Spontaneous magnetization of the Ising model on a layered square lattice

We have studied the Ising model on a layered square lattice with four different coupling constants and two different magnetic moments. The partition function at zero magnetic field is derived exactly. We propose a formula for the spontaneous magnetization which agrees with the exact low-temperature series expansion up to the 16th order and reduces to the exact result of Au-Yang and McCoy in a special case.     

Absence of a spontaneous long-range order in a mixed spin-(1/2, 3/2) Ising model on a decorated square lattice due to anomalous spin frustration driven by a magnetoelastic coupling

The mixed spin-(1/2, 3/2) Ising model on a decorated square lattice, which takes into account lattice vibrations of the spin-3/2 decorating magnetic ions at a quantum-mechanical level under the assumption of a perfect lattice rigidity of the spin-1/2 nodal magnetic ions, is examined via an exact mapping correspondence with the effective spin-1/2 Ising model on a square lattice. Although the considered magnetic structure is in principle unfrustrated due to bipartite nature of a decorated square lattice, the model under investigation may display anomalous spin frustration driven by a magnetoelastic coupling. It turns out that the magnetoelastic coupling is a primary cause for existence of the frustrated antiferromagnetic phases, which exhibit a peculiar coexistence of antiferromagnetic long-range order of the nodal spins with a partial disorder of the decorating spins with possible reentrant critical behavior. Under certain conditions, the anomalous spin frustration caused by the magnetoelastic coupling is responsible for unprecedented absence of spontaneous long-range order in the mixed-spin Ising model composed from half-odd-integer spins only.     

Ising model on mixed two-dimensional lattices

We inform results on physical and topological magnitudes related to the ground level of Ising model on mixed two-dimensional lattices of coordination numbers 4 (Kagomé lattices) and 5 (five-point star lattices). We consider little clusters of size N, where N represents the total number of spins, subject to periodic boundary conditions. On these systems we randomly distribute ±J nearest-neighbor interactions (+J: antiferromagnetic, −J: ferromagnetic (F)). Concentration x of F interactions is varied in the interval (0,1). Two different methods are used to obtain results reported here. First, a numerical method related to multiple replicas. Second, an analytical method based on probabilistic analysis of flat and curved plaquettes. Both methods are complementary to each other. Initially, this study is restricted to calculate frustration of plaquettes and bonds, energy and bond order parameter at T=0. The results of magnitudes informed here are compared with the similar ones obtained for honeycomb, square and triangular lattices.     

The transverse spin-1 Ising model with random interactions

The phase diagrams of the transverse spin-1 Ising model with random interactions are investigated using a new technique in the effective field theory that employs a probability distribution within the framework of the single-site cluster theory based on the use of exact Ising spin identities. A model is adopted in which the nearest-neighbor exchange couplings are independent random variables distributed according to the law P(J_ij)=pδ(J_ij−J)+(1−p)δ(J_ij−αJ). General formulae, applicable to lattices with coordination number N, are given. Numerical results are presented for a simple cubic lattice. The possible reentrant phenomenon displayed by the system due to the competitive effects between exchange interactions occurs for the appropriate range of the parameter α.     

Spin-1/2 Ising model on a AFM/FM two-layer Bethe lattice in a staggered magnetic field

A bilayer spin-1/2 Ising model consisting of two superposed Bethe lattices with antiferromagnetic/ferromagnetic interactions is studied by the use of exact recursion relations in a pairwise approach in the presence of an external staggered magnetic field. Besides the ground state phase diagrams calculated in different possible planes of the model parameters space, the thermal variations of the order-parameters and the free energy are investigated to obtain the temperature-dependent phase diagrams of the model for different values of the coordination numbers q. Our calculations reveal that depending on the strength of the model parameters, the model exhibits a variety of interesting phase transitions and therefore phase diagrams.     

The effective-field theory studies of critical phenomena in a mixed spin-1 and spin-2 Ising model on honeycomb and square lattices

An effective-field theory with correlations has been used to study critical behaviors of a mixed spin-1 and spin-2 Ising system on a honeycomb and square lattices in the absence and presence of a longitudinal magnetic field. The ground-state phase diagram of the model is obtained in the longitudinal magnetic field (h) and a single-ion potential or crystal-field interaction (Δ) plane. The thermal behavior of the sublattice magnetizations of the system are investigated to characterize the nature of (continuous and discontinuous) of the phase transitions and obtain the phase transition temperature. The phase diagrams are presented in the (Δ/|J|, k_BT/|J|) plane. The susceptibility, internal energy and specific heat of the system are numerically examined and some interesting phenomena in these quantities are found due to the absence and presence of the applied longitudinal magnetic field. Moreover, the system undergoes second- and first-order phase transition; hence, the system gives a tricritical point. The system also exhibits reentrant behavior.     

Frustration and quantum phase transitions in an anisotropic antiferromagnet on a square lattice

We study the effect of frustration between nearest and next-nearest neighbors of the quantum S=1 anisotropic Heisenberg model on a square lattice using the bond operator technique. A single-site anisotropy term induces a quantum phase transition in the system. We calculate the effect of zero-temperature quantum fluctuations on the magnetization for the Néel and collinear antiferromagnetic phases.     

Phase diagram of a diluted triangular lattice Ising antiferromagnet in a field

Magnetization processes and phase transitions in a geometrically frustrated triangular lattice Ising antiferromagnet in the presence of an external magnetic field and a random site dilution are studied by the use of an effective-field theory with correlations. We find that the interplay between the applied field and the frustration-relieving dilution results in peculiar phase diagrams in the temperature-field-dilution parameter space.     

