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Moscow University Physics Bulletin

The Critical Behavior of the Antiferromagnetic Ising Model with Long-Range Interaction Effects

С. В. Белим, И. Б. Ларионов

Published Jul 1, 2018• 3 citations
Publisher

Abstract

In this paper, the antiferromagnetic Ising model with ferromagnetic long-range interaction is modeled by the Monte Carlo method. The case of ferromagnetic long-range forces decreasing by a power law is considered. The dependence of the phase-transition temperature on long-range interaction parameters is obtained. The phase diagram was constructed at different values of long-range interaction parameters. Conditions for the existence of the frustrated state of the system were revealed.

DOI
10.3103/s0027134918040021

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In this paper, the antiferromagnetic Ising model with ferromagnetic long-range interaction is modeled by the Monte Carlo method. The case of ferromagnetic long-range forces decreasing by a power law is considered. The dependence of the phase-transition temperature on long-range interaction parameters is obtained. The phase diagram was constructed at different values of long-range interaction parameters. Conditions for the existence of the frustrated state of the system were revealed.
In this article, spin systems with competing interactions in an external magnetic field are investigated. The Ising model with ferromagnetic short-range forces and long-range antiferromagnetic forces is considered. Long-range forces decrease according to the power law. Long-range interaction is characterized by the rate of decrease with distance and relative intensity. The Metropolis algorithm is used for computer modeling. Two order parameters for ferromagnetic and antiferromagnetic ordering are used. The theory of finite size scaling is used. The transition temperature was defined based on Binder's cumulants for each of the order parameters. The phase diagram for the system is constructed at various values of the long-range interaction parameters. As the temperature decreases, the system can transfer from the paramagnetic phase into the ferromagnetic and antiferromagnetic ones. The type of the low-temperature phase depends on the intensity of long-range forces and does not depend on the rate of decrease of the interaction with distance. There is a boundary value of the intensity of long-range forces, below which the ferromagnetic phase is implemented, and above it is the antiferromagnetic phase. The change in the phase diagram under the influence of an external magnetic field is investigated. In an external magnetic field, the ferromagnetic phase transition becomes spread. The temperature of the antiferromagnetic phase transition in an external field decreases according to the square law. The boundary value of the intensity of long-range forces linearly grows with increasing the intensity of the external magnetic field. An increase in the boundary value in a magnetic field leads to a change in the type of the low-temperature phase. This effect is shown as a phase transition from the antiferromagnetic to the ferromagnetic phase in the external magnetic field. A comparison with the data of real experiments is carried out.
Critical behavior of Ising ferromagnetic systems with the competition between short- and long-range interactions near the point of a second-order phase transition has been investigated. The effects of long-range interaction lead to the need to calculate the influence of next-nearest spins, which lie inside the sphere of a specific radius. The study was carried out for a power law of decreaseing the exchange integral with distance for a three-dimensional Ising model with a cubic lattice. A comparison with the results of a field-theoretical approach and data of real experiments has been carried out. It has been shown that the critical temperature grows linearly with an increase in the parameters of long-range interaction. Critical exponents have been calculated at different parameters of long-range interaction. It is shown that at small dimensions of a system it is the short-range interaction that exerts the dominant effect on the critical behavior of the system. With an increase in the linear dimension of the system, long-range forces begin to dominate.
We study an Ising model in one dimension with short-range ferromagnetic and long-range (power law) antiferromagnetic interactions. We show that the zero temperature phase diagram in a (longitudinal) field HHH involves a sequence of up and down domains whose size varies continuously with HHH, between $\ensuremath{-}{H}_{c}$ and Hc{H}_{c}Hc​, which represent the edges of the ferromagnetic up and down phases. The implications of long-range interaction in many body systems are discussed.
Critical behavior of the one- and two-dimensional general continuous Ising models with long-range ferromagnetic interactions decaying as ${1/r}^{d+\ensuremath{\sigma}}$ is studied using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between $\ensuremath{-}1$ and 1. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents which depend on $\ensuremath{\sigma}.$ Results for various values of $\ensuremath{\sigma}$ will be shown and compared to predictions from renormalization-group theory. Though there is an agreement with the overall tendency predicted, there are several fundamental differences. Discussion is given.
The article deals with two-dimensional and three-dimensional Ising models with the long-range spin interactions. The intensity of interaction between the spins relies decreasing with distance r as a power law r-d-σ with dimensional d and parameter σ. The research are conducted by Monte-Carlo method with Metropolis algorithm using parallel computing techniques. On the basis of nu- merical simulation the dependence of the phase transition temperature on the parameter σ is found. It is shown that at phase transition temperature decreases with increasing σ.
Critical exponents for a one-dimensional general continuous Ising model with long-range ferromagnetic interactions decaying as 1/r1+σ are calculated using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between −1 and 1. The critical point behavior is investigated. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents; the singularities in the specific heat and susceptibility depend on σ. For σ=1, there are extremely weak finite size effects: the first-order and the second-order cumulants of the order parameter yield ν=2.42(1). The susceptibility exponent γ=2.259(9). Results for σ=0.7 will be shown and discussed.