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arXiv:cond-mat/0404231 · Brazilian Journal of Physics

Probability distribution of the order parameter

P. H. L. Martins, J. A. Plascak

Published Jun 1, 2004• 16 citations
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Abstract

The probability distribution of the order parameter is analyzed in order to obtain the criticality of magnetic systems. Monte Carlo simulations have been employed by using single spin flip Metropolis algorithm aided by finite-size scaling and histogram reweighting techniques. A method is proposed to obtain this probability distribution even when the transition temperature of the model is unknown. A test is performed on the two-dimensional spin-1/2 and spin-1 Ising model and the results show that the present procedure can be quite efficient and accurate to describe the criticality of the system.

DOI
10.1590/s0103-97332004000300021

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The probability distribution of the order parameter is analyzed in order to obtain the criticality of magnetic systems. Monte Carlo simulations have been employed by using single spin flip Metropolis algorithm aided by finite-size scaling and histogram reweighting techniques. A method is proposed to obtain this probability distribution even when the transition temperature of the model is unknown. A test is performed on the two-dimensional spin-1/2 and spin-1 Ising model and the results show that the present procedure can be quite efficient and accurate to describe the criticality of the system.
The two-dimensional spin-1 Baxter-Wu model is studied by using Monte Carlo simulations. The standard single-spin-flip Metropolis algorithm is used to generate the configurations from which the order parameter, specific heat and magnetic susceptibility are measured. The finite-size scaling procedure is employed in order to get the critical behavior. Extensive simulations show that the critical exponents are different from those of the spin-1/2 model suggesting that the spin-1 model is in a different universality class.
We apply extensive Monte Carlo simulations to study the probability distribution P(m) of the order parameter m for the simple cubic Ising model with periodic boundary condition at the transition point. Sampling is performed with the Wolff cluster flipping algorithm, and histogram reweighting together with finite-size scaling analyses are then used to extract a precise functional form for the probability distribution of the magnetization, P(m), in the thermodynamic limit. This form should serve as a benchmark for other models in the three-dimensional Ising universality class.
Recent developments in the use of histograms (measured discrete probability distributions) now make it possible to determine critical properties with high accuracy using Monte Carlo (MC) simulations combined with finite-size scaling analyses. The advantages and limitations of the use of histograms for extracting the maximum information from MC simulation data are described. Results from extensive MC studies of the critical properties of two important magnetic systems, the d=3 ferromagnetic Ising and Heisenberg models, are presented. It is shown that the careful use of histogram techniques can provide results comparable or superior to these obtained with other numerical methods while maintaining all the advantages of standard MC techniques.
The two-dimensional Baxter-Wu model with spin-1, in the presence of a crystal field, is studied by using Monte Carlo simulations. The standard single-spin-flip Metropolis algorithm is used to generate the configurations from which the order parameter, specific heat and magnetic susceptibility are measured. Single histogram techniques are used to get the results close to the phase transitions. The finite-size scaling procedure is employed in order to obtain the critical behavior. The simulations have shown that the critical exponents are different from the spin-1/2 case and are crystal field dependent.
To demonstrate the application of computational physics in modern statistical physics, this paper investigates the critical behavior of a finite-size ferromagnetic system based on the two-dimensional square-lattice Ising model.A nearest-neighbor Ising Hamiltonian is constructed, and periodic boundary conditions are adopted to reduce finite-boundary effects.The Metropolis Monte Carlo algorithm is used to sample spin configurations at different temperatures, while energy, magnetization, specific heat, magnetic susceptibility and Binder cumulant are calculated to describe the transition from an ordered state to a disordered state.The simulation results show that the system exhibits clear long-range order at low temperatures, where the order parameter remains high.As temperature increases, thermal fluctuations become stronger, magnetization decreases rapidly, and the system gradually evolves into a disordered state.The specific heat curve shows a pronounced peak at approximately (T = 2.269), while the susceptibility peak shifts slightly toward the high-temperature side due to finite-size effects and the discrete temperature grid.Overall, the simulated critical region is consistent with the theoretical critical temperature of the two-dimensional square-lattice Ising model.Spin-configuration analysis further reveals enhanced fluctuations and domain reconstruction near the critical region.In addition, a thermalization criterion and an extended sampling test for L = 48 near the critical region are added to examine the influence of finite sampling and equilibration.These results indicate that Monte Carlo simulation of the two-dimensional Ising model can effectively reproduce continuous phase transition, critical fluctuation and finite-size effects, providing a basic computational case for understanding critical phenomena in complex systems and data-driven physical analysis.
The magnetization probability density in d = 2 and 3 dimensional Ising models in slab geometry of volume [Formula: see text] is computed through Monte-Carlo simulation at the critical temperature and zero magnetic field. The finite-size scaling of this distribution and its dependence on the system aspect-ratio [Formula: see text]and boundary conditions are discussed. In the limiting case [Formula: see text] of a macroscopically large slab ([Formula: see text]) the distribution is found to scale as a Gaussian function for all tested system sizes and boundary conditions.