An explanation of the phenomenon of polytypism

Based on the analogy between polytypes and spin-half Ising chains, polytypes can be considered as different phases of a spin-half Ising system with competing nearest neighbour and next nearest neighbour interactions operating in a single direction. It is known that such an Ising system exhibits extremely rich and complicated phase behaviour. This behaviour is shown to be very similar to the phase behaviour exhibited by polytypes.     

Monte carlo calculations near a uniaxial Lifshitz point

The Monte Carlo method is applied to a threedimensional Ising model with nearest neighbour ferromagnetic interactions and next nearest neighbour antiferromagnetic interactions along one axis only. Special emphasis is given to the critical behaviour near the Lifshitz point.Some parts of this paper have been presented at the 13th International Conference on Statistical Physics, Haifa, August 1977     

Order disorder transitions in Ising models in transverse fields with second neighbour interactions

The (ferromagnetic) order-disorder transitions in a class of Ising models with second neighbour interaction in transverse fields is studied using the path integral method. Within the limitations of the method, the critical fields at zero temperature are estimated for different systems.     

The ground state and its entropy for a class of one-dimensional classical lattice systems

We calculate explicitly the zero-temperature limit of the finite temperature equilibrium state and its entropy for a class of one-dimensional Ising models with an interaction of range n. For that purpose we extend the usual transfer matrix method to ground states. The interaction consists of a ferromagnetic nearest neighbour term and a competing antiferromagnetic interaction with the nth neighbour.     

Additional Long-Range Interactions on Diamond-Type Hierarchical Lattices

The Ising model with additional next-nearest neighbour (n.n.n.) interactions in generator of diamond-type hierarchical lattice is studied. It is found that phase transition occurs only when the nearest neighbour (n.n.) interaction K₁ and n.n.n. interaction K₂ fulfils a certain relation and the universal class of phase transition is not changed by the long-range interactions.     

Layered Ising models on triangular and honeycomb lattices

We study inhomogeneous Ising models on triangular and honeycomb lattices. The nearest neighbour couplings can have arbitrary strength and sign such that the coupling distribution is translationally invariant in the direction of one lattice axis, i.e. the models have a layered structure. By using a transfer matrix method we derive closed form expressions for the partition functions and free energies. The critical temperatures are calculated. Phase transitions at a finite critical temperature are universally of Ising type. Models with no phase transition may show different behaviour atT=0, which is explicitly shown for fully frustrated models on square, triangular and honeycomb lattices. Finally, generalizations to layered Ising models on more general lattices are discussed.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köln     

Non-universality for ising-like spin systems

Renormalisation calculations are carried out which reasonably reproduce the interaction-dependent critical point exponents of the Baxter model. Similar calculations indicate non-universal behaviour of a nearest and next-nearest neighbour pair-interaction Ising model in two dimensions.     

Multicritical behaviour of the antiferromagnetic spin- Blume-Capel model

The multicritical behaviour of the spin- Ising model on the square lattice with nearest-neighbour antiferromagnetic exchange interaction (J<0), a crystal-field interaction (D) and an external magnetic field (H) is studied within mean field approximation (M.F.A.). The phase diagram exhibits a rich variety of behaviour: second order, first order and critical points of different order.     

Non-Equilibrium Behaviour in Unidirectionally Coupled Map Lattices

A spatially one dimensional coupled map lattice with a local and unidirectional coupling is introduced. This model is studied analytically by a perturbation theory that is valid for small coupling strength. In parameter space three phases with different ergodic behaviour are observed. Via coarse graining the deterministic model is mapped to a stochastic spin model that can be described by a master equation. Because of the anisotropic coupling non-equilibrium behaviour is found on the coarse grained level. However, the stationary statistical properties of the spin dynamics can still be described with a nearest neighbour Ising model whereby the ordering is predominantly antiferromagnetic.     

Biquadratic exchange in the diluted Ising ferromagnet

Effects of biquadratic exchange on Curie temperature and specific heat have been studied in the light of a new effective field theory for the Ising Spin System. The nearest neighbour interaction is taken into account.     

Monte-Carlo phase diagram of an Ising system with isotropic competing interaction

An Ising model is studied here with isotropic nearest and next nearest neighbour competing interaction along the three axes of a simple cubic system. The phase diagram for this model is determined by Monte-Carlo simulation through a study of the correlation functions.     

