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Anisotropic Effect on Local Magnetic Properties of 3D Extended Ising Model

Anouar Elidrysy, Said Harir, A. Zouhair, Y. Boughaleb

Published Aug 14, 2020• 11 citations
Publisher

Abstract

The magnetic properties of anisotropic 3D Ising model on a cubic lattice are studied by Monte Carlo simulation. In particular, we have considered an extended 3D Ising model with spatially uniaxial anisotropic bond randomness on the simple cubic lattice parameterized by exchange interaction parameter [Formula: see text], anisotropy parameter [Formula: see text] and external longitudinal magnetic field [Formula: see text]. The obtained numerical data clearly point out a second-order ferromagnetic-paramagnetic phase transition belonging in the same universality class with the 3D random Ising model at critical temperature [Formula: see text] that is strongly correlated to [Formula: see text] and [Formula: see text]. Especially, in the limit, [Formula: see text], the spin ½ cubic lattice becomes a collection of noncorrelated Ising chains, whereas in the other limit, [Formula: see text], the system becomes a stack of noncorrelated Ising square lattice.

DOI
10.1142/s2010324720500150

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The magnetic properties of anisotropic 3D Ising model on a cubic lattice are studied by Monte Carlo simulation. In particular, we have considered an extended 3D Ising model with spatially uniaxial anisotropic bond randomness on the simple cubic lattice parameterized by exchange interaction parameter [Formula: see text], anisotropy parameter [Formula: see text] and external longitudinal magnetic field [Formula: see text]. The obtained numerical data clearly point out a second-order ferromagnetic-paramagnetic phase transition belonging in the same universality class with the 3D random Ising model at critical temperature [Formula: see text] that is strongly correlated to [Formula: see text] and [Formula: see text]. Especially, in the limit, [Formula: see text], the spin ½ cubic lattice becomes a collection of noncorrelated Ising chains, whereas in the other limit, [Formula: see text], the system becomes a stack of noncorrelated Ising square lattice.
We study the ±J three-dimensional (3D) Ising model with a spatially uniaxial anisotropic bond randomness on the simple cubic lattice. The ±J random exchange is applied on the xy planes, whereas, in the z direction, only a ferromagnetic exchange is used. After sketching the phase diagram and comparing it with the corresponding isotropic case, the system is studied at the ferromagnetic-paramagnetic transition line using parallel tempering and a convenient concentration of antiferromagnetic bonds (p(z)=0;p(xy)=0.176). The numerical data clearly point out a second-order ferromagnetic-paramagnetic phase transition belonging in the same universality class with the 3D random Ising model. The smooth finite-size behavior of the effective exponents, describing the peaks of the logarithmic derivatives of the order parameter, provides an accurate estimate of the critical exponent 1/ν=1.463(3), and a collapse analysis of magnetization data gives an estimate of β/ν=0.516(7). These results are in agreement with previous papers and, in particular, with those of the isotropic ±J three-dimensional Ising model at the ferromagnetic-paramagnetic transition line, indicating the irrelevance of the introduced anisotropy.
Using the Monte Carlo Simulation (MCS) method combined with the Metropolis algorithm, we explore the magnetic properties of a two-dimensional (2D) spin system with spins 1, 2 and 3. The adapted simulation to the Ising model aims to study the impact of the correction term a of the interaction with second neighbors and the crystal field D on the critical temperature [Formula: see text] of the 2D anisotropic square lattice for different integer values of spins. The obtained results demonstrate that the critical temperature [Formula: see text] is significantly affected by the competing interactions between the anisotropy parameter, second-neighbor interaction and the crystal field. The considered model exhibits a critical phase transition regardless of the spin S value where the critical temperature increases with the value of S. Note that this phase transition occurs under the condition [Formula: see text]. Moreover, the obtained results show that the complex interactions give rise to a variety of phase diagrams, thereby revealing the richness of possible magnetic behaviors in the studied system.
