Studi Napoli, manuscript PHYSICAL REVIEW E JANUARY 1999VOLUME 59, NUMBER 1 This result supports the argument suggested in Ref. @4#, recently verified by more rigorous analysis in Ref. @5#, and by numerical simulations on a generalization of the SG model in Refs. @7,8#. According to these analyses, in the SG the onset T* of the nonexponential relaxation should be greater than or equal to the Griffiths temperature T G . This behavior is caused by the existence of unfrustrated ferromagnetic-type clusters of interactions, the same as those responsible for the Griffiths singularity @6#. The presence of nonexponential relaxation in this approach is therefore a di- rect consequence of the quenched disorder. Another mechanism leading to nonexponential relaxation in frustrated systems, such as SG?s, has been suggested by several authors @9?11#. According to these arguments the onset T* of nonexponential relaxation is greater than or equal to the percolation transition T p of the Kasteleyn- Fortuin and Coniglio-Klein ~KFCK! clusters @12,13#. How- dimensions using better statistics, finding that T* is numeri- cally consistent with T p . To clarify the role of percolation we study also the 3D q-bond FF percolation model. For q52 this model is ob- tained applying the KFCK cluster formalism to the FF Ising model ~see Sec. II!. We simulate it using the ??bond flip?? dynamics @7#. In this way the percolation properties of the model are stressed, and the appearance of nonexponential relaxation functions at T p are more evident. In both these cases we find that the relaxation functions exhibit an exponential long time behavior at high tempera- tures. Below the percolation temperature T p of the KFCK clusters, which is higher than the transition temperature T c of the model, the long time regime of the relaxation functions becomes nonexponential and is well approximated by a stretched exponential. Our results are consistent with the pic- ture suggested by Campbell et al. @9# in the space of configu- Percolation transition and the onset of nonexponential Annalisa Fierro, Giancarlo Franzese, Antonio Dipartimento di Scienze Fisiche, Universita ` Degli Padiglione 19, 80125 and INFM, Unita ` di ~Received 11 March 1998; revised We numerically study the dynamical properties of fully results obtained support the hypothesis that the percolation sponds to the onset of stretched exponential autocorrelation namical behavior may be due to the ??large scale?? effects old. Moreover, these results are consistent with the picture ~1987!# in the space of configurations. @S1063-651X~98 PACS number~s!: 05.50.1q I. INTRODUCTION At low temperature, spin glasses ~SG?s! undergo a transi- tion characterized by the divergence of the nonlinear suscep- tibility. Moreover, the relaxation functions of the system be- come nonexponential at temperatures higher than the transition temperature T sg . This behavior has been observed in canonical metallic and insulating spin glasses, investigated by neutron and hyperfine techniques @1#. In the Ising SG model, studied with spin-flip Monte Carlo dynamics, both in two @2# and three dimensions @3#, nonex- ponential relaxation functions have been observed below some temperature T* higher than T sg . Moreover in the three-dimensional ~3D! system Ogielski @3# observed that the long time regime of the relaxation functions is well approxi- mated by the following function: f~t!5 f 0 t 2x exp@2~t/t! b #. ~1! Fitting the data with this function, he obtained that the onset of nonexponential relaxation is consistent with the Griffiths temperature T G , that coincides with the critical temperature of the ferromagnetic model. PRE 591063-651X/99/59~1!/60~7!/$15.00 relaxation in fully frustrated models de Candia, and Antonio Coniglio di Napoli ??Federico II,?? Mostra d?Oltremare, Napoli, Italy Napoli, Italy received 30 July 1998! frustrated models in two and three dimensions. The transition of the Kasteleyn-Fortuin clusters corre- functions in systems without disorder. This dy- of frustration, present below the percolation thresh- suggested by Campbell et al. @J. Phys. C 20, L47 !07412-1# ever, in frustrated systems with disorder, T p is less than but close to T G ; therefore its eventual effects are hidden by those related to T G . A way to verify if percolation mechanisms can play a role in the dynamical transition of frustrated systems is to con- sider frustrated models without disorder, where the Griffiths