Self Organized Critical Dynamic on the Sierpinski Carpet

The mapping between the Ising model and the BCP model unveils new relationships among the emergent critical exponents. To elucidate this, it is pertinent to remember that in our Ising-BCP model, each bond is randomly occupied with a probability which depends on the temperature (see equation (1)). At the critical probability $p_{c}$, corresponding to the Ising critical temperature $T_{c}$, the percolation threshold is reached. This is the point where a vast or infinite cluster first spans the lattice. Below $p_{c}$, clusters tend to be small and disconnected. Above it, a dominant large cluster overtakes the system, resulting in a more uniform structure. At $p_{c}$, however, the cluster structure is complex and highly irregular. Indeed, it displays a complex, self-similar structure over several scales. The mass of the infinite cluster (number of occupied sites) scales with its linear size as

where $d_{f}$ is known as its fractal dimension.

The fractal dimension is related to some other critical exponents. To introduce these relations, using the percolation formalism, one can define the average density of points connected to one cluster as

where $d$ is the dimension of the lattice. Then, the mass of the largest cluster, $M_{*}$, will be

Near the critical point, the average density of points will follow a power law given by

In this scenario, $P$ corresponds to the order parameter, equivalent to the magnetization in the Ising formalism. Consequently, the exponent $\beta$ remains identical in both formulations. If $L$ is of the same order than the correlation length, $\xi$, it follows that

Finally, using the fact that

one gets

Equation (8) establishes a connection between the fractal dimension $d_{f}$ of the largest cluster and the Ising critical exponents $\beta$ and $\nu$ within a finite lattice of linear dimension $L$ munoz1999avalanche ; suzuki1983phase ; coniglio1983droplet .

On the other hand, the critical percolation threshold can also be characterized by taking into account the distribution of clusters sizes. Near the critical point, the clusters size extend up to a maximum cluster size denoted as $s_{max}$. The distribution of these cluster sizes is modulated by a decaying function expressed as

where $n_{s}$ is the number of clusters with size $s$. Moreover,

where $\tau$ and $\sigma$ are critical exponents. According to stauffer2018introduction ; munoz1999avalanche , the following relations are satisfied

and

Therefore,

The Sierpinski carpet (see Figure 2) is a classic example of a fractal and is constructed using an iterative process of eliminating repetitive patterns. For Sierpinski fractals, one can define the generating cell as $SP_{s}(l^{d},N_{\text{occ}},k)$, where $d$ is the dimension of the initial hypercube (in this context, a square), $l^{d}$ is the number of hypercubes the initial one is divided into, $N_{\text{occ}}$ is the number of hypercubes preserved post-elimination and $k$ is the number of iterations of this iterative process. The true fractal is obtained when $k\to\infty$, that is, in the thermodynamic limit monceau2004direct . The Haussdorf dimension is given by $D_{f}=\log(N_{\text{occ}})/\log(l)$. In the case of the SC, $D_{f}=\log(8)/\log(3)$.

To isolate an arbitrarily large bounded set of points connected to an arbitrary point in the Sierpinski carpet, the number of significant bonds that must be cut grows as a power of the size of this bounded set. Consequently, the SC exhibits infinite ramification. Previous studies have shown that the Ising model on fractals with infinite ramification displays a second-order phase transition gefen1980critical . Moreover, for the case of the SC, this has been confirmed through numerical methods.

Over the last few years, several studies have been conducted to investigate the critical properties of the Ising model on the SC using Monte Carlo simulations. Most of these studies have employed the finite-size scaling (FSS) formalism to calculate a value for the critical temperature and critical exponents PhysRevB.58.14387 ; monceau1998magnetic . However, there is ongoing debate about the effectiveness of this method, with some authors asserting that each stage of segmentation signifies an autonomous thermodynamic system. This challenges the notion of treating it as a scaled iteration of the previous one, which is the fundamental idea behind FSS. In response to the earlier discussion, alternative methods have been developed for calculating critical parameters. Among these are the slope method pruessner2001monte and the short-time dynamics (STD) method bab2005critical . One approach used to assess the accuracy of simulations involves the Ising model hyperscaling law. Once the critical exponents are determined, it becomes possible to estimate the effective dimension of the system using the formula

where $\nu$ is the critical exponent associated with the correlation length, $\beta$ is the critical exponent associated with the magnetization and $\gamma$ is the critical exponent associated with the magnetic susceptibility. Different simulations have provided evidence that this law holds in fractal dimensions. Bab et al. bab2005critical and Monceau et al. monceau1998magnetic provided a summary of the most significant results according to different methods. Table 1 compiles the results most pertinent to our study.

