem. 100 (1996) 13 200. [3] A. Coniglio, II Nuovo Cimento, D 16 (1994) 1027. [4] It can be shown that this constraint can be induced by the t:~j when these variables represent interactions between the particles internal degree of freedom. See M. Nicodemi and A. Coniglio, to be published. [5] S. Scarpetta, A. de Candia and A. Coniglio, to be published. [6] A. Coniglio and H.J. Herrmann, Physica A 225 (1996) 1, M. Nicodemi A. Coniglio and H.J. Herrmann, to Phys. Rev. Lett., submitted. [7] A. Coniglio, F. di Liberto, G. Monroy and F. Peruggi, Phys. Rev. B 44 (1991) 12605. [8] A connection between the dilute spin glass at zero temperature and the ½ state Potts model has been discussed by A. Aharony and Pfeuty J. Phys. C 12 (19791 L125. [9] If in a given configuration of bonds there is more than one path of bonds connecting i and /, it is straightforward to show that the resulting phase qii is path independent since the configuration does not contain frustrated loops (Fig. 2(b)). [10] Y. Melt, A. Aharony and A. B. Harris, Europhys. Lett. 10 (1989) 275. and references therein. [I 1] S.G. Glotzer and A. Coniglio, J. Comput. Math. Sci. 4 (1995) 324, N. Jan, S.C. Glotzer, P.H. Poole and A. Coniglio, Fractals, 3 (1990) 465. [12] C.M. Fortuin and P.W. Kasteleyn, Physica 57 (1972) 536. [13] A. Coniglio and W. Klein, J. Phys. A 13 (1980) 2775. [14] V. Cataudella, A. Coniglio, L. de Arcangelis and F. di Liberto, Physica A 192 (1993) 167. [15] Note that in the limit q--~ 0 we obtain the frustrated tree percolation, which allows the same configurations as in the random tree percolation since the trees do not contain frustration. Therefore the frustrated q-Potts model (4) and the ferromagnetic q-Potts model coincide in the limit q ~ 0. [16] S.R. McKay, A.N. Berker and S. Kirkpatrick, Phys. Rev. Lett. 48 (19821 767; and J. Appl. Phys. 53, (1982) 1873. [17] In the Hamiltonian (4) the term 2 has been added so that the energy between two sites is zero if the interaction is satisfied. With this choice the partition function is given exactly by (5). Otherwise the partition function would have been multiplied by an unessential factor. [I 8] Further details will appear in U. Pezzella and A. Coniglio, to be published. [19] L. de Arcangelis, A. Coniglio and F. Peruggi, Europhys. Lett. 14 (19911 515. [20] R.H. Swendsen and J.S. Wang, Phys. Rev. Lett. 58 (1987) 86. [21] N. Kawashima and A.P. Young, Phys. Rev. B 53 (1996) R484. [22] V. Cataudella, Physica A 183 (1991) 249. [23] A.T. Ogielski, Phys. Rev. B 32 (1985) 7384.
362 [24] [25] [26] [27] [28]
U Pezzella, A. Coniolio/Physica A 237 (1997) 353 362 A. Fierro, A. De Candia and A. Coniglio, to be published. See also I.A. Campbell and L. Bernardi, Phys. Rev. B 50 (1994) 12643. T. Tomida and T. Egami, Phys. Rev. B 52 (1995) 3290. D. Kivelson et al., Physica A 219 (1995) 27. M. Adam, M. Delsanti, J.P. Munch and D. Durand, Phys. Rev. Lett. 61 (1988) 706.
