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International Conference on Information Technology and Nanotechnology (ITNT)

Studies of 3D-Ising model with long-range interaction

Boris Kryzhanovsky, Olga G. Maksimova, Leonid B. Litinskii, V. I. Egorov

Published Sep 20, 2021• 1 citation
Publisher

Abstract

We use the Monte-Carlo simulations to study thermodynamic properties of Ising models on cubic lattices accounting for interaction with next-nearest neighbors. The results of simulations are in good agreement with an analytic formula that defines the critical inverse temperature as a function of the interaction parameter. The behaviors of the internal energy and the heat capacity indicate that at the critical point the heat capacity has an infinite singularity.

DOI
10.1109/itnt52450.2021.9649257

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For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent $2-\sqrt{2} \approx 0.5858,$ comparable to the exponent $4/7 \approx 0.5714$ found from renormalization arguments.

•Jun 1, 2006•0 citations
We use the Monte-Carlo simulations to study thermodynamic properties of Ising models on cubic lattices accounting for interaction with next-nearest neighbors. The results of simulations are in good agreement with an analytic formula that defines the critical inverse temperature as a function of the interaction parameter. The behaviors of the internal energy and the heat capacity indicate that at the critical point the heat capacity has an infinite singularity.
Phase transitions and critical properties of the frustrated Ising model on a cubic lattice with next-nearest-neighbor intralayer interactions are investigated by the replica Monte Carlo method. Estimations are made for the magnitude relation of the next-nearest-neighbor and nearest-neighbor exchange interactions r = J2/J1 in the value ranges of r ∈ [0.0, 1.0]. The phase diagram of the dependence of critical temperature on the next-nearest-neighbor interaction has been plotted. The static critical exponents of the heat capacity, the susceptibility, the ordering parameter and the correlation length as well as the Fisher exponent are calculated by means of the finite-size scaling theory. The universality class of the critical behavior of this model is revealed to remain within the limits of values r ∈ [0.0, 0.4]. It is found that the change in the next-nearest-neighbor interaction value in the range of r ∈ [0.9, 1.0] leads to another critical behavior.
The heat capacity of the S= ∞ Ising ferromagnet arrayed on a simple cubic lattice is investigated by analysis of Monte Carlo data for the internal energy. A function is least-squares fitted to the energy data close to the critical temperature T c . The function is consistent with the expected singularity of the heat capacity C(t) = A ± | t | -α (1 + D ± | t | Δ 1 ) + B ± , where t = (T - T c )/T c and the ± superscripts refer to t > 0 and t < 0, respectively. The results of the data analysis show that the relation B + = B - is supported and that A + /A - accords with the expected universal value for a system with Ising symmetry. The ratio of the confluent singularity amplitudes D + /D - is somewhat lower than the expected universal value. The amplitudes of the leading and confluent singularities of the order parameter are also reported.
The replica Monte Carlo method has been used to investigate the critical behavior of a threedimensional antiferromagnetic Ising model on a body-centered cubic lattice, taking into account interactions of the adjacent behind neighbors. Investigations are carried out for the ratios of the values of exchange interactions behind the nearest and next nearest neighbors k = J 2/J 1 in the range of k ∈ [0.0, 1.0] with the step Δk = 0.1. In the framework of the theory of finite-dimensional scaling the static critical indices of heat capacity α, susceptibility γ, of the order parameter β, correlation radius ν, and also the Fisher index η are calculated. It is shown that the universality class of the critical behavior of this model is kept in the interval of k ∈ [0.0, 0.6]. It is established that a nonuniversal critical behavior is observed in the range k ∈ [0.8, 1.0].
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858,2-\sqrt{2} \approx 0.5858,2−2​≈0.5858, comparable to the exponent 4/7≈0.57144/7 \approx 0.57144/7≈0.5714 found from renormalization arguments.