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Physical review. E

Critical properties of a two-dimensional Ising magnet with quasiperiodic interactions

Gladstone Alencar Alves, M.S. Vasconcelos, T. F. A. Alves

Published Apr 12, 2016• 4 citations
Publisher

Abstract

We address the study of quasiperiodic interactions on a square lattice by using an Ising model with ferromagnetic and antiferromagnetic exchange interactions following a quasiperiodic Fibonacci sequence in both directions of a square lattice. We applied the Monte Carlo method, together with the Metropolis algorithm, to calculate the thermodynamic quantities of the system. We obtained the Edwards-Anderson order parameter q_{EA}, the magnetic susceptibility χ, and the specific heat c in order to characterize the universality class of the phase transition. We also use the finite size scaling method to obtain the critical temperature of the system and the critical exponents β,γ, and ν. In the low-temperature limit we obtained a spin-glass phase with critical temperature around T_{c}≈2.274, and the critical exponents β,γ, and ν, indicating that the quasiperiodic order induces a change in the universality class of the system. Also, we discovered a spin-glass ordering in a two-dimensional system which is rare and, as far as we know, the unique example is an under-frustrated Ising model.

DOI
10.1103/physreve.93.042111

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We address the study of quasiperiodic interactions on a square lattice by using an Ising model with ferromagnetic and antiferromagnetic exchange interactions following a quasiperiodic Fibonacci sequence in both directions of a square lattice. We applied the Monte Carlo method, together with the Metropolis algorithm, to calculate the thermodynamic quantities of the system. We obtained the Edwards-Anderson order parameter q_{EA}, the magnetic susceptibility χ, and the specific heat c in order to characterize the universality class of the phase transition. We also use the finite size scaling method to obtain the critical temperature of the system and the critical exponents β,γ, and ν. In the low-temperature limit we obtained a spin-glass phase with critical temperature around T_{c}≈2.274, and the critical exponents β,γ, and ν, indicating that the quasiperiodic order induces a change in the universality class of the system. Also, we discovered a spin-glass ordering in a two-dimensional system which is rare and, as far as we know, the unique example is an under-frustrated Ising model.
Abstract We investigated the Ising model on a square lattice with ferro and antiferromagnetic interactions modulated by the quasiperiodic Octonacci sequence in both directions of the lattice. We have applied the replica exchange Monte Carlo (parallel tempering) technique to calculate the thermodynamic quantities of the system. We obtained the order parameter, the associated magnetic susceptibility ( χ ) and the specific heat ( c ) in order to characterize the universality class of the phase transition. Also, we use the finite size scaling method to obtain the critical temperature of the system and the critical exponents β , γ and ν . In the low temperature limit we have obtained a continuous transition with critical temperature around T c ≈ 1.413 . The system obeys the Ising universality class with logarithmic corrections. We found estimatives for the correction exponents β ^ , γ ^ and λ ^ by using the finite size scaling technique.
We investigated the critical behavior of the Ising model in a triangular lattice with ferro and anti-ferromagnetic interactions modulated by the Fibonacci sequence, by using finite-size numerical simulations. Specifically, we used a replica exchange Monte Carlo method, known as Parallel Tempering, to calculate the thermodynamic quantities of the system. We have obtained the staggered magnetization qqq, the associated magnetic susceptibility (χχχ) and the specific heat ccc, to characterize the universality class of the system. At the low-temperature limit, we have obtained a continuous phase transition with a critical temperature around Tc≈1.4116T_{c} \approx 1.4116Tc​≈1.4116 for a particular modulation of the lattice according to the Fibonacci letter sequence. In addition, we have used finite-size scaling relations with logarithmic corrections to estimate the critical exponents βββ, γγγ and ννν, and the correction exponents β^\hatββ^​, γ^\hatγγ^​, α^\hatαα^ and λ^\hatλλ^. Our results show that the system obeys the Ising model universality class and that the critical behavior has logarithmic corrections.
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 frustrated antiferromagnetic XY model on the triangular lattice is a simple model describing the critical phenomena of magnetic systems. By using the Monte Carlo simulation method associated with the Metropolis algorithm, we investigate phase transitions of the system at critical temperatures and near critical temperatures. To achieve a scaling form of important parameters, the finite size effect must be taken into account. Our results show clearly that the broken symmetries of the system occur at two separate critical temperatures, the Kosterlitz-Thouless temperature and the Ising-like temperature (TI). However these temperatures lie close to each other about 0.6% compared with the lower temperature. At TI, the reflection symmetry breaking transition occurs and it is known as the Ising-like transition. While at the lower temperature, the rotational symmetry breaking transition occurs and it is known as the Kosterlitz-Thouless transition. Moreover, we found the asymptotic behavior of the correlation length of the Ising-like transition with the critical exponent, ν = 0.74. Taken into account the finite size effect, this value approaches 0.83 which differs from the value, ν = 1, for the Ising universality class. It is likely that the Ising-like transition may not belong to the Ising universality class. Also the other parameters such as the energy of the system, the order parameters, the specific heat, the magnetic susceptibility and the correlation functions have been calculated and their properties near the critical temperatures and at the critical temperatures have been studied.
In this work, we simulated a quenched random mixed-spin Ising model on the square lattice. The model system consists of two different particles with spins σ = 1/2 (states ±1/2) and S = 1 (states ±1, 0). These particles are randomly distributed on the lattice, and we considered only nearest-neighbor interactions. This model can represent a random magnetic binary alloy AxB1-x, obtained from the high-temperature quenching of a liquid mixture. We performed Monte Carlo simulations for several lattice sizes and temperatures, and we found its critical temperature through the reduced fourth-order cumulant. We also determined the magnetization, the susceptibility, and the specific heat as a function of temperature. We used finite-size scaling arguments to estimate the critical exponents β, γ, and ν of the model. We showed that the quenched model is in the same universality class of the two-dimensional pure Ising model. We also investigated the sample to sample fluctuations that occur in the values of the thermodynamic quantities in order to describe the self-averaging properties at the critical point.