Physica A 237 (1997) 353 362
Spin glasses and frustrated percolation: a renormalization group approach Umberto Pezzella*, Antonio Coniglio INFM and INFN Sezione di Napoli, Dipartimento di Scienze Fisiche, Universita' di Napoli. 1-80125 Napoli, Italy Received II October 1996
Abstract
We develop a Hamiltonian formalism for the frustrated percolation model. This is a cluster model which has recently received attention for its relevance to glassy systems and granular materials. The exact solution of the model on the frustrated hierarchical lattice of Mc Kay et al. exhibits a percolation transition with critical exponents of the ferromagnetic ½-state Potts model and a spin glass transition characterized by chaotic renormalization group trajectories. Finally, we discuss the model in light of recent experiments. PA CS: 05.50
One of the features of many complex systems is the appearance of many minima in the free energy, which allows the system to exist in one of the many different states for a given set of external parameters. The corresponding rough landscape in configurational space induces a complex dynamical behavior, characterized by relaxation phenomena occurring on many time scales. At low temperatures, the system may become trapped in one of the minima and exhibit non-ergodic behavior. One mechanism leading to a landscape with many minima is based on the concept of frustration. In spin glasses [1] (SG), frustration occurs when the ferromagnetic and antiferromagnetic interactions are distributed in such a way that not all the spins can satisfy all the interactions simultaneously. In glass-forming systems [2] without underlying crystalline order, frustration arises because the molecules are intrinsically unable to fon-n close-packed configurations at low temperature or high density, while for systems with underlying crystalline order, frustration arises when the local arrangement of molecules kinetically prevents all the molecules from reaching the crystalline state. Examples of * Corresponding author. 0378-4371/97/$17.00 Copyright (~ 1997 Published by Elsevier Science B.V. All rights reserved Pll S03 7 8 - 4 3 7 1 ( 9 6 ) 0 0 4 3 4 - 7
354
u. Pezzella, A. Coniglio/Physica A 237 (1997) 353-362
other systems where frustration is present are proteins, liquid crystals, granular media and many others. In order to understand the complex behavior of frustrated systems, recently, a model has been introduced in both site and bond version [3], which contains as essential ingredient only geometrical frustration. Since percolation plays an important role, this model has been called frustrated percolation (FP). To define site FP, let us consider a regular lattice and randomly assign to each edge of the lattice a quenched interaction eij = +1 with equal probability. A loop is considered frustrated if the product of the signs along the loop is - 1 . We now randomly introduce particles on the vertices of this lattice, with the constraint [4] that frustrated loops cannot be fully occupied Fig. l(a). Frustration therefore induces defects or holes in the system, since a frustrated loop must contain at least one hole. The dramatic effect of frustration on particle motion can be seen by allowing the particles to diffuse such that no frustrated loops become completely occupied. At low particle densities, motion is not inhibited by frustration because of the abundance of holes. However, at high densities, a given particle can diffuse through the system only by a large scale, cooperative rearrangement of many particles. The site version, which has been proposed as the model for glass-fomaing liquids, exhibits a complex dynamical behavior characterized [5] by a two-step relaxation function, typical of glass-forming systems. Recently, the site version has also been applied to granular material [6] to explain the complex properties of such systems. In the bond version [7], we assign to each edge of the lattice a random bond, each with probability p under the constraint that a frustrated loop cannot be fully occupied Fig. l(b). That is, to a given bond configuration C, we assign zero weight W ( C ) = 0 if the configuration C contains at least one fully occupied frustrated loop and a weight given by W ( C ) = plCl(1 - p)lAI
(1)
(a)
I
Q,,-
0
0
II ~
'
(b)
I !
,,,~..,.~. I I ~
II ~
Ii
II
Fig. I. A configuration of (a) sites and (b) bonds in the frustrated percolation model. Straight lines and wavy lines correspond,respectively,to ci/= +1, and cii - -1. In (a) particles arc represented by filled circles. In (b) present bonds are represented by thick lines.
U. Pezzella, A. Coniglio/Physica A 237 (1997) 3 5 3 - 3 6 2
355
if the bond configuration C does not contain a fully occupied jrustrated loop. Here, [C 1 is the number of present bonds and [A[ the number of absent bonds. The FP interpolates between standard percolation, in which all configurations with and without loops are allowed, and Tree Percolation in which only loopless configurations are allowed. However, as we will show now, the model is much richer due to interfering effects caused by frustration. In fact, the bond FP is related to the i J [sing spin-glass model, just as the random bond percolation is related to the Ising model. We have studied the static properties of the bond FP model via a renormalization group calculation on a hierarchical lattice, using a Hamiltonian formalism which will be introduced in the next section. We find two transitions: a percolation transition with critical exponents of the ferromagnetic ½-state Potts model [8] (as opposed to standard percolation which is described by the ferromagnetic l-state Potts model), and in the percolating phase a second-transition in the same universality class of the Ising spinglass transition. Each critical point is characterized by a diverging length, associated with the quantities Pij = P+ + P~, ,q, i =
-
~2)
p;.
