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arXiv:2411.06819 · Springer proceedings in physics

Monte Carlo Simulation of An-Isotropic Ising Model Using Metropolis and Wolff Algorithm

Basit Iqbal, Kingshuk Sarkar

Published Dec 28, 2024• 1 citation
arXivHTMLPDF
DOI
10.1007/978-981-96-0828-7_2

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We show that addition of Metropolis single spin flips to the Wolff cluster-flipping Monte Carlo procedure leads to a dramatic increase in performance for the spin-1/2 Ising model. We also show that adding Wolff cluster flipping to the Metropolis or heat bath algorithms in systems where just cluster flipping is not immediately obvious (such as the spin-3/2 Ising model) can substantially reduce the statistical errors of the simulations. A further advantage of these methods is that systematic errors introduced by the use of imperfect random-number generation may be largely healed by hybridizing single spin flips with cluster flipping.
The Ising models S=1/2 and S=1 are studied by efficient Monte Carlo schemes on the (3,4,6,4) and the (3,3,3,3,6) Archimedean lattices. The algorithms used, a hybrid Metropolis-Wolff algorithm and a parallel tempering protocol, are briefly described and compared with the simple Metropolis algorithm. Accurate Monte Carlo data are produced at the exact critical temperatures of the Ising model for these lattices. Their finite-size analysis provide, with high accuracy, all critical exponents which, as expected, are the same with the well-known 2D Ising model exact values. A detailed finite-size scaling analysis of our Monte Carlo data for the S=1 model on the same lattices provides very clear evidence that this model obeys, also very well, the 2D Ising model critical exponents. As a result, we find that recent Monte Carlo simulations and attempts to define effective dimensionality for the S=1 model on these lattices are misleading. Accurate estimates are obtained for the critical amplitudes of the logarithmic expansions of the specific heat for both models on the two Archimedean lattices.
We apply numerical simulations to study the criticality of the 3D Ising model with random site quenched dilution. The emphasis is given to the issues not being discussed in detail before. In particular, we attempt a comparison of different Monte Carlo techniques, discussing regions of their applicability and advantages/disadvantages depending on the aim of a particular simulation set. Moreover, besides evaluation of the critical indices we estimate the universal ratio Γ+/Γ−Γ^+/Γ^-Γ+/Γ− for the magnetic susceptibility critical amplitudes. Our estimate Γ+/Γ−=1.67±0.15Γ^+/Γ^- = 1.67 \pm 0.15Γ+/Γ−=1.67±0.15 is in a good agreement with the recent MC analysis of the random-bond Ising model giving further support that both random-site and random-bond dilutions lead to the same universality class.