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 $$\mathcal {O}(\log N)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 nonequilibrium phenomena in disordered spin systems. Graphical abstract