Here p+ ( p g ) is the probability that (1) sites i and j are connected by at least one path of bonds, and (2) the phase qii defined as the product over all the signs ~:..... along the path connecting i and j is +1 ( - 1 ) [9] (Fig. 2). The length ~p associated with the pair connectedness function ~ diverges at the percolation critical point, while the length ~ associated with O~i diverges at the second critical point (the bar represents the average over all possible interaction configurations {~:ij}). Since Pij gives the probability that i and j belong to the same cluster of sites connected by bonds, ~p corresponds to the linear dimension of these clusters. What is the physical meaning of the second diverging length? The second transition, like the quantum percolation transition [10], occurs at a bond density higher than the usual percolation transition, due to the interference of paths with different phases. At high density the number of allowed configurations is extremely reduced and most of lhe a)
b~
Fig. 2. Examples of clusters in frustrated percolation. Straight lines and wavy lines correspond, respectively. to ~:ij = +1 and ~:iJ --1. Thick lines are present bonds. The phase *lij = +1 in (a) and (b) and ~lii I in (c). -
356
U. Pezzella, A. ConiolioIPhysica A 237 (1997) 353-362
configurations connecting a given pair of sites i and j will have a common path. We call this path a "quasi-frozen" path, since in a dynamical sequence of configurations exploring the allowed phase space, this path will be present most of the time. In conclusion, the major contribution to 9i) is due to those configurations in which i and j are connected by quasi-frozen bonds [11] and 19~ijl roughly coincides with the probability that i and j belong to the same quasi-frozen cluster, and the length associated with 9 2 roughly represents the linear dimension of these clusters. Hamiltonian formalism. Consider the following hamiltonian: qJ - H = ~ - ~_~[6~a,(eijS,.Sj + 1) - 2], @)
(4)
where each site contains both an Ising variable S/ = + 1 and an s-state Potts variable ai = 1. . . . . s. As before, the e0 = + l are randomly distributed quenched variables. Note that for s = 1 (6~,,j = 1), the Hamiltonian (4) coincides with the spin glass model. It has been shown that, for a given set of interactions {~S}, the partition function for this Hamiltonian can be expressed, using the cluster formalism of Fortuin-Kasteleyn [12] and Coniglio-Klein [13], in terms of the bond FP configurations:
= X
= Z
{s,o,}
(s)
c
where Wq(C) = 0 for a bond configuration C which contains at least one fully occupied frustrated loop and Wq(C) = plCl(1 - p)lAlq N(C)
(6)
for a bond configuration C which does not contain a fully occupied frustrated loop. Here q = 2s, p = 1 e -qlU, N ( C ) is the number of clusters in the bond configuration C and [C[ and IAI are the numbers respectively of present and absent bonds. Moreover, following the Fortuin and Kasteleyn [12] (FK) formalism for the ferromagnetic Ising model, it is possible to show [14] that the pair correlation functions of the spin variables and Potts variables are related to the connectivity quantities (2) and (3). More precisely, for each fixed realization of {e,ij} -
Oij ~ (SiSj} = Vii-{+)- P{i{ ).
(7)
The angular brackets stand for an average over all configurations of spins and Potts variables {Si, at} weighted with the Boltzmann factor e -I~H, where H is given by (4). p{+) (/ ( p l f ) ) is the probability that (1) sites i and j are connected by at least one path of bonds, and (2) the phase ~lij along the path connecting i and j is +1 ( - 1 ) . Now the bond configurations are weighted according to the weights Wq(C) defined above. Similarly ((saael - 1)(sra,!
-
1)} = (s - 1)p{j in + (s
-
112 i,¢ , ;'is,
(8)
where ,_}j o fin (pi~ f ) is the probability that i and j are in the same finite (infinite) cluster.
U. Pezzella, A. Coni.qliolPhysica A 237 (1997) 353
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By taking the limit q ~ 2 (s ~ l) in (5), we obtain the FK cluster formulation of the partition function of the spin glass mode [7], while in the limit q ~ 1 (s ~ ½) Eq. (5) gives the partition function of the bond FP model (1) [15]. In the ferromagnetic case (~:i,j = 1 ) Eq. (5) reproduces the FK cluster formulation of the partition function of the q-state Potts model. In this case Pij(-) = 0 and Pij(+) = P~/ = p.lji,; ,j + P,:/i,;! • In this formulation it is clear how the FP model is closely related to the SG model. The only difference is that the bond configurations are weighted by (6) with q 1 for FP and q 2 for SG. The Hamiltonian formalism (4) allows to study FP using a renormalization group approach. We, in fact, apply an exact renormalization group to the Hamiltonian (4) on the hierarchical lattice introduced by Mc Kay et ah [16] (Fig. 3). The renormalization group generates a new term 26~;~/ in the Hamiltonian (4). So the Hamiltonian which keeps the same form under renormalization apart from an unessential constant [17] is the following: --#
= ~-~ (J (eijsisj + I)4-).)6,7~<. (,,/)
(9)
For 2 -+ vc the Potts variables are all identical and the Hamiltonian (4) reduces to SG model, while for J = 0 it reduces to the s-state Potts model. After decimating the internal sites, we have obtained [18] renormalization group equations in the variables x = e -qJ,'kBT and y = e->/'~Brwhich exhibit for any s > 0, in the plane 0~
-
o
o
(c) o
o
N~ o
Fig. 3. Construction of hierarchical lattice (c); made of N8 ~ units (a) and N c ~ units (b) in parallel. Straight lines and wiggly lines represent, respectively, ferromagnetic and antiferromagnetic interactions, n is the number of antiferromagnetic interactions in (a) and ml + m2 is the number of ferromagnetic interactions in (b). We have chosen NB -- 31, N c = 1, ml -- m2 -- 7.
U. Pezzella, A. Coniglio/Physica A 237 (1997) 353-362
358
Xp , ~ :
,
,
,
,
,
,i
\
(sG)
~(s=fl k
\ \ \ \
\ \ "..
i\
I
0
,
,
,'..~Y
Xs9
(a)
1
×
Xp
Xc
//
(p)
\,~,. ,yFs : in)
(sG) \ \ \ \ \ \
0
(b)
I
,
,
T ×sg ×
Fig. 4. Fixed points (y = 0, x = XSG) and (x = 1, y = yp(S)) and chaotic band (bold lines on the y = 0 axis). Also are shown schematically the two critical lines (continuous) and the renormalization group trajectories (dashed) for (a) spin glass (s = I) and (b) frustrated percolation (s = ½). In (c) is shown for s = ½ the chaotic behavior of the renormalization group trajectories. Starting anyway in the spin-glass phase they end up in the chaotic band.
