Two-layer Ising Model Research Articles

The two-layer Ising model on a sequence of diamond-like hierarchical lattices

The two-layer Ising model on diamond-like hierarchical lattices of different structures with equal fractal dimension was investigated. We applied the renormalization group transformations for the considered lattices. The critical values of the shift exponent φ were computed for various intralayer interaction values (J1 and J2). For the simplest diamond-like hierarchical lattice φ≈2.35 at J1=J2. As the structure of the lattice becomes more complex, the shift exponent value increases. At J1=0.5J2 the value φ≈0.5 was obtained, which is consistent with the data for the square lattice. The graphs of the heat capacity, magnetization and magnetic susceptibility were plotted at J1=0.5J2. They showed the absence of the second phase transition at an arbitrarily weak coupling between the layers.

Classical lattice models with single-node interactions on hierarchical lattices: The two-layer Ising model

A general approach is proposed for renormalization group transformations at arbitrary hierarchical lattices with two root nodes and the presence of single-node interactions (interactions between layers, magnetic field, chemical potential, etc.). The effectiveness of the proposed approach was shown for the two-layer Ising model in a zero magnetic field on the simplest representative of folded square hierarchical lattices. The phase diagram was investigated and the shift exponent (φ) was calculated at various values of the interaction energy in each layer (J1,J2) and between the layers (J3). The value φ≈ 2.41 was obtained for identical interactions in the layers (J1= J2). In the remaining cases (J1≠J2) the shift exponent turned out to be close to 0.5, which is consistent with the data for the square lattice. The exceptional case is J1 > 0, J2> 0, and J1 ≠J2, where the transition shift exponent in the second layer takes the value 2.57.

Investigation of phase diagrams for cylindrical Ising nanotube using cellular automata

Recent developments in the field of applied nanoscience and nanotechnology have heightened the need for categorizing various characteristics of nanostructures. In this regard, this paper establishes a novel method to investigate magnetic properties (phase diagram and spontaneous magnetization) of a cylindrical Ising nanotube. Using a two-layer Ising model and the core-shell concept, the interactions within nanotube has been modelled. In the model, both ferromagnetic and antiferromagnetic cases have been considered. Furthermore, the effect of nanotube's length on the critical temperature is investigated. The model has been simulated using cellular automata approach and phase diagrams were constructed for different values of inter- and intra-layer couplings. For the antiferromagnetic case, the possibility of existence of compensation point is observed.

Complexity of possibly gapped histogram and analysis of histogram.

We demonstrate that gaps and distributional patterns embedded within real-valued measurements are inseparable biological and mechanistic information contents of the system. Such patterns are discovered through data-driven possibly gapped histogram, which further leads to the geometry-based analysis of histogram (ANOHT). Constructing a possibly gapped histogram is a complex problem of statistical mechanics due to the ensemble of candidate histograms being captured by a two-layer Ising model. This construction is also a distinctive problem of Information Theory from the perspective of data compression via uniformity. By defining a Hamiltonian (or energy) as a sum of total coding lengths of boundaries and total decoding errors within bins, this issue of computing the minimum energy macroscopic states is surprisingly resolved by applying the hierarchical clustering algorithm. Thus, a possibly gapped histogram corresponds to a macro-state. And then the first phase of ANOHT is developed for simultaneous comparison of multiple treatments, while the second phase of ANOHT is developed based on classical empirical process theory for a tree-geometry that can check the authenticity of branches of the treatment tree. The well-known Iris data are used to illustrate our technical developments. Also, a large baseball pitching dataset and a heavily right-censored divorce data are analysed to showcase the existential gaps and utilities of ANOHT.

Calculation of critical properties for the anisotropic two-layer Ising model on the Kagome lattice: Cellular automata approach

The critical point (Kc) of the two-layer Ising model on the Kagome lattice has been calculated with a high precision, using the probabilistic cellular automata with the Glauber algorithm. The critical point is calculated for different values of the inter- and intra-layer couplings (K1≠K2≠K3≠Kz), where K1, K2 and K3 are the nearest-neighbor interactions within each layer in the 1, 2 and 3 directions, respectively, and Kz is the intralayer coupling. A general ansatz equation for the critical point is given as a function of the inter- and intra-layer interactions, ξ=K3/K1,σ=K2/K1 and ω=Kz/K1 for the one- and two-layer Ising models on the Kagome lattice.