Competing Ising interactions and chaotic glass-like behaviour on a Cayley tree

For the Ising model on a Cayley tree with competing nearest neighbour coupling J and next nearest neighbour coupling J′, we find in addition to the expected paramagnetic, ferromagnetic and antiferromagnetic phases, an intermediate range of J′/J < 0 values where the local magnetization has chaotic oscillatory glass-like behaviour.     

Ground-state properties of a 2-dimensional Ising-system with competing interactions

We analyze ground-state properties of a 2-dimensional spin –1/2 Ising System on a square lattice with interactions up to the third-next neighbour in the presence of a homogeneous magnetic field. The interactions may be competing or not. Our analysis is complete in the sense that we can give the ground-state energy for all combinations of the interaction parameters and magnetic field, i.e. we can calculate the magnetization and two-spin-correlation functions up to third-next neighbour.     

Toom’s Model with Glauber Rates: An Exact Solution Using Elementary Methods

A simple spatially two-dimensional stochastic cellular automaton with asymmetric coupling and synchronous updating according to Glauber rates is considered. While detailed balance is violated it is still possible to compute analytically the stationary probability distribution by elementary means. The stationary distribution can be written as a canonical equilibrium distribution of a spin system on a triangular lattice with nearest neighbour coupling. Thus, the cellular automaton shows a nonequilibrium phase transition with Ising critical behaviour.     

Anomalous Universality in the Anisotropic Ashkin–Teller Model

The Ashkin–Teller (AT) model is a generalization of Ising 2–d to a four states spin model; it can be written in the form of two Ising layers (in general with different couplings) interacting via a four–spin interaction. It was conjectured long ago (by Kadanoff and Wegner, Wu and Lin, Baxter and others) that AT has in general two critical points, and that universality holds, in the sense that the critical exponents are the same as in the Ising model, except when the couplings of the two Ising layers are equal (isotropic case). We obtain an explicit expression for the specific heat from which we prove this conjecture in the weakly interacting case and we locate precisely the critical points. We find the somewhat unexpected feature that, despite universality, holds for the specific heat, nevertheless nonuniversal critical indexes appear: for instance the distance between the critical points rescale with an anomalous exponent as we let the couplings of the two Ising layers coincide (isotropic limit); and so does the constant in front of the logarithm in the specific heat. Our result also explains how the crossover from universal to nonuniversal behaviour is realized.Partially supported by NSF Grant DMR 01–279–26     

Phase transition in the Blume-Capel model with second neighbour interaction

A two dimensional antiferromagnetic spin-1 Ising model with negative next- nearest neighbour interaction (J ₂ <0) and under an external magnetic field is investigated by two methods: The mean-field theory and Finite-Size-Scaling based on transfer matrix (TMFSS) calculations. The ground state diagrams exhibit several new phases including frustrated ones. At finite temperature we obtain by these two methods quite rich phase diagrams, with several multicritical points. While Mean field approximation yields phase diagrams which are sometimes even qualitatively incorrect, accurate results are obtained from transfer matrix finite size scaling calculations. For a certain range of interaction parameters, the model is shown to violate the ordinary universality hypothesis. Received: 3 November 1997 / Revised: 31 March 1998 / Accepted: 7 April 1998     

Monte Carlo study of the two-step spin transition in [FeXZn1∓X(2-pic)3]Cl2 · EtOH

The Monte Carlo method has been applied to reproduce the two-step spin transition in [Fe_xZn_1?x(2-pic)₃]Cl₂ · EtOH (2-pic = 2-picolylamine). The partition function of the Hamiltonian has been factorized in two parts describing the inner degrees of freedom of the spin transition molecules and the interaction between them. The factorization is equivalent to an effective Hamiltonian which can be mapped on an Ising system in an applied magnetic field. The interaction consists of an infinite range and a short range interaction part. Qualitatively the two-step transition under pressure and the Zn diluted system have been well reproduced with infinite range and the next nearest neighbour interaction of ferromagnetic type and the nearest neighbour interaction of antiferromagnetic type in a simple cubic lattice.     

Monte Carlo study of an Ising gauge system

The phase diagram of the 3-d Ising gauge model with additional ferromagