Invaded Cluster Dynamics for Frustrated Models

The Invaded Cluster (IC) dynamics introduced by Machta et al. [Phys. Rev. Lett. 75 2792 (1995)] is extended to the fully frustrated Ising model on a square lattice. The properties of the dynamics which exhibits numerical evidence of self-organized criticality are studied. The ﬂuctuations in the IC dynamics are shown to be intrinsic of the algorithm and the ﬂuctuation-dissipation theorem is no more valid. The relaxation time is found very short and does not present critical size dependence.


I. INTRODUCTION
Recently a new type of cluster Monte Carlo (MC) dynamics, the Invaded Cluster (IC) dynamics, based on invaded percolation has been introduced by Machta et al. for the ferromagnetic Ising model. [1] The IC dynamics is based on the Kastelein-Fortuin and Coniglio-Klein (KF-CK) cluster formulation of the Ising model [2] and has been shown to be even more efficient than the Swendsen-Wang (SW) dynamics [3] in equilibrating the system at the critical temperature. The IC dynamics has the advantage that the value of the critical temperature does not need to be known a priori. In fact the dynamics itself drives the system to the critical region as in self-organized critical (SOC) systems.
The aim of this paper is to extend the IC dynamics to frustrated systems where KF-CK clusters percolate at a temperature T p , higher than the critical temperature T c . In particular we will consider the fully frustrated (FF) Ising model on a square lattice where it has been shown numerically [4] that k B T p /J ≃ 1.69 (k B is the Boltzmann constant and J is the strength of the interaction) and T c = 0. We will use two definitions of clusters. The first definition is based on the KF-CK clusters. In this case the IC dynamics leads to a SOC percolating state at temperature T p corresponding to the percolation of KF-CK clusters.
The second definition is more general [5] and reduces to the cluster of Kandel, Ben-Av and Domany (KBD) [6] in the FF Ising model. In this case the IC dynamics leads to a SOC state at the thermodynamical critical temperature T c = 0.
In sec. II we review the definition and the results of IC dynamics on ferromagnetic Ising model. Then we extend the IC dynamics to the FF Ising model using the KF-CK cluster in sec. III and the KBD clusters in sec. IV where we study also the equilibrium relaxation of the proposed dynamics.

II. IC DYNAMICS FOR FERROMAGNETIC ISING MODEL
The rules which define the IC dynamics for the ferromagnetic Ising model are very simple. Let us start from a given spin configuration. As first step all the pairs of nearest neighbour (nn) spins are ordered randomly. Then, following the random order, a bond is activated between the nn spin pairs only if the two spins satisfy the interaction. The set of activated bonds partitions the lattice into clusters which are classified every time a new bond is activated. When one of the clusters spans the system the procedure is stopped and the spins belonging to each cluster are reversed all together with probability 1/2. The spin configuration obtained can be used as starting point for the next application of the dynamical rule. Successively, the algorithm has been generalized to different stopping rules [7,8] and to different model (the Potts model [7] and the Widom-Rowlinson fluid [9]).
The rational which is behind this rule is the well known mapping between the ferromagnetic Ising model and the correlated bond percolation. [2] In this framework bonds are introduced in the ferromagnetic Ising model between parallel nn pairs of spins with probability p = 1 − exp(−2βJ) where β = 1/(k B T ). The clusters (KF-CK clusters), defined as maximal set of connected bonds, represent sets of correlated spins and percolate exactly at the critical temperature, T p = T c . The well known SW cluster dynamics [3] uses these clusters to sample very efficiently the phase space at any temperature. In this dynamics at each MC step the KF-CK clusters are constructed and the spins belonging to each cluster are reversed all together with probability 1/2. This produces a new spin configuration on which a new cluster configuration can be build up. The IC dynamics is very similar to SW dynamics and differs only on the way clusters are constructed. Since the dynamic rule introduced by Machta et al. builds up clusters which, by definition, percolate through the system, it is expected that the average properties of the clusters are the same as the KF-CK clusters at the critical temperature. In fact, in Ref. [1] it has been shown that the ratio of activated bond to satisfied interactions, in the large L limit (L is the linear lattice size), is very close to the critical probability p c = 1 − exp(−2J/(k B T c )) with which KF-CK clusters 3 are constructed and it has a vanishing standard deviation. Furthermore, the estimated mean energy also tends in the large L limit to a value very close to the critical equilibrium value E(T c ) with the finite size behaviour expected in the ferromagnetic Ising model at the critical point. On the other hand the energy fluctuation C = E 2 − E 2 is not related to the specific heath. In fact, it has been found that C diverges linearly with L and not logarithmically as the specific heat does. The latter result points to the fact that the IC dynamics does not sample the canonical ensemble in finite volume and, therefore, energy fluctuations and specific heat are not any longer related by the fluctuation-dissipation theorem. Since the fluctuation C 1/2 /L 2 → 0 for L → ∞, it is generally assumed that in the thermodynamic limit the IC dynamics is equivalent to the canonical ensemble even if a rigorous proof is still lacking.