LL Pezzella, A. ConigliolPhysica A 237 (1997) 353 362
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0.8
0.6
0.4
0.2
0
(c)
0
02
0.4
0.6
08
Xg
x
Fig. 4 (continued)
the line y = 0 below the SG transition xsc we find a chaotic band as found by Mc Kay et al. [16] Starting anywhere in the SG phase, the renormalization group trajectories end up in the chaotic band with chaotic behavior. (Fig. 4(c)) The fixed point structure leads to two critical lines (Fig. 4(a) and (b)), one governed by the critical behavior of the Ising spin glass fixed point separating the spin-glass phase (SG) from the percolating phase (P); the other by the ferromagnetic s-state Potts model separating the (P) phase from the nonpercolating phase (NP). The intersections of the critical lines with y = 1 give the two critical points Xp(S) and x~(s) with
x~(s = 1) = xsc. The fact that the percolation transition of FP is described by a ferromagnetic ½-state Potts model, is consistent with the observation that such model interpolates between random percolation s = 1 and tree percolation s - 0, of the ferromagnetic s-state Potts model. In general, the RG analysis applied to the Hamiltonian (8) and therefore to the model (6) with q = 2s predicts a disordered high-temperature phase above Tp(q), a percolating phase between Tp(q) and Tc(q) and a spin-glass phase below T,.(q). At T~(q) the transition is for all q in the same universality class of the Ising SG transition. The critical behavior at Tp(q) depends on q and corresponds to the same universality class of the ferromagnetic s = q/2-state Potts model. For the spin-glass model (q = 2) the percolation temperature Tp(2) corresponds to the divergence of the clusters introduced by Coniglio and Klein [13] for the Ising model. However, in the lsing model the
360
U. Pezzella, A. Coniglio/Physica A 237 (1997) 353 362
percolation temperature coincides with the Ising critical temperature. In spin glasses instead, we find the percolation temperature Tp(2) larger than the SG temperature TsG = Tc(2). This result is consistent with numerical simulations [19], and explains why the cluster dynamics introduced by Swendsen and Wang [20] is not efficient at low temperature since an infinite cluster is present already above the spin glass transition Tsc. For the spin glass, numerically it is found Tsc ~ 1.1 [21] whereas the percolation transition Tp(2) ~- 3.95 [19]. In 2-d Tsc = 0 and Tp(2) ~_ 1.80 [22] and for the 2-d fully frustrated model Tc = 0 and Tp ~_ 1.69 [22]. If we look at the partition function (5), one is lead to conclude that there is a singularity at Tp(q) for any q (including q = 2) due to the singularity in the cluster number N ( C ) , as shown by the renormalization group calculation included q = 2. This consideration would predict a singularity in the spin-glass free energy at Tp(2) well above the spin-glass transition. However, this singularity has never been observed. To understand this puzzle better, we have applied the renormalization group calculation to the spin glass Hamiltonian obtained from (4) putting 6~,~j = 1. In this case we find the spin glass singularity at Tsc and no singularity at Tp(2). This result is interpreted by postulating the following critical behavior of the free energy of the frustrated Potts model (4) near Tp(q) : Fq(T) - Fq(Tp) ,-~ A ( q ) ( T - Tp(q)) 2-~(q),
where A(q) is an amplitude which vanishes as q approaches 2, and c~(q) is the specific heat critical exponent of the feromagnetic q-state Potts model [14]. This result is also consistent with relation (8) which shows that in the limit s = 1 (q = 2) the right-hand side vanishes. However, it is interesting to note that Tp in the 3-D [19,23] SG, in the 2-D fully frustrated model [24] and in the site-frustrated percolation model [5] numerically was found very close to the temperature below which the autocorrelation functions decay with stretched exponentials. This seems to suggest that the percolation transition may coincide with the onset of the stretched exponential behavior [25]. Since FP seems to well-describe the glass transitions in glass forming liquids [3], the model suggests that the presence of a percolation-type transition may be a general feature below which frustration effect starts to be manifested. This transition may be responsible for various precursor phenomena, such as the onset of stretched exponential behavior, breakdown of Stokes Einstein relation, presence of spatial heterogeneity. The presence of a percolation transition well above the glass transition has recently been discovered by Tomida and Egami [26] in a molecular dynamics simulation of monoatomic liquids. It is also interesting to note that Kivelson et al. [27] showed that the viscosity of 15 glass-forming liquids could be collapsed on one single curve, by assuming only one characteristic temperature well above the glass transition. Finally, we note that FP may also turn out to be a useful model for complex systems such as gels and rubber, where both connectivity properties and frustration properties are important. Large polymer molecules, in fact, cannot easily reach a closed packed configuration.