The algorithm converges towards temperatures close to those reported in Table 1, in both cases, with fluctuations that decrease as the lattice size increases. Figure 3.A illustrates the performance of the algorithm in the SC when initiated in two distinct regimes: $T<T_{c}$ and $T>T_{c}$. An effective approach to characterize the algorithm’s behavior near the critical point involves examining the number of clusters normalized by the total number of spins in the lattice as a function of the distance to the critical temperature. Figure 3.A depicts the distribution of the number of clusters for $T<T_{c}$ and Figure 3.B for $T>T_{c}$. In the first regime, there is an exponential decay as the algorithm moves away from the critical temperature. In contrast, in the second regime, there is a linear increase as the algorithm moves away from the critical temperature. This difference in cluster behavior is intricately linked to the magnetic state of the model and the nature of correlations among clusters. In the ferromagnetic phase, observed for $T<T_{c}$, the model exhibits pronounced magnetic correlations attributed to the long-range order among magnetic moments. This results in the emergence of a single infinite cluster that increasingly dominates the space as the temperature continues to decrease. Conversely, in the paramagnetic phase, which occurs for $T>T_{c}$, the model is characterized by thermal noise that overpowers magnetic correlations. As the temperature increases, thermal fluctuations grow more influential, further promoting the fragmentation of clusters. This behavior signifies a transition to short-range order among magnetic moments. In Domb_1977 ; Domb_1975 , Domb et al. identified a discontinuity in the average cluster size at the critical point, demonstrating results consistent with these findings and broadening the discussion of this behavior.

Naturally, it is expected that the fractal structure had influenced the distribution of the clusters. This is because, unlike a compact model like a square, the formation of clusters is conditioned by the availability of spins, as the number of neighbors for each spin is variable and depends on its location. Figures 4 and 5 show the behavior of clusters across three different types of lattices: the SC lattice, the stochastic version of the SC lattice and a square lattice. Considering that all three lattices have the same number of spins ($N=4096$), it is expected that as the number of clusters increases, the average cluster size decreases. Then, to quantify a difference in the distribution of clusters, the following quantity is introduced

where $S_{mc}$ is the mean size of the clusters and $N_{c}$ the number of clusters. It is noteworthy that these two quantities are interrelated. Specifically, $S_{mc}\cdot N_{c}=N$, where $N$ represents the total number of spins. Subsequently,

As $N$ is constant for the three cases, a large ratio indicates the presence of larger and more interconnected clusters. Meanwhile, a small ratio indicates that the clusters are smaller and more fragmented. In terms of assessing the algorithm’s performance, a large $\eta$ ratio correlates with a higher efficiency in steering the system towards its critical state with minimal computational overhead. This efficiency stems from the reduced necessity to inspect and manage a myriad of smaller clusters. Large interconnected clusters simplify the system’s dynamics, making it easier and less computationally intensive to monitor and adjust the system’s parameters to reach and maintain criticality. Conversely, a small $\eta$ ratio, indicative of numerous small and fragmented clusters, demands greater computational resources for constant inspection and adjustment, thereby slowing down the process of achieving a critical state. This highlights the dual benefit of focusing on large cluster formations: they not only signal proximity to criticality but also enhance computational efficiency by simplifying the system’s management.

Figure 4 illustrates the behavior of the quantity $\eta$ on a logarithmic scale, plotted as a function of the distance to the critical temperature, for temperatures $T$ less than the critical temperature $T_{c}$, in the three different lattice geometries. It’s crucial to underscore that for each lattice geometry, the algorithm identifies a unique critical temperature $T_{c}$ from which the distance to the critical point is measured. Each behavior is then quantitatively described by introducing the exponent $H$ as follows

For the SC, we find $H_{SC}=9.12\pm 0.07$. In contrast, the stochastic SC and the square lattice exhibit $H_{R}=6.27\pm 0.04$ and $H_{S}=5.68\pm 0.05$, respectively. These variations in $H$ values elucidate the relative interconnectivity of clusters and the facility with which each system gravitates towards a critical state. More specifically, an elevated $H$ value indicates a more rapid increase in the quantity $\eta$ as the temperature approaches the critical threshold $T_{c}$ from below, signifying more pronounced correlations or larger clusters as the phase transition is neared. This suggests a heightened tendency for the infinite cluster to fragment into a densely interconnected structure as it nears the percolation threshold, where it ultimately adopts a fractal dimension.

Meanwhile, for temperatures $T$ greater than the critical temperature $T_{c}$, the number of clusters increases linearly with temperature. According to equation (16), the quantity $\eta$ inversely depends on the square of the number of clusters. Then, in this regime, the relationship governing $\eta$ can be expressed as

Here, $h^{\prime}$ represents the rate at which the number of clusters decreases as the temperature increases and $N_{c_{0}}$ accounts for the baseline number of clusters present at the critical temperature.