Study of Ising model on the rectangular-triangular lattice

The Ising model is studied on a new type of lattice which is named the rectangular-triangular lattice. The critical temperature for the ferromagnetic lattice is calculated exactly and it is shown that the antiferromagnetic lattice does not order at any temperature. Ground state properties are investigated and some features of frustration on the antiferromagnetic Ising lattice outlined.     

Susceptibility of the Kagomé lattice Ising model

We explicitly calculate the zero-field magnetic susceptibility of the anisotropic Kagomé lattice Ising model on two different varieties of the parameter space. One of them is the limitH=0 of the solubility condition, obtained in a previous paper by Giacomini, for the model with magnetic field. The other one is the disorder variety of the model, for which a dimensional reduction occurs. These varieties do not contain any nontrivial critical behavior of the model. A functional relation is also established, which relates the zero-field susceptibility for ferromagnetic and competing interactions.     

Mixed spin-1 and spin-3/2 Ising system with two alternative layers of a honeycomb lattice within the effective-field theory

An effective-field theory with correlations is developed for a mixed spin-1 and spin-3/2 Ising system with two alternative layers of a honeycomb lattice. Spin-1 atoms and spin-3/2 atoms are distributed in alternative layers of a honeycomb lattice. We consider that the nearest-neighbor spins of each layer are coupled ferromagnetically and the interaction between the vertically aligned spins and adjacent spins are coupled either ferromagnetically or antiferromagnetically depending on the sign of the bilinear exchange interactions. We investigate the temperature dependence of the total magnetization to find the compensation points and to determine the type of compensation behavior. We present the phase diagrams in different planes for h=0, and the phase diagrams contain the paramagnetic, nonmagnetic and ferrimagnetic phases. The system also presents a tricritical behavior besides multicritical point (A), isolated critical point (C) and double critical end point (B) depending on the interaction parameters.     

The random-field Ising model with asymmetric bimodal probability distribution

The Ising model, in the presence of a random field, is investigated within the mean-field approximation based on Landau expansion. The random field is drawn from the bimodal probability distribution P(h)=pδ(h−h₀)+(1−p)δ(h+h₀), where the probability p assumes any value within the interval [0,1], asymmetric distribution. The prevailing transitions are of second-order but, for some values of p and h₀, first-order phase transitions take place for smaller temperatures and higher h₀, thus confirming the existence of a tricritical point. Also, the possible reentrant phenomena in the phase diagram (T−h₀ plane) occur for appropriate values of p and h₀. Using the variational principle, we determine the equilibrium equation for magnetization and solve it for both transitions and at the tricritical point in order to determine the magnetization profile with respect to h₀.     

On the Ising model for the simple cubic lattice

The Ising model was introduced in 1920 to describe a uniaxial system of magnetic moments, localized on a lattice, interacting via nearest-neighbour exchange interaction. It is the generic model for a continuous phase transition and arguably the most studied model in theoretical physics. Since it was solved for a two-dimensional lattice by Onsager in 1944, thereby representing one of the very few exactly solvable models in dimensions higher than one, it has served as a testing ground for new developments in analytic treatment and numerical algorithms. Only series expansions and numerical approaches, such as Monte Carlo simulations, are available in three dimensions. This review focuses on Monte Carlo simulation. We build upon a data set of unprecedented size. A great number of quantities of the model are estimated near the critical coupling. We present both a conventional analysis and an analysis in terms of a Puiseux series for the critical exponents. The former gives distinct values of the high- and low-temperature exponents; by means of the latter we can get these exponents to be equal at the cost of having true asymptotic behaviour being found only extremely close to the critical point. The consequences of this for simulations of lattice systems are discussed at length.     

The anti-ferromagnetic Ising model on the simplest pure Husimi lattice: An exact solution

The anti-ferromagnetic spin-1/2 Ising model on the pure Husimi lattice with three sites in the elementary polygon (p=3$p=3$) and the coordination number z=4$z=4$ is investigated. It represents the simplest approximation of the anti-ferromagnetic Ising model on the two-dimensional kagome lattice which takes into account effects of frustration. The exact analytical solution of the model is found and discussed. It is proven that the model does not exhibit the first order as well as the second order phase transitions. A detailed analysis of the magnetization properties is performed and the existence of the magnetization plateaus for low temperatures is shown. All possible ground states of the model are found and discussed.     

Critical behaviour of the ferromagnetic Ising model on a triangular lattice

We use a new updated algorithm scheme to investigate the critical behaviour of the two-dimensional ferromagnetic Ising model on a triangular lattice with the nearest neighbour interactions. The transition is examined by generating accurate data for lattices with L= 8, 10, 12, 15, 20, 25, 30, 40 and 50. The updated spin algorithm we employ has the advantages of both a Metropolis algorithm and a single-update method. Our study indicates that the transition is continuous at T_c= 3.6403({2}). A convincing finite-size scaling analysis of the model yields υ=0.9995(21), β / υ = 0.12400({17}), γ / υ = 1.75223(22), γ '/υ=1.7555(22), α/υ= 0.00077(420) (scaling) and α / υ = 0.0010(42) (hyperscaling). The present scheme yields more accurate estimates for all the critical exponents than the Monte Carlo method, and our estimates are shown to be in excellent agreement with their predicted values.     

The density of states for an antiferromagnetic Ising model on a triangular lattice

The Wang-Landau algorithm is an efficient Monte Carlo approach to the density of states of a statistical mechanics system. The estimation of state density would allow the computation of thermodynamic properties of the system over the whole temperature range. We apply this sampling method to study the phase transitions in a triangular Ising model. The entropy of the lattice at zero temperature as well as other thermodynamic properties is computed. The calculated thermodynamic properties are explained in the context of the magnetic phase transition.