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Physics Procedia

Simulated Tempering and Magnetizing Simulations of the Ising Model

Tetsuro Nagai, Yuko Okamoto

Published Jan 1, 2012• 4 citations
Publisher

Abstract

We carried out two-dimensional simulated tempering (ST) simulations of the two-dimensional Ising model with several lattice sizes. We aim to investigate the applicability of the two-dimensional ST to systems with phase transi–tions, and to study crossover of critical scaling behaviors. In the two-dimensional ST simulations, not only temperature but also external field is treated as a dynamical variable. Thus, this method can be referred to as “Simulated Tempering and Magnetizing (STM)”. As has been discussed by previous studies, the ST method is not always compatible with first-order phase transitions. This is also true in the magnetizing process. Processes of flipping the total magnetization hardly occurred during the simulations under Tc in large lattice-size simulations. However, the phase changed through the high temperature region. Therefore, the dimensional extension let us avoid the diffculty of the first-order phase transitions and study a wide area of the phase diagram. We finally study the crossover behavior of the phase transitions with respect to the temperature and external field. The crossover behavior was clearly observed in the simulations in the agreement with the theoretical implications

DOI
10.1016/j.phpro.2012.05.016

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We carried out two-dimensional simulated tempering (ST) simulations of the two-dimensional Ising model with several lattice sizes. We aim to investigate the applicability of the two-dimensional ST to systems with phase transi–tions, and to study crossover of critical scaling behaviors. In the two-dimensional ST simulations, not only temperature but also external field is treated as a dynamical variable. Thus, this method can be referred to as “Simulated Tempering and Magnetizing (STM)”. As has been discussed by previous studies, the ST method is not always compatible with first-order phase transitions. This is also true in the magnetizing process. Processes of flipping the total magnetization hardly occurred during the simulations under Tc in large lattice-size simulations. However, the phase changed through the high temperature region. Therefore, the dimensional extension let us avoid the diffculty of the first-order phase transitions and study a wide area of the phase diagram. We finally study the crossover behavior of the phase transitions with respect to the temperature and external field. The crossover behavior was clearly observed in the simulations in the agreement with the theoretical implications
We have performed two-dimensional simulated tempering (ST) simulations of the two-dimensional Ising model with different lattice sizes in order to investigate the two-dimensional ST's applicability to dealing with phase transitions and study the crossover of critical scaling behavior. The external field, as well as the temperature, was treated as a dynamical variable updated during the simulations. Thus this simulation can be referred to as simulated tempering and magnetizing (STM). We also performed simulated magnetizing (SM) simulations, in which the external field was considered as a dynamical variable and temperature was not. As discussed in previous studies, the ST method is not always compatible with first-order phase transitions. This is also true in the magnetizing process. Flipping of the entire magnetization did not occur in the SM simulations under the critical temperature T{c} in large-lattice-size simulations; however, the phase changed through the high-temperature region in the STM simulations. Thus the dimensional extension let us eliminate the difficulty of the first-order phase transitions and study a wide area of the phase space. We discuss how frequently parameter-updating attempts should be made for optimal convergence. The results favor frequent attempts. We finally study the crossover behavior of the phase transitions with respect to the temperature and external field. The crossover behavior is clearly observed in the simulations, in agreement with the theoretical implications.
We applied the simulated tempering and magnetizing (STM) method to the two-dimensional three-state Potts model in an external magnetic field in order to perform further investigations of the STM's applicability. The temperature as well as the external field are treated as dynamical variables updated during the STM simulations. After we obtained adequate information for several lattice sizes LLL (up to 160×160160\times 160160×160), we also performed a number of conventional canonical simulations of large lattices, especially in order to illustrate the crossover behavior of the Potts model in external field with increasing LLL. The temperature and external field for larger lattice size simulations were chosen by extrapolation of the detail information obtained by STM. We carefully analyzed the crossover scaling at the phase transitions with respect to the lattice size as well as the temperature and external field. The crossover behavior is clearly observed in the simulations in agreement with theoretical predictions.
We apply the simulated tempering and magnetizing (STM) method to the two-dimensional three-state Potts model in an external magnetic field in order to investigate STM’s applicability further. The temperature and the external field are treated as dynamical variables updated during the STM simulations. On the basis of adequate information obtained by STM for several lattice sizes L × L (up to 160×160), we also perform a number of conventional canonical simulations of larger lattices in order to illustrate the crossover behavior of the Potts model in an external field with increasing L . The temperature and external field for the larger lattice size simulations are chosen by extrapolation of the detailed information obtained by STM. We present a careful analysis of the crossover-scaling behavior at the phase transitions with respect to the lattice size as well as the temperature and external field. The crossover behavior is clearly observed in the simulations in agreement with theoretical predictions.
We apply simulated tempering and magnetizing (STM) Monte Carlo simulations to the two-dimensional threestate Potts model in an external magnetic field in order to investigate the crossover scaling behaviour in the temperature-field plane at the Potts critical point and towards the Ising universality class for negative magnetic fields.Our data set has been generated by STM simulations of several square lattices with sizes up to 160×160 spins, supplemented by conventional canonical simulations of larger lattices at selected simulation points.We present careful scaling and finite-size scaling analyses of the crossover behaviour with respect to temperature, magnetic field and lattice size.
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.
In this work, Markov Chain-Monte Carlo technique was used to study the phase transition in two and three dimensional Ising Model (IM) in a square and cubic lattice. The study of temperature dependence of average magnetization and specific heat in different magnetic fields has been carried out in the 3x3 and 3x3x3 lattice with periodic boundary. Critical temperature point kBTc / J for 2D and 3D Ising Model has been observed at around 2.2 and 4.3 respectively at zero field. Our work satisfies Onsager’s critical value in 2D IM. The simulation suggests bifurcation in average magnetization below critical temperature Tc. Temperature plays the role of increasing randomness of spins. We found that Ising Model in small lattice size still retains interesting features like spontaneous magnetization and symmetry breaking below Tc at B = 0. At a non-zero field, the likelihood of spins to prefer certain alignment depends on the direction of the external field and magnitude of magnetization depends on magnitude of field ±B. Specific heat Cv, which gives us fluctuation of energy at particular temperature, has been found to attain maximum value at critical regions.