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Journal of the Physical Society of Japan

Two-Dimensional Ising Ferromagnet with Second-Neighbor Interaction. II. Monte Carlo Method

Sei-ichi Takase

Published May 15, 1976• 5 citations
Publisher

Abstract

The static properties are investigated by computer simulation for the square Ising ferromagnet in which the nearest-neighbor interactions are ferromagnetic and the second interactions antiferromagnetic. Energy and magnetization at various temperatures are obtained by Monte Carlo calculations performed on the lattice with 20×20 spins. The Curie point determined as the inflection point of the energy-vs-temperature curve agrees with the result by Fan and Wu rather than other analytical approximations. The behavior of the spontaneous magnetization M 0 ( T ) for T → T c -0 as characterized by the critical exponent β is shown by comparing the data with an analytical equation for the magnetization. The exponent β remains probably unchanged from the original value 1/8.

DOI
10.1143/jpsj.40.1240

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The static properties are investigated by computer simulation for the square Ising ferromagnet in which the nearest-neighbor interactions are ferromagnetic and the second interactions antiferromagnetic. Energy and magnetization at various temperatures are obtained by Monte Carlo calculations performed on the lattice with 20×20 spins. The Curie point determined as the inflection point of the energy-vs-temperature curve agrees with the result by Fan and Wu rather than other analytical approximations. The behavior of the spontaneous magnetization M 0 ( T ) for T → T c -0 as characterized by the critical exponent β is shown by comparing the data with an analytical equation for the magnetization. The exponent β remains probably unchanged from the original value 1/8.
The square Ising ferromagnet is investigated by computer simulation in the region of competing interactions (nearest-neighbor interaction K >0 and second-neighbor one K '<0). Spontaneous magnetization M 0 and energy E are calculated from advanced Monte Carlo calculations. The estimates of M 0 and of the Curie point T c ( K '/ K ) determined by analyzing the data on E are compared with the results from analysis of low-temperature series based on the assumption β=1/8. The values of T c ( K '/ K ) are obtained from padé-approximant method for low-temperature series. The value of β does not vary from the Onsager's value 1/8.
Monte Carlo simulations are performed on the square lattice Ising system with both the nearest neighbor and the two-dimensional dipolar interactions in order to investigate the critical property of ferromagnetic phase transitions in this 2-dimensional system. Applying the finite size scaling theory to Monte Carlo data just near the transition point, the critical indices of this dipolar system are calculated and α=0.31±0.04, β =0.20±0.02, γ=1.08±0.20 and ν=0.72±0.10 are obtained. However, in the temperature region considerably far from the transition point, different values of α, β and γ seem to be adequate, which are rather close to those of the rigorous result of the 2-dimensional Ising model.
Using the Monte Carlo Simulation (MCS) method combined with the Metropolis algorithm, we explore the magnetic properties of a two-dimensional (2D) spin system with spins 1, 2 and 3. The adapted simulation to the Ising model aims to study the impact of the correction term a of the interaction with second neighbors and the crystal field D on the critical temperature [Formula: see text] of the 2D anisotropic square lattice for different integer values of spins. The obtained results demonstrate that the critical temperature [Formula: see text] is significantly affected by the competing interactions between the anisotropy parameter, second-neighbor interaction and the crystal field. The considered model exhibits a critical phase transition regardless of the spin S value where the critical temperature increases with the value of S. Note that this phase transition occurs under the condition [Formula: see text]. Moreover, the obtained results show that the complex interactions give rise to a variety of phase diagrams, thereby revealing the richness of possible magnetic behaviors in the studied system.
The critical properties of an antiferromagnetic Ising model on a square lattice with interactions of the next-to-nearest neighbors are investigated by a replica Monte-Carlo method. Using the finite-size scaling theory the static critical exponents of specific heat, ordering parameter, susceptibility, correlation radius as well as the Fisher exponent are calculated. An analysis of data is performed both with and without taking into account a correction to the finite-size scaling. It was found that in the model under consideration the second order phase transition is observed. It is shown that this model belongs to the new class of universality of critical behavior.
The two-dimensional square Ising ferromagnet with the nearest-neighbor interaction - J and the second-neighbor interaction - J ' is investigated starting from the identity < s i >=<tanh (ΣK i j s j )>. The Hartree approximation is used for treating the many-spin correlation functions. The critical point thus determined has two values for J '/ J <0, and is corrected by the study of the spontaneous magnetization and the free energy to give a resonable function T c ( J '/ J ). Our approximation may be valid at least qualitatively as a whole.