phase does not exist. In particular, we have considered fully frustrated ~FF! spin systems @14#, where ferromagnetic and antiferromagnetic interactions are distributed in a regular way on the lattice, in such a way that no unfrustrated region ~no Griffiths phase! exists, but the percolation temperature of KFCK clusters is still defined. In a previous paper @7# we studied the 2D FF Ising model. We found numerically that the model exhibits a nonexponen- tial relaxation below the percolation temperature T p of the KFCK clusters. Moreover the long time regime of these functions is well approximated by a Kohlrausch-Williams- Watts function, also known as the ??stretched exponential?? f~t!5 f 0 exp@2~t/t! b #. ~2! In this paper we analyze, with conventional spin flip, the dynamical behavior of the FF Ising model in three and two 60 ©1999 The American Physical Society rations, and can be interpreted considering that T p corre- sponds to a thermodynamic transition in a generalized frustrated model @8#. In Sec. II we present the ??q-bond frustrated percolation?? model, and in Sec. III we study the percolation properties of this model on a FF cubic lattice. We find that the percolation transition is in the same universality class of the q/2-state ferromagnetic Potts model, confirming the results obtained in the disordered version of the model in 2D @8#. In Sec. IV we study the FF Ising model dynamical properties with conven- tional spin flip, and in Sec. V we present the relaxation func- tions obtained simulating the FF q-bond percolation model for q52, with the ??bond flip?? dynamics. In Sec. VI we show the connection with the Campbell scenario @9#, and in Sec. VII we give conclusions. II. ??q-BOND FRUSTRATED PERCOLATION?? MODEL The FF Ising spin model is defined by the Hamiltonian H52J ( ^ij& ~e ij S i S j 21!, ~3! where e ij are quenched variables which assume the values 61. The ferromagnetic and antiferromagnetic interactions are distributed in a regular way on the lattice ~see Fig. 1!. Using the KFCK cluster formalism for frustrated spin Hamiltonians @15#, it is possible to show that the partition function of the model Hamiltonian in Eq. ~3! is given by Z5 ( C * e mn~C!/k B T q N~C! , ~4! where q52 is the multiplicity of the spins, k B is the Boltz- mann constant, m5k B T ln(e qJ/k B T 21), and n(C) and N(C), respectively, are the number of bonds and the number of clusters in the bond configuration C. The summation ( C * extends over all the bond configurations that do not contain a ??frustrated loop,?? that is, a closed path of bonds which con- tains an odd number of antiferromagnetic interactions. Note that there is only one parameter in the model, namely, the temperature T, ranging from 0 to ?. The parameter m, that FIG. 1. Distribution of interactions for the FF model. Straight lines and wavy lines correspond, respectively, to e ij 51 and 21. PRE 59 PERCOLATION TRANSITION can assume positive or negative values, plays the role of a chemical potential. Varying q, we obtain an entire class of models differing by the ??multiplicity?? of the spins, which we call the q-bond FF percolation model. More precisely, for a general value of q, the model can be obtained from a Hamiltonian @16# H52sJ ( ^ij& @~e ij S i S j 11!d s i s j 22#, ~5! in which every site carries two types of spin, namely, an Ising spin and a Potts spin s i 51,...,s with s5q/2. For q 51 the factor q N(C) disappears from Eq. ~4!, and we obtain a simpler model in which the bonds are randomly distributed under the conditions that the bond configurations do not con- tain a frustrated loop. For q!0 we recover the tree percola- tion, in which all loops are forbidden, be they frustrated or not @17#. When all the interactions are positive ~i.e., e ij 51) the sum in Eq. ~4! contains all bond configurations without any restriction. In this case the partition function coincides with the partition function of the ferromagnetic q-state Potts model, which in the limit q51 gives the random bond per- colation @17#. From renormalization group @18#, mean field @19# and nu- merical results @8,20#, we expect that the model in Eq. ~5! exhibits two critical points: the first at a temperature T p (q), corresponding to the percolation of the bonds on the lattice, in the same universality class of the ferromagnetic q/2-state Potts model; the other at a lower temperature T c (q), in the same universality class as the FF Ising model. III. STATIC PROPERTIES In this section we analyze the percolation properties of the model defined by Eq. ~5! for q52, on a FF cubic lattice. After preliminary runs with spin-flip dynamics on systems with lattice sizes L510 and 20, and with statistics of 5310 3 thermalization steps and 5310 6 acquisition steps, we found that the percolation transition occurs well above the critical temperature T c 51.35 @21#~in the following the tem- peratures will be given in J/k B units!. Then we simulated the model for L530? 80, by Swendsen-Wang cluster dynamics @22#, that turns out to be very efficient for the temperature regime of interest, allowing one to consider only 5310 4 ac- quisition steps. At every step we evaluated the percolation probability P512 ( s sn s , ~6! and the mean cluster size S5 ( s s 2 n s , ~7! where n s is the density of finite clusters of size s. Around the percolation temperature, the averaged quanti- ties P(T) and S(T), for different values of the lattice size L, should obey to the finite size scalings @23# P~T!5L 2b/n F P @L 1/n ~T2T p !#, ~8a! 61AND THE ONSET OF... S~T!5L g/n F S @L 1/n ~T2T p !#, ~8b! CANDIA, where b, g, and n are critical exponents, and F P (x) and F S (x) are universal functions of an adimensional quantity x. Standard scaling analysis results are summarized in Fig. 2. We obtained T p 53.81760.005, n50.8860.06, b/n50.46 60.04, and g/n52.0360.03. The values of the critical ex- ponents coincide, within the errors, with the random bond percolation exponents @23#. As we expect, the q52 bond frustrated percolation model is in the same universality class of the q/251 state ferromagnetic Potts model. IV. RELAXATION FUNCTIONS OF THE FULLY FRUSTRATED ISING SPIN MODEL In this section we present our results in the study of the FF Ising model, defined by the Hamiltonian in Eq. ~5! for q52, simulated by spin-flip dynamics. For each temperature T, 16 different runs were made, varying the random number generator seed, on a FF cubic lattice of size L530. We took about 10 4 steps for thermalization, and about 10 5 steps for acquisition, calculating at each step the energy E(t). The relaxation function of the energy is defined as f~t!5 ^dE~t!dE~0!& ^~dE! 2 & , ~9! FIG. 2. Finite size scaling of ~a! P(T) and ~b! S(T), for the q 52 model, and for lattice sizes L530, 40, 50, 60, 70, and 80. 62 FIERRO, FRANZESE, de where dE(t)5E(t)2^E&. For each value of T, we averaged the 16 functions calculated and evaluated the error as a stan- dard deviation of the mean. Here a unit of time is considered to be one Monte Carlo step, that is L d single spin update trials. In Fig. 3 we show the results for T54.0, 3.5, 3.0, 2.0, and 1.5. We also observe a two step decay for high temperatures. For all the temperatures we fit the long time tail of the re- laxation functions with the empirical formula proposed in Eq. ~1! by Ogielski. The temperature dependence of exponents b(T) is pre- sented in Fig. 4. Note that b(T) increases as function of T from the value b50.5860.03 for T51.5 to the value b 51 for T53.7 and 4.0. We do not observe any regular be- havior in the temperature dependence of exponent x(T). We estimated the errors on parameters as the range where we obtain a good fit of the relaxation function. As we can see in Fig. 4, these results are consistent, within the errors, with the FIG. 3. Relaxation functions f (t) of energy as a function of time t for the d53 FF Ising model, with spin flip dynamics and lattice size L530, for temperatures ~from left to right! T54.0, 3.5, 3.0, 2.0, and 1.5. FIG. 4. Stretching exponents b(T) as a function of T/T , the PRE 59AND CONIGLIO p ratio of temperature over percolation temperature, for the d53FF Ising model, with spin flip dynamics and lattice size L530. scenario in which the onset of the stretched exponential re- laxation coincides with the percolation temperature T p 53.81760.005 ~see Sec. III!. We also simulated the FF Ising model on a square lattice of size L560. We calculated the relaxation functions of the energy. Averages were made over 16 different random gen- erator seeds, and between 10 5 and 10 6 steps for acquisition were taken, after about 10 4 steps for thermalization. In Fig. 5 we show the relaxation functions obtained for T52.5, 2.0, 1.8, 1.5, and 1.0. For all temperatures we fit the long time tail of the relaxation functions with Eq. ~1!. The temperature dependence of exponents b(T) is shown in Fig. 6. Note that b(T) increases as function of T from the value