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Adam et al. [28] have recently reported a dynamical study of a polymer cluster solution near the gelation threshold. In this experiment, they first obtained a cluster distribution by chemical gelation, then by dilution they changed the polymer concentration C. They found two critical concentrations: C* the concentration when the polymer chains start to interpenetrate and a second freezing concentration C/. Between the two critical concentrations they found the stretched exponential relaxation, and for large concentration, inside the glassy phase, power-law relaxation behavior. The presence of the two critical concentrations and the similarity with the spin-glass phenomenology is strongly suggestive of a description in terms of the FP model. We would like to thank A. De Candia, M. Nicodemi and S. Glotzer for interesting discussions. This work has been supported by CNR. References [1] K. Binder and A.P. Young, Rev. Mod. Phys. 58 (1986) 801; M. Mezard, G. Parisi and M.A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 19871. [2] For a review see for example M.D. Ediger, C.A. Angell and S.R. Nagel, J. Phys. Chem. 100 (1996) 13 200. [3] A. Coniglio, II Nuovo Cimento, D 16 (1994) 1027. [4] It can be shown that this constraint can be induced by the t:~j when these variables represent interactions between the particles internal degree of freedom. See M. Nicodemi and A. Coniglio, to be published. [5] S. Scarpetta, A. de Candia and A. Coniglio, to be published. [6] A. Coniglio and H.J. Herrmann, Physica A 225 (1996) 1, M. Nicodemi A. Coniglio and H.J. Herrmann, to Phys. Rev. Lett., submitted. [7] A. Coniglio, F. di Liberto, G. Monroy and F. Peruggi, Phys. Rev. B 44 (1991) 12605. [8] A connection between the dilute spin glass at zero temperature and the ½ state Potts model has been discussed by A. Aharony and Pfeuty J. Phys. C 12 (19791 L125. [9] If in a given configuration of bonds there is more than one path of bonds connecting i and /, it is straightforward to show that the resulting phase qii is path independent since the configuration does not contain frustrated loops (Fig. 2(b)). [10] Y. Melt, A. Aharony and A. B. Harris, Europhys. Lett. 10 (1989) 275. and references therein. [I 1] S.G. Glotzer and A. Coniglio, J. Comput. Math. Sci. 4 (1995) 324, N. Jan, S.C. Glotzer, P.H. Poole and A. Coniglio, Fractals, 3 (1990) 465. [12] C.M. Fortuin and P.W. Kasteleyn, Physica 57 (1972) 536. [13] A. Coniglio and W. Klein, J. Phys. A 13 (1980) 2775. [14] V. Cataudella, A. Coniglio, L. de Arcangelis and F. di Liberto, Physica A 192 (1993) 167. [15] Note that in the limit q--~ 0 we obtain the frustrated tree percolation, which allows the same configurations as in the random tree percolation since the trees do not contain frustration. Therefore the frustrated q-Potts model (4) and the ferromagnetic q-Potts model coincide in the limit q ~ 0. [16] S.R. McKay, A.N. Berker and S. Kirkpatrick, Phys. Rev. Lett. 48 (19821 767; and J. Appl. Phys. 53, (1982) 1873. [17] In the Hamiltonian (4) the term 2 has been added so that the energy between two sites is zero if the interaction is satisfied. With this choice the partition function is given exactly by (5). Otherwise the partition function would have been multiplied by an unessential factor. [I 8] Further details will appear in U. Pezzella and A. Coniglio, to be published. [19] L. de Arcangelis, A. Coniglio and F. Peruggi, Europhys. Lett. 14 (19911 515. [20] R.H. Swendsen and J.S. Wang, Phys. Rev. Lett. 58 (1987) 86. [21] N. Kawashima and A.P. Young, Phys. Rev. B 53 (1996) R484. [22] V. Cataudella, Physica A 183 (1991) 249. [23] A.T. Ogielski, Phys. Rev. B 32 (1985) 7384.
362 [24] [25] [26] [27] [28]
U Pezzella, A. Coniolio/Physica A 237 (1997) 353 362 A. Fierro, A. De Candia and A. Coniglio, to be published. See also I.A. Campbell and L. Bernardi, Phys. Rev. B 50 (1994) 12643. T. Tomida and T. Egami, Phys. Rev. B 52 (1995) 3290. D. Kivelson et al., Physica A 219 (1995) 27. M. Adam, M. Delsanti, J.P. Munch and D. Durand, Phys. Rev. Lett. 61 (1988) 706.