Exploring the role of connectivity in disordered system

Abstract

We study a minimal model of disordered systems, the random field Ising model (RFIM) on a generalized Petersen Graph, GP(N,k). This graph has a connected inner and outer loop, where both the loops consist of N nodes constituting a total of 2N nodes. The parameter k satisfies the condition 1<=k<=N/2, such that any site i in the inner loop has i-k and i+k as its two nearest neighbours, apart from its connection to a node on the outer loop. Thus, each node in GP(N,k) has coordination number z=3, and by varying k different connections between the nodes in the inner loop can be obtained. The objective is to study whether different connectivity between nodes in these graphs affects the system's response to an external field when the coordination number is fixed. This is of interest because critical behaviour is absent for z<=3 on a random graph which has been solved exactly as well as on the honeycomb lattice in the context of RFIM. Using single-spin-flip Glauber dynamics at zero temperature, we compare the system's response with the known case of a z=3 random graph and the generalized Petersen graph for various connectivity k, albeit for the same z. Our study finds the absence of critical behaviour on GP(N,k) highlighting the importance of coordination number over varying connectivity between the nodes. Additionally, we explore the case of directed GP(N,k) and compare it with the undirected GP(N,k) results.

Figures (8)

• Figure 1: Schematic representation of $GP(N,k)$: (a) $GP(10,1)$ and (b) $GP(10,2)$. $k$ determines how nodes in the inner loop are connected in $GP(N,k)$.
• Figure 2: Hysteresis loops $m(h)$ vs $h$ on $GP(N=49999,k=1)$ for different disorder strength $\sigma$. Solid line with stars, circles, squares and triangles correspond to $\sigma=0.5,1, 1.5$ and 2, respectively. The $m(h)$ curves varies smoothly for all the values of $\sigma$ and the hysteresis loops become narrower with increasing $\sigma$.
• Figure 3: Plot of $m(h)$ vs $h$ on random graph with $z=3$ and $GP(N,k)$ for $k = 1, 2, 3, 4, 5, 10, 1000$ and $10000$. The main plot and inset show results for $\sigma=1.2$ and $\sigma=2.2$ respectively. The $m(h)$ curves of $GP(N,k)$ for different values of $k$ spread out for small $\sigma$ values, while $m(h)$ curves gradually approaches the $m(h)$ curve of random graph and collapses onto each other for larger $\sigma$ values. The simulation data is obtained on $N=10^6$ for random graph and $N=499999$ for $GP(N,k)$.
• Figure 4: Comparison of $m(h)$ vs $h$ curves obtained from simulation on $GP(N=499999,k=10)$ and analytical solution for $\sigma=1.8,1.9$ and 2.2. The solid lines in the plot correspond to the theoretical predictions and the markers correspond to the simulation results.
• Figure 5: Log-log plot of the integrated avalanche size distribution for $k = 1, 2, 3, 4, 5, 10, 1000$ and $10000$. The data is obtained on $GP(N=499999,k)$ for $\sigma=1.2$ averaged over 100 independent configurations. The avalanche size distribution for different values of $k$ deviates from power law.