In a previous work, a realistic three-dimensional 3-D Ising model of FeCl2 was studied. It consists of triangular lattice planes stacked along the c axis of the crystal, with the following interactions: a nearest neighbor (nn) ferromagnetic one competing with a next-nearest-neighbor (nnn) antiferromagnetic interaction, which causes some degree of frustration in the plane, the planes being weakly coupled by an antiferromagnetic interaction. In fact, the latter is a superexchange interaction, and we have taken into account all the equivalent superexchange paths in the sense of the Anderson rule, which amounts to 12 neighbors in each one of the adjacent planes. In this work we present the preliminary results of a study of the critical exponents of this model using the Monte Carlo histogram method and finite size scaling. The interest of such study lies in the layered structure of this system, which might have an effect on its effective dimensionality at the transition point, and also in the effects of frustration, as well as those of superexchange interactions on the critical exponents. We also show the results obtained by the same method, on the critical exponents of the diluted system in the zero and low magnetic field.
Phase transitions and critical characteristics of the layered antiferromagnetic Ising model in the case of a cubic lattice with next-nearest-neighbor intralayer interactions are studied in the framework of the Monte Carlo method implementing the replica algorithm. The characteristics of the phase transitions are analyzed within the histogram method and with the Binder cumulants. For the model under study, it is found that the transition from the superantiferromagnetic phase to the paramagnetic one is a second order phase transition. The static critical exponents for the specific heat α, susceptibility γ, order parameter β, correlation radius ν, and the Fisher exponent η are calculated using the finite-size scaling theory. It is shown that the three-dimensional Ising model for the cubic lattice with next-nearest-neighbor interactions belongs to the same universality class of critical behavior as the completely frustrated three-dimensional Ising model.
The critical properties of the antiferromagnetic layered Ising model on a cubic lattice with regard to the nearest-neighbor and next-nearest-neighbor interactions are investigated by the Monte Carlo method using the replica algorithm. The investigations are carried out for the ratios of exchange nearest-neighbor and next-nearest-neighbor interactions r = J 2/J 1 in the range of 0 ≤ r ≤ 1.0. Using the finite-size scaling theory, the static critical indices of specific heat α, order parameter β, susceptibility γ, correlation radius ν, and Fisher index η are calculated. It is shown that the universality class of the critical behavior of this model is retained in the range of 0 ≤ r ≤ 0.4. It is established that the change in the next-nearest-neighbor interaction value in this model in the range of r > 0.8 leads to the same universality class as the three-dimensional fully frustrated Ising model on the cubic lattice.
Critical properties of the Ising model on a stacked triangular lattice, with antiferromagnetic first- and second-neighbor in-plane interactions, are studied by extensive histogram Monte Carlo simulations. The results, in conjunction with the recently determined phase diagram, strongly suggest that the transition from the period-3 ordered state to the paramagnetic phase remains in the XY universality class. This conclusion is in contrast with a previous suggestion of mean-field tricritical behavior.
Phase transitions and critical properties of the frustrated Ising model on a cubic lattice with next-nearest-neighbor intralayer interactions are investigated by the replica Monte Carlo method. Estimations are made for the magnitude relation of the next-nearest-neighbor and nearest-neighbor exchange interactions r = J2/J1 in the value ranges of r ∈ [0.0, 1.0]. The phase diagram of the dependence of critical temperature on the next-nearest-neighbor interaction has been plotted. The static critical exponents of the heat capacity, the susceptibility, the ordering parameter and the correlation length as well as the Fisher exponent are calculated by means of the finite-size scaling theory. The universality class of the critical behavior of this model is revealed to remain within the limits of values r ∈ [0.0, 0.4]. It is found that the change in the next-nearest-neighbor interaction value in the range of r ∈ [0.9, 1.0] leads to another critical behavior.
The critical relaxation from the low-temperature ordered state of the three-dimensional Ising model with competing interactions on a simple cubic lattice has been studied for the first time using the short-time dynamics method. Competition between exchange interactions is due to the ferromagnetic interaction between the nearest neighbors and the antiferromagnetic interaction between the next nearest neighbors. Particles containing 262144 spins with periodic boundary conditions have been studied. Calculations have been performed by the standard Metropolis Monte Carlo algorithm. The static critical exponents of the magnetization and correlation radius have been calculated. The dynamic critical exponent of the model under study has been calculated for the first time.
In this review, we show our results with new interpretation on the critical exponents of thin films obtained by high-performance multi-histogram Monte Carlo simulations. The film thickness
Nz consists of a few layers up to a dozen of layers in the
z direction. The free boundary condition is applied in this direction while in the
xy plane periodic boundary conditions are used. Large
xy plane sizes are used for finite-size scaling. The Ising model is studied with nearest-neighbor (NN) interaction. When
Nz=1, namely the two-dimensional (2D) system, we find the critical exponents given by the renormalization group. While, for
Nz>1, the critical exponents calculated with the high-precision multi-histogram technique show that they deviate slightly but systematically from the 2D values. If we use these values of critical exponents in the hyperscaling relation with
d=2, then the hyperscaling relation is violated. However, if we use the hyperscaling relation and the critical exponents obtained for
Nz>1 to calculate the dimension of the system, we find the system dimension slightly larger than 2. This can be viewed as an "effective" dimension. More discussion is given in the paper. We also show the cross-over between the first- and second-order transition while varying the film thickness in an antiferromagnetic FCC Ising frustrated thin film. In addition, we will show evidence that when a 2D system has two order parameters of different symmetries with a single transition, the critical exponents are new, suggesting a universality class of coupled two-symmetry breakings. In this case, the 2D hyperscaling does not hold. Another case is the 3D Ising model coupled to the lattice vibration: the critical exponents deviate from the 3D Ising ones, the results suggest the violation of the hyperscaling.