b50.6160.05 for T50.8 to the value b51 for T>2.0. As we can see in Fig. 6, our estimate of the onset of the stretched exponential relaxation is also consistent, within the errors, with the percolation temperature T p 51.701 @7#. Within the errors, the exponent x(T) increases as function of T from the value x50.460.2 for T50.8 to the value x 51.660.4 for T52.5. FIG. 5. Relaxation functions f (t) of energy as a function of time t for the d52 FF Ising model, with spin flip dynamics and lattice size L560, for temperatures ~from left to right! T52.5, 2.0, 1.8, 1.5, and 1.0. FIG. 6. Stretching exponents b(T) as a function of T/T , the PRE 59 PERCOLATION TRANSITION p ratio of temperature over percolation temperature, for the d52FF Ising model, with spin flip dynamics and lattice size L560. V. RELAXATION FUNCTIONS OF THE ??q-BOND FRUSTRATED PERCOLATION?? MODEL In this section we analyze the dynamical behavior of the model defined by Eq. ~4! with q52, simulated by the bond flip dynamics @7#. The dynamics is carried out in the follow- ing way: at each step we choose at random a particular edge on the lattice; we calculate the probability P of changing its state, that is, of creating a bond if the edge is empty, and of destroying the bond if the edge is occupied; and, finally, we change the state of the edge with probability P. For each temperature T, 16 different runs were made, varying the random number generator seed, on a FF cubic lattice of size L530. We took about 10 3 steps for thermali- zation, and between 10 4 and 10 5 steps for acquisition, calcu- lating at each step the density of bonds r(t). The relaxation function of the density of bonds is defined as f~t!5 ^dr~t!dr~0!& ^~dr! 2 & , ~10! where dr(t)5r(t)2^r&. For each value of T, we averaged the 16 functions calculated and evaluated the error as a stan- dard deviation of the mean. We consider a unit of time to consist of G^r& 21 single update trials, where G53L 3 is the number of edges on the lattice. In Fig. 7 we show the results obtained for temperatures T54.0, 3.5, 3.0, and 2.5. For T54.0 and 3.5 we fitted the calculated points with the function in Eq. ~1!. The value of b extracted from the fit is equal to one within the error, and the value of x is zero. Thus for these temperatures the relaxation is purely exponential. For T,3.5 we observe a two step decay, and only the long time regime of the relaxation functions could be fitted by Eq. ~1!. The value of b extracted is less than 1, showing that stretched exponential relaxation has appeared for these FIG. 7. Relaxation functions f (t) of bond density as function of time t, for the q52 FF bond percolation model, on a d53 lattice of size L530, for temperatures ~from left to right! T54.0, 3.5, 3.0, and 2.5. 63AND THE ONSET OF... temperatures. In Fig. 8 the values of b(T) as function of the ratio T/T p are shown, with errors estimation. The exponent CANDIA, x(T) becomes nonzero only for T52.5, for this value of temperature we obtain x51.160.1. As we can see in Fig. 8, our estimate of the onset of stretched exponential relaxation is consistent, within the errors, with the percolation tempera- ture T p 53.81760.005 of the KFCK clusters. VI. CONNECTION WITH THE RANDOM WALK PICTURE In this section we make a connection between our model and the random walk picture of Campbell et al. @9#, which we will briefly illustrate. Consider an hypercube in a D-dimensional space. Each summit is occupied with a prob- ability p. On such a dilute lattice, a random walker is al- lowed to diffuse, like the ??ant?? on a percolating cluster in the de Gennes picture. The mean square displacement after a time t is given by r 2 ~t![ K ( i51 D x i ~t!2x i ~0! 2 L D , ~11! where D is the hypercube dimension, x is a D-dimensional vector of components 0 and 1 that identify the hypercube 2 D summits, and x(t) indicates the ??ant?? position at the time t. Campbell et al. suggested in the Ising SG model that an accessible region in the space of configurations, compact at high temperature, becomes ramified at a temperature T*, and that a complex space of configurations is responsible for the appearing of nonexponential relaxation. They also supposed that this temperature T* is the percolation temperature of the KFCK clusters. The idea is that the diffusive ant mimics quite well the evolution in the space of configurations in the SG model. In the study of the random walk on a randomly occupied hypercube, they found that for p