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arXiv:2112.11569 · SPIN

Nonequilibrium Study of the J1−J2 Ising Model with Random J2 Couplings in the Square Lattice

Octavio D. Rodrı́guez Salmon, Minos A. Neto, T. Lobo, Francisco Dinóla Neto

Published Jul 21, 2023• 0 citations
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Abstract

We studied the phase transitions of the [Formula: see text] spin-1/2 Ising model in the square lattice by considering [Formula: see text] fixed and [Formula: see text] as random interactions following discrete and continuous probability distribution functions. The configuration of [Formula: see text] in the lattice evolves in time through a competing kinetics using the Metropolis algorithm leading to a steady state without reaching the free-energy minimum. So, different phase diagrams were obtained according to the distribution of [Formula: see text]. For fixed and random values of [Formula: see text], we noted that the amplitude of the correlation length is different from the Ising value at the critical point of the superantiferromagnetic–paramagnetic frontier.

DOI
10.1142/s2010324723500248

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Abstract. We use improved Monte-Carlo algorithms to study the antiferromagnetic 2D-Ising model with competing interactions J1 on nearest neighbour and J2 on next-nearest neighbour bonds. The finite-temperature phase diagram is divided by a critical point at J2 = J1/2 where the groundstate is highly degenerate. To analyse the phase boundaries we look at the specific heat and the energy distribution for various ratios of J2/J1. We find a first order transition for small J2> J1/2 and the transition temperature suppressed to TC = 0 at the critical point. 1.
The phase diagram of the Ising square lattice with first- and second-nearest-neighbor antiferromagnetic interactions (J1{\mathit{J}}_{1}J1​ and J2{\mathit{J}}_{2}J2​) is calculated within the framework of the cluster-variation method. The Hamiltonian is solved within the nine-point-cluster approximation. In contrast to previous studies we find that the phase transition from the superantiferromagnetic to the paramagnetic state is of first order in the range 0.5R=J1{\mathit{J}}_{1}J1​/${\mathit{J}}_{2}$1.144. The non-Ising critical behavior observed in that range of interactions might be due to that phenomenon.
The effects of locally random magnetic fields are considered in a nonequilibrium Ising model defined on a square lattice with nearest-neighbor interactions. In order to generate the random magnetic fields, we have considered random variables { h } that change randomly with time according to a double-Gaussian probability distribution, which consists of two single-Gaussian distributions, centered at + h o and − h o , with the same width σ. This distribution is very general and can recover in appropriate limits the bimodal distribution ( ) and the single-Gaussian one ( h o = 0). We performed Monte Carlo simulations in lattices with linear sizes in the range L = 32–512. The system exhibits ferromagnetic and paramagnetic steady states. Our results suggest the occurrence of first-order phase transitions between the above-mentioned phases at low temperatures and large random-field intensities h o for some small values of the width σ. By means of finite size scaling, we estimate the critical exponents in the low-field region, where we have continuous phase transitions. In addition, we show a sketch of the phase diagram of the model for some values of σ.
The effects of locally random magnetic fields are considered in a nonequilibrium Ising model defined on a square lattice with nearest-neighbors interactions. In order to generate the random magnetic fields, we have considered random variables {h} that change randomly with time according to a double-gaussian probability distribution, which consists of two single gaussian distributions, centered at +ho and −ho, with the same width σ. This distribution is very general, and can recover in appropriate limits the bimodal distribution (σ → 0) and the single gaussian one (ho = 0). We performed Monte Carlo simulations in lattices with linear sizes in the range L = 32 − 512. The system exhibits ferromagnetic and paramagnetic steady states. Our results suggest the occurence of first-order phase transitions between the above-mentioned phases at low temperatures and large random-field intensities ho, for some small values of the width σ. By means of finite size scaling, we estimate the critical exponents in the low-field region, where we have continuous phase transitions. In addition, we show a sketch of the phase diagram of the model for some values of σ. PACS numbers: 05.10.Ln, 05.50.+q, 64.60.De, 75.10Hk, 75.40.MgFirst-order phase transition in a 2D random-field Ising model with conflicting dynamics2 1.
We study the square-lattice Ising model with nearest-neighbor ( J 1 ) and next nearest-neighbor ( J 2 ) interactions, using the Wang–Landau Monte Carlo algorithm and the partition function zeros in the complex temperature plane (Fisher zeros). Here, the competition between J 1 and J 2 interactions is defined by the coupling ratio R = J 2 / J 1 . For J 2 > 0, the critical behavior is trivial and is the same as the square-lattice Ising model with only J 1 interaction. On the other hand, the new phase of superantiferromagnet appears for J 2 < 0. However, the nature of the transition into the superantiferromagnetic phase is still controversial in the literature. To resolve the controversy, we estimate the critical temperature T c ( R ) and thermal scaling exponent y t ( R ) from the Fisher zeros, as a function of R . The estimated values of T c ( R ) agree well with the results in the literature. We obtain the simple critical behavior with a fixed value of the thermal scaling exponent, y t ( R ) = 1, for and J 2 < 0 (Ising antiferromagnet). For and J 2 < 0 (Ising superantiferromagnet), however, it is shown that y t ( R ) continuously varies depending on R values, and the estimated values of y t ( R ) are clearly less than that of the first-order phase transition, y t = 2.