Our results show that the normalized values for $h^{\prime}$ are $h^{\prime}_{SC,N}=-0.352\pm 0.003$, $h^{\prime}_{R,N}=-0.311\pm 0.009$, and $h^{\prime}_{S,N}=-0.330\pm 0.005$, as illustrated in Figure 5. The steeper slope for the SC indicates that, as the temperature decreases, large clusters form more readily compared to the stochastic SC and the square lattices. However, in this regime, the differences are less significant than in the low-temperature regime because, at higher temperatures, clusters are smaller and less influenced by the lattice geometry.

For $T<T_{c}$, it is evident that the stochastic version of the SC exhibits slower convergence compared to the traditional SC but faster than that of a square lattice. This indicates that the stochasticity slightly hinders the convergence, but the system still gains advantages from the underlying fractal structure. Meanwhile, for $T>T_{c}$, the stochastic version of the SC shows slower convergence than both the traditional SC and the square lattice, primarily due to the randomness in its structure. At higher temperatures, where spins form much smaller clusters, this randomness hinders the convergence compared to the more compact square lattice. These findings suggest that the fast convergence observed in the classic version of the SC is primarily due to its hierarchical nature, which is diminished when randomness is introduced.

Finite-size scaling (FSS) analysis was conducted to investigate the behavior of the system as a function of lattice size for both the SC lattice and the square lattice. For the $H$ coefficients, the extrapolated values in the thermodynamic limit are presented in Figure 6, yielding $H_{SC}^{\infty}=8.71$ and $H_{S}^{\infty}=5.98$. The results reveal distinct behaviors for each lattice type, which can be attributed to the different relationships between lattice size and the number of spins in each case. In the SC, as $L$ increases, the number of spins $N$ grows more slowly compared to the square lattice. Then, the negative finite-size correction arises because the compactness of the clusters becomes slightly less pronounced as the system size increases. In contrast, in the square lattice, clusters become relatively more compact as the system grows, leading to a positive correction. For the coefficient $h^{\prime}$, its variation across different lattice sizes was minimal, converging quickly to $h^{\prime\infty}_{SC,N}=-0.35$ and $h^{\prime\infty}_{S,N}=-0.33$.

To enhance our understanding of the algorithm’s performance on the SC, we numerically calculated the fractal dimension of the incipient infinite cluster $d_{f}$, describing its spatial scaling behavior; and the critical exponent $\tau$, detailing the size distribution of clusters at criticality.

Figure 7.A presents an estimation of the fractal dimension of the incipient infinite cluster $d_{f}$ for the case of the SC. After the algorithm converged to the critical temperature, the mass of the infinite cluster, $M_{*}(L)$, was calculated. Following equation (2), this process was repeated for various lattice sizes $L$. It is important to note that the increase in linear sizes actually corresponds to the increase in the iterations of the fractal construction ($k$), meaning that only lattices scaling as powers of 3 can be constructed. The graph clearly illustrates, through a logarithmic scale, a linear trend whose slope aligns with the sought-after value. Figure 7.B, on the other hand, presents the cluster size distribution for the Sierpinski carpet with $k=5$. To derive these values, at the critical point predicted by the algorithm, the frequency $n_{s}$ of each cluster size $s$ was computed. As specified by equation (9), the slope value on a logarithmic scale determine the critical exponent $\tau$. The regression analysis was concentrated on the region where the decay is clearest.

Considering the critical exponents $\beta$ and $\nu$ specifically for free boundary conditions –since the concept of an infinite cluster is only meaningful in this context– (see Table 1), and using equation (8), we obtain $D_{\text{eff}}=1.86\pm 0.03$. Conversely, using equation (13), we obtain $D_{\text{eff}}=1.97\pm 0.09$. Both values are consistent with the fractal dimension of the Sierpinski carpet $D_{f}\approx 1.892$. This implies that the hyperscaling relations also hold for non-integer dimensions and that the mapping between the Ising and BCP model, upon which our algorithm is based, is effective in this fractal lattice. Table 2 shows the fractal dimension of the percolating clusters and the $\tau$ exponent for different dimensionalities.

In addition, as detailed in Section II, we initially described an avalanche as the flip of a single spin. However, since spins are interconnected according to Coniglio and Klein bonds, a more precise definition of an avalanche would be the simultaneous flip of all spins within a connected cluster. This process, where the entire cluster flips together based on the Metropolis rule, underpins the Swendsen-Wang algorithm, which is adept at simulating Ising dynamics and is employed in our simulations. Therefore, under this definition, the distribution of cluster sizes directly reflects the avalanche size distribution, which is depicted in Figure 7.B. It is clear that the distribution of avalanches in the system follows a power-law behavior, confirming that there is no characteristic scale for the avalanches, a hallmark of SOC. Furthermore, the fractal dimension of the avalanches can be inferred from the magnetic susceptibility exponent $\gamma$. This is because the number of spins influenced by the flip of a spin or a cluster scales as $L^{\gamma/\nu}$ in a system of size $L$. The values shown in Table 1 thereby provide additional insight into the characterization of SOC behavior.