We have studied the anisotropic three-dimensional nearest-neighbor Ising model with competitive interactions in an uniform longitudinal magnetic field HHH. The model consists of ferromagnetic interaction Jx(Jz)J_{x}(J_{z})Jx​(Jz​) in the x(z)x(z)x(z) direction and antiferromagnetic interaction JyJ_{y}Jy​ in the yyy direction. We have compared our calculations within a effective-field theory in clusters with four spins (EFT-4) in the simple cubic (sc) lattice with traditional Monte Carlo (MC) simulations. The phase diagrams in the h−kBT/Jxh-k_{B}T/J_{x}h−kB​T/Jx​ plane (h=H/Jxh=H/J_{x}h=H/Jx​) were obtained for the particular case λ1=Jy/Jx(λ2=Jz/Jx)=1λ_{1}=J_{y}/J_{x} (λ_{2}=J_{z}/J_{x})=1λ1​=Jy​/Jx​(λ2​=Jz​/Jx​)=1 (anisotropic sc). Our results indicate second-order frontiers for all values of HHH for the particular case λ2=0λ_{2}=0λ2​=0 (square lattice), while in case λ1=λ2=1λ_{1}=λ_{2}=1λ1​=λ2​=1, we observe first- and second-order phase transitions in the low and high temperature limits, respectively, with presence of a tricritical point. Using EFT-4, a reentrant behavior at low temperature was observed in contrast with results of MC.
The effects of bond randomness on the universality aspects of the simple cubic lattice ferromagnetic Blume-Capel model are discussed. The system is studied numerically in both its first- and second-order phase transition regimes by a comprehensive finite-size scaling analysis. We find that our data for the second-order phase transition, emerging under random bonds from the second-order regime of the pure model, are compatible with the universality class of the 3d random Ising model. Furthermore, we find evidence that the second-order transition emerging under bond randomness from the first-order regime of the pure model belongs to a new and distinctive universality class. The first finding reinforces the scenario of a single universality class for the 3d Ising model with the three well-known types of quenched uncorrelated disorder (bond randomness, site and bond dilution). The second amounts to a strong violation of the universality principle of critical phenomena. For this case of the ex-first-order 3d Blume-Capel model, we find sharp differences from the critical behaviors, emerging under randomness, in the cases of the ex-first-order transitions of the corresponding weak and strong first-order transitions in the 3d three-state and four-state Potts models.
The effects of bond randomness on the universality aspects of the simple cubic lattice ferromagnetic Blume-Capel model are discussed. The system is studied numerically in both its first- and second-order phase transition regimes by a comprehensive finite-size scaling analysis. We find that our data for the second-order phase transition, emerging under random bonds from the second-order regime of the pure model, are compatible with the universality class of the 3d random Ising model. Furthermore, we find evidence that the second-order transition emerging under bond randomness from the first-order regime of the pure model belongs to a new and distinctive universality class. The first finding reinforces the scenario of a single universality class for the 3d Ising model with the three well-known types of quenched uncorrelated disorder (bond randomness, site and bond dilution). The second amounts to a strong violation of the universality principle of critical phenomena. For this case of the ex-first-order 3d Blume-Capel model, we find sharp differences from the critical behaviors, emerging under randomness, in the cases of the ex-first-order transitions of the corresponding weak and strong first-order transitions in the 3d three-state and four-state Potts models.
The effects of bond randomness on the universality aspects of the simple cubic lattice ferromagnetic Blume-Capel model are discussed. The system is studied numerically in both its first- and second-order phase transition regimes by a comprehensive finite-size scaling analysis. We find that our data for the second-order phase transition, emerging under random bonds from the second-order regime of the pure model, are compatible with the universality class of the 3d random Ising model. Furthermore, we find evidence that, the second-order transition emerging under bond randomness from the first-order regime of the pure model, belongs to a new and distinctive universality class. The first finding reinforces the scenario of a single universality class for the 3d Ising model with the three well-known types of quenched uncorrelated disorder (bond randomness, site- and bond-dilution). The second, amounts to a strong violation of universality principle of critical phenomena. For this case of the ex-first-order 3d Blume-Capel model, we find sharp differences from the critical behaviors, emerging under randomness, in the cases of the ex-first-order transitions of the corresponding weak and strong first-order transitions in the 3d three-state and four-state Potts models.