Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rejection-free Glauber Monte Carlo for the 2D Random Field Ising Model via Hierarchical Probabilistic Counters

Luca Cattaneo, Federico Ettori, Giovanni Cerri, Paolo Biscari, Ezio Puppin

Abstract

We present an efficient Monte Carlo algorithm for the simulation of the two-dimensional Random Field Ising Model (RFIM). The method combines the event-driven, rejection-free character of the Bortz Kalos-Lebowitz (BKL) algorithm with Glauber transition probabilities, introducing hierarchical probabilistic counters to perform spin selection in O(log N) operations. This enables efficient sampling of the system's dynamics, especially in the low-temperature and low-disorder regime, where traditional Metropolis updates suffer from critical slowing down. Furthermore, this approach allows a proper dynamical simulation of the Ising system's behavior even in the presence of a Random Field (RF), unlike the BKL method. RFIM simulations with Gaussian field distributions reproduce the expected reduction of the pseudo-critical temperature with increasing disorder. Benchmarking shows speedups exceeding two orders of magnitude compared to the Metropolis algorithm in the low-temperature regime. The proposed method provides an efficient and dynamically faithful tool for studying both equilibrium and non-equilibrium phenomena in disordered spin systems.

Rejection-free Glauber Monte Carlo for the 2D Random Field Ising Model via Hierarchical Probabilistic Counters

Abstract

We present an efficient Monte Carlo algorithm for the simulation of the two-dimensional Random Field Ising Model (RFIM). The method combines the event-driven, rejection-free character of the Bortz Kalos-Lebowitz (BKL) algorithm with Glauber transition probabilities, introducing hierarchical probabilistic counters to perform spin selection in O(log N) operations. This enables efficient sampling of the system's dynamics, especially in the low-temperature and low-disorder regime, where traditional Metropolis updates suffer from critical slowing down. Furthermore, this approach allows a proper dynamical simulation of the Ising system's behavior even in the presence of a Random Field (RF), unlike the BKL method. RFIM simulations with Gaussian field distributions reproduce the expected reduction of the pseudo-critical temperature with increasing disorder. Benchmarking shows speedups exceeding two orders of magnitude compared to the Metropolis algorithm in the low-temperature regime. The proposed method provides an efficient and dynamically faithful tool for studying both equilibrium and non-equilibrium phenomena in disordered spin systems.
Paper Structure (7 sections, 7 equations, 11 figures)

This paper contains 7 sections, 7 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The BKL or N-Fold Way Algorithm
  3. The algorithm
  4. Validation and Performance Comparison with Metropolis
  5. Validation
  6. Performance comparison
  7. Conclusions

Figures (11)

  • Figure 1: Schematic representation of how the hierarchical cumulative counter vectors are built and used in the present algorithm. Lowercase letters indicate the transition probability of the spins $p_s$, while uppercase letters represent the cumulative counters $P_i$.
  • Figure 2: Inset: Average magnetization per site $<|M|>$ versus temperature $T$ for different lattice sizes $L$ for the 2D Ising Model with no RF applied. Main figure: Susceptibility $\chi$ versus temperature $T$ for different lattice sizes $L$ for the 2D Ising Model with no RF applied. Circles in the figure represent the average of the simulation results. A solid line connects the simulation values for visualization purposes, while the areas around it show the standard error of the mean.
  • Figure 3: Inset: Average magnetization per site $<|M|>$ versus temperature $T$ for different lattice sizes $L$ for the 2D RFIM with $\sigma = 1.0$. Main figure: Susceptibility $\chi$ versus temperature $T$ for different lattice sizes $L$ for the 2D RFIM with $H = 0$ and $\sigma^2 = 1.0$. Circles in the figure represent the average of the simulation results. A solid line connects the simulation values for visualization purposes, while the areas around it show the standard error of the mean.
  • Figure 4: Speedup Factor $F$ in the simulation of the magnetization reversal of a RFIM with lattice size $L = 100$ and an applied Gaussian RF with $H = 1.5$. Different colors show the performance when different variance values $\sigma^2$ of the applied field are used. Circles in the figure show the average of the simulation results, while a solid line connects the simulation values for visualization purposes.
  • Figure 5: Speedup Factor $F$ in the simulation of the magnetization reversal of a RFIM with lattice size $L = 60$ and an applied Gaussian RF with $H = 1.5$. Different colors show the performance when different variance values $\sigma^2$ of the applied field are used. Circles in the figure show the average of the simulation results, while a solid line connects the simulation values for visualization purposes.
  • ...and 6 more figures