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Journal of Physics Conference Series

Phase Transition in 2D Ising Model with Next-Neighbor Interaction

Boris Kryzhanovsky, Leonid B. Litinskii

Published Nov 1, 2019• 2 citations
Publisher

Abstract

Abstract We used a Monte Carlo method to analyze a planar Ising model taking into account a ferromagnetic interaction with next neighbors. We found that independent of the value of the additional interaction, the character of singularity of the heat capacity did not change. Namely, at the critical point the heat capacity has a logarithmic singularity. We obtained a semi-theoretical formula relating the value of the critical temperature with the value of the additional ferromagnetic interaction.

DOI
10.1088/1742-6596/1391/1/012077

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Abstract We used a Monte Carlo method to analyze a planar Ising model taking into account a ferromagnetic interaction with next neighbors. We found that independent of the value of the additional interaction, the character of singularity of the heat capacity did not change. Namely, at the critical point the heat capacity has a logarithmic singularity. We obtained a semi-theoretical formula relating the value of the critical temperature with the value of the additional ferromagnetic interaction.
The critical behavior at the phase transition of the ferromagnetic two-dimensional anisotropic Ising model with next-nearest neighbor (NNN) couplings in the presence of the field is determined using mainly Monte Carlo (MC) method. This method is used to investigate the phase diagram of the model and to verify the existence of a divergence at null temperature which often appears in two-dimensional systems. We analyze also the influence of the report of the NNN interactions [Formula: see text] and the magnetic field [Formula: see text] on the critical temperature of the system, and we show that the critical temperature depends on the magnetic field for positive values of the interaction. Finally, we have investigated other thermodynamical qualities such as the magnetic susceptibility [Formula: see text]. It has been shown that their thermal behavior depends qualitatively and quantitatively on the strength of NNN interactions and the magnetic field.
Abstract In the present paper we analyzed a change in the density of states of a two-dimensional Ising model when a next-next-neighbor interaction is introduced. In other words, we examined two-dimensional lattices with diagonal connections. The same as in a three-dimensional model in this case each spin has 6 connections. Since the model is planar, we can calculate the free energy and other characteristics of the system using a polynomial algorithm. We performed computer simulations using the Kasteleyn–Fisher algorithm, which allowed us to study changes of critical values and the density of states when a long-range interaction is taken into account. From the obtained results it follows that the interactions of the type analyzed here result in quantitative changes of the system’s characteristics, but they did not change them qualitatively. In particular, we again obtained a logarithmic divergence of the heat capacity in the critical point.
The phase transitions in two-dimensional antiferromagnetic Ising model are studied on a square lattice by taking the interactions of next-nearest neighbors into account. The model is investigated on basis of the replica Monte Carlo algorithm and the histogrammic analysis of data. The diagram of the critical temperature dependence on an interaction value of next-nearest neighbors is plotted. The studied model reveals the phase transition of second order and the frustration points depending on relative interaction values and magnetic field. A effect, heat capacity splitting near the frustration points, is found.
In the framework of the spin flip dynamics algorithm with an original expression for the energy, an influence of the interaction of second, third neighbors, four-particle interaction, temperature and size of a 1D Ising ferromagnet on the heat capacity critical exponent is studied. Periodic border conditions are used. The critical index α is calculated by the finite size scaling method. It is established that the growth of energy parameters of the multi-spin interaction significantly affects the heat capacity exponent. Interaction energy of third neighbors decreases the value of α to a greater extent than the second-neighbor interaction J2 and four-particle one J1 – 4. Different characters of the influence of pair and four-particle interactions on α are revealed. Energy parameters are established for which this critical exponent attains negative values. For instance, taking into account the interaction energy of third neighbors, with an approach to the critical region, the heat capacity critical exponent has a minimum near negative value of α = –0.0043 ± 0.0005. With increasing number of nodes in the 1D Ising chain, heat capacity critical exponent increases. Extrapolation according to the inverse size of the system in the thermodynamic limit gives the values α = 0.19 ± 0.02, α = 0.17 ± 0.02, α = 0.20 ± 0.02, when one takes into account the interaction energy of second neighbors J2 = 0.1, third neighbors J3 = 0.1 and four-particle interaction J1 – 4 = 0.1, respectively. Note that the second neighbor and four-particle interaction energies do not change the sign of the critical exponent for N > 6, and when approaching the critical region the critical exponent α tends to zero. The results obtained are compared with the values for Ising model with "broken ends" boundary conditions, as well as with other known results for two-dimensional and three-dimensional cases.