III. IC DYNAMICS FOR FF ISING MODEL WITH KF-CK CLUSTERS
We now extend the IC dynamics to frustrated systems where the KF-CK clusters percolate at a temperature T p which is higher than the critical one T c . In particular we will consider the FF Ising model on square lattice where [10] T p ≃ 1.69 [4] and T c = 0. The FF Ising model is defined by the Hamiltonian where S i takes the values ±1 and ǫ i,j assumes the value −1 on even columns and is +1 otherwise.
In the simplest extension of the IC dynamics we introduce bonds at random between spins satisfying the interaction (i.e. ǫ ij S i S j = 1) until a spanning cluster is found. As in the ferromagnetic case, at this point the spins belonging to each cluster are reversed all together with probability 1/2. The procedure is iterated until equilibrium is reached. We have done simulations performing measurements over 5 · 10 3 MC sweeps after discarding the first 10 3 for equilibration for systems with sizes L ranging from 16 to 300, and over 3.75 · 10 3 MC sweeps after discarding the first 750 for L = 350.
In Fig.1.a we show as functions of the system size L the ratio of activated bonds N b to The results obtained show that the dynamics has not driven the system into the critical thermodynamical state at T = 0, but to the percolation critical state at T = T p . In order to study the convergence as function of L we have extracted for the IC dynamics at each L an effective temperature T IC (L) (see Tab.1) by using f = 1 − exp(−2/T IC ), then we have compared the IC energy density to the analytical energy density ǫ(T IC ) at T IC ( Fig.1.b). This analysis clearly show that the L → ∞ limit is reached in the IC dynamics very slowly. In particular the energy is systematically larger than that obtained by SW dynamics. Furthermore, since the dynamics build up, by definition, percolating clusters the mean cluster size S/L 2 of IC clusters diverges in the following way S/L 2 = AL γp/νp where γ p /ν p = 1.78 ± 0.11 is a good approximation of the exponent γ p /ν p = 1.792 of Random Bond Percolation [11] and A = 0.022 ± 0.013 (Fig.2). This result is in excellent agreement with the behaviour found for KF-CK clusters for both the critical exponent and the prefactor A. [4] This critical behaviour drives also energy and magnetization fluctuations in a critical regime, i.e., E 2 − E 2 ∼ L 2.9 and M 2 − M 2 ∼ L 3.8 (Fig.3). This result is in strong contrast with the behaviour of specific heat and magnetic susceptibility at T = T p which behave as L 2 . In fact T p is not a critical thermodynamic temperature and both specific heat and magnetic susceptibility are finite at T = T p . This again stresses the fact that the IC dynamics does not sample the canonical ensemble in finite volume and shows that the spin configurations visited by the IC dynamics with a finite probability correspond to a much larger range of energies and magnetization than in any ordinary dynamics.
It is interesting to note that the straightforward application of the IC dynamics to more 5 complex systems as the Ising spin glass (SG) model could allow to sample very efficiently the equilibrium spin configurations of the system at T = T p where the KF-CK clusters percolate (in 2d SG T p ≃ 1.8 [4], in 3d SG T p ≃ 3.95 [12]) with a dynamics which exhibits SOC.
However since T p is usually much larger than the spin glass critical temperature T SG (in 2d T SG = 0, in 3d T SG ≃ 1.11 [13]), any other dynamics does not suffer for slowing down.

IV. IC DYNAMICS FOR FF ISING MODEL WITH KBD CLUSTERS
We ask now how to build up an IC dynamics able to drive a system to the frustrated thermodynamical critical temperature. From the analysis and the argument given for the ferromagnetic case we understand that the cluster definition needs to be modified in such a way that i) the frustrated system can still be mapped on the corresponding correlated percolation model and ii) the clusters percolate at the thermodynamical critical temperature.
[14] The first conditions is always satisfied by the KF-CK clusters [15] but not the second one.
A general procedure to construct such clusters in a systematic way was suggested in Ref. [5].
In particular for the FF model this procedure leads to the cluster algorithm ). Thus we expect to recover the right behavior only for sizes fairly large.
We have studied the energy and magnetization fluctuations also in this case (Fig.7) and found that E 2 − E 2 ∼ L 2.1 and M 2 − M 2 ∼ L 3.1 . These exponents, as in the previous cases do not agree with those expected for specific heat and magnetic susceptibility. As in the ferromagnetic case we obtain that energy and magnetization fluctuations are larger in 7 the IC dynamics than in the canonical ensemble.
We have also studied the equilibrium relaxation of the magnetization of the proposed IC dynamics where the time t is measured in MC steps. As shown in Fig.8 φ(t) vanishes in few MC steps.
The integrated autocorrelation time τ defined as is reported in Tab The extension with KBD clusters, whose percolation point coincide in the large L limit with the critical point of the FF system, has properties very similar to those obtained in the ferromagnetic model: it drives the system to the critical region without a previous knowledge of the critical temperature and gives a reasonably good estimation of the average energy at the critical point. The estimated integrated autocorrelation time is smaller than that obtained in the KBD dynamics. We have also stressed that the extension has been 8 possible since there exists a percolation model into which the FF square Ising model can be exactly mapped. The extension to other frustrated systems, such as spin glass, in principle can be done using the systematic procedure suggested in Ref. [5]. The computation have been done on DECstation 3000/500 with Alpha processor.

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[14] The two conditions i) and ii) have been discussed in detail in Ref. [5]. In general these two conditions are not sufficient to be sure that percolation clusters represent spincorrelated regions, i.e. to be sure that the set of percolation critical exponents coincides with the set of thermodynamical critical exponents (ν p = ν, γ p = γ and so on). However in many cases it has been shown [5] that these two conditions allow a dramatic decrease of the critical slowing down.
[16] In Ref. [5] it has been also numerically studied a FF model (called asymmetric FF model) where the antiferromagnetic interaction strength is X times the ferromagnetic interaction strength, with 0 ≤ X < 1, showing that it is always possible to find a cluster definition (different from KF-CK one and KBD one) such that T p = T c , ν p = ν, γ p = γ and so on, within the numerical precision. Let's note that in this paper we are considering the symmetric FF model (X = 1) where only T p = T c and ν p = ν still holds within the numerical precision (see text).