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Journal of Statistical Physics

The stochastic models method applied to the critical behavior of Ising lattices

Hagai Meirovitch, Z. Alexandrowicz

Published Feb 1, 1977• 17 citations
Publisher
DOI
10.1007/bf01418747

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Journal of Experimental and Theoretical Physics•Dec 1, 2004•35 citations
A markoffian stochastic model of interacting Ising spins is discussed near its critical point. The long time behavior of relaxation of its magnetization and energy is investigated near T c by the use of high-temperature series expansion and the ratio method. The numerical estimate of the critical exponent Δ M M of slowing down of magnetization is made, the results of which show Δ M M ≃2 for the simple square and triangular lattices, and Δ M M ≃1.4 for the simple cubic lattice. The exponent Δ E E of slowing down of energy is also estimated, which gives the result Δ E E ≃2 for the simple square and triangular lattices. The method to derive high-temperature expansion of an n -spin correlation function <σ 1 σ 2 …σ n > for an Ising model of spin 1/2 by the use of an equation of detailed balance is presented and applied to these estimates.
The method of 'symmetrized' stochastic truncation is applied to the (2+1)-dimensional Ising model, to calculate the ground-state energy, mass gap, magnetization and susceptibility on lattices of sizes up to 6*6 sites. The method is shown to be a powerful and accurate Monte Carlo technique, provided the lattice size is not too large, and the model is suitably 'convergent'. Finite-size scaling estimates of the critical indices are nu =0.627(2), alpha =0.12(2), beta =0.324(3), and delta =1.23(1). An estimate of the combination gamma / nu +2 beta / nu =3.000(1) is obtained, in outstanding agreement with the expected hyperscaling relation.
A Monte Carlo method is applied to simulate the static critical behavior of a cubic-lattice 3D Ising model for systems with quenched disorder. Numerical results are presented for the spin concentrations of p = 1.0, 0.95, 0.9, 0.8, 0.6 on L × L × L lattices with L = 20–60 under periodic boundary conditions. The critical temperature is determined by the Binder cumulant method. A finite-size scaling technique is used to calculate the static critical exponents α, β, γ, and ν (for specific heat, susceptibility, magnetization, and correlation length, respectively) in the range of p under study. Universality classes of critical behavior are discussed for three-dimensional diluted systems.