Obtaining critical point and shift exponent for the anisotropic two-layer Ising and Potts models: Cellular automata approach

Using probabilistic cellular automata with the Glauber algorithm, we have precisely calculated the critical points for the anisotropic two-layer Ising and Potts models (Kx≠Ky≠Kz) of the square lattice, where Kx and Ky are the nearest-neighbor interactions within each layer in the x and y directions, respectively and Kz is the inter-layer coupling. A general equation is obtained as a function of the inter- and intra-layer interactions (ξ,σ) for both the two-layer Ising and Potts models, separately, where ξ=Kz/Kx and σ=Ky/Kx. Furthermore, the shift exponent for the two-layer Ising and Potts models is calculated. It was demonstrated that in the case of σ=1 for the two-layer Ising model, the value of ϕ=1.756(±0.0078) supports the scaling theories’ prediction that ϕ=γ. However, for the unequal intra-layer couplings for the two-layer Ising model and also in the case of both equal and unequal intra-layer interactions for the two-layer Potts model, our results are different from those obtained from the scaling theories. Finally, an equation is obtained for the shift exponent as a function of intra-layer couplings (σ) for the two-layer Ising and Potts models.

Phenomenological renormalization group approach to the anisotropic two-layer Ising model

The anisotropic two-layer Ising model is studied by the phenomenological renormalizaiton group method. It is found that the anisotropic two-layer Ising model with symmetric couplings belongs to the same universality class as the two dimensional Ising model.

Calculation of the Critical Temperature for the Anisotropic Two-Layer Ising Model Using the Transfer Matrix Method

A new finite-size scaling approach based on the transfer matrix method is developed to calculate the critical temperature of an anisotropic two-layer Ising ferromagnet on strips of r-wide sites of square lattices. The reduced internal energy per site has been accurately calculated for the ferromagnetic case, with the nearest-neighbor couplings Kx and Ky (where Kx and Ky are the nearest-neighbor interactions within each layer in the x and y directions, respectively) and with interlayer coupling Kz, using different size-limited lattices. The calculated energies for different lattice sizes intersect at various points when plotted versus the reduced temperature. It is found that the location of the intersection point versus the lattice size can be fitted to a power series in terms of the lattice sizes. The power series is used to obtain the critical temperature of the unlimited two-layer lattice. The results obtained are in good agreement with the accurate values reported by others.

Critical exponents of the two-layer Ising model

The symmetric two-layer Ising model (TLIM) is studied by the corner transfer matrix renormalisation group method. The critical points and critical exponents are calculated. It is found that the TLIM belongs to the same universality class as the Ising model. The shift exponent is calculated to be 1.773, which is consistent with the theoretical prediction 1.75 with 1.3% deviation.

SHORT-TIME CRITICAL DYNAMIC PROCESS OF TWO-LAYER ISING MODEL

Monte Carlo simulations of critical dynamics has been used to study the short-time process and scaling behavior of two-layer Ising model. The result shows that there are critical points even when the coupling between the two layers is not equal to zero; the short-time dynamic critical exponent θ was determined; the traditional critical exponent 1/νz was estimated also.Our result encouraged the supposition of the existence of the critical line and evidenced that this model was likely a weak universal model.

Extension of a new analytical method for locating critical temperatures

We investigate recently proposed method for locating critical temperatures and introduce some modifications which allow to formulate exact criterion for any self-dual model. We apply the modified method for the Ashkin-Teller model and show that the exact result for a critical temperature is reproduced. We test also a two-layer Ising model for the presence of eventual self-duality.

Shift exponent and breakdown of universality for the two-layer Ising model on a square lattice

By using a β-function for a spin- 3 2 Ising model, we investigate the critical phenomena of a general two-layer Ising model of square lattice. We find that the value of the shift exponent ψ is 1.73 ± 0.04 for the system with the same intra-layer interaction and 0.5 for the system with different intra-layer interactions. We also find an evidence that the value of critical exponent ν exceeds 1 in some region of interaction-parameter space, where the degeneracy of the ground state is fourfold.

Formula omitted] ising model and two-layer Ising model

We present a spin- 3 2 Ising model which is equivalent to a “two-layer” Ising model. We find a solvable spin- 3 2 Ising model and show that a system may have several critical exponents η corresponding to correlation functions of different Ising-type variables. We find two phase-transition temperatures in some of the systems and clarify the nature of phases. In a system with competing antiferromagnetic and ferromagnetic interactions on a triangular lattice, there are two critical temperatures and the value of η is 1 4 at a finite critical temperature and 1 2 at the critical zero temperature. We calculate the critical temperature as a function of ratio of interactions by using the interfacial approximation for a system with competing antiferromagnetic and ferromagnetic interactions on a square lattice; it turns out that the shift exponent is 0.5.