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arXiv:1805.05725

Logarithm corrections in the critical behavior of the Ising model on a triangular lattice modulated with the Fibonacci sequence

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

Published May 15, 2018• 0 citations
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Abstract

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 $q$, the associated magnetic susceptibility ($χ$) and the specific heat $c$, 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 $T_{c} \approx 1.4116$ 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 $β$, $γ$ and $ν$, and the correction exponents $\hatβ$, $\hatγ$, $\hatα$ and $\hatλ$. Our results show that the system obeys the Ising model universality class and that the critical behavior has logarithmic corrections.

DOI
10.48550/arxiv.1805.05725

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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.
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 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 Phase transitions and critical and thermodynamic properties of the three-dimensional antiferromagnetic Ising model on a layered triangular lattice with variable interlayer exchange interaction are studied by the replica algorithm of the Monte Carlo method. The studies are carried out for the ratios of the intralayer J1 and interlayer J2 exchange interactions in the range of r = J2/J1 = 0.01–1.0. It is established that a second-order phase transition is observed in the considered r interval. Using the finite size scaling theory, the static critical exponents of the heat capacity α, susceptibility γ, order parameter β, correlation radius ν, and Fisher index η are calculated. It is shown that the universality class of the critical behavior of this model is preserved in the interval of 0.05 < r ≤ 1.0. It was found that with a further decrease in the r value, a crossover from the three-dimensional critical behavior to the quasi-two-dimensional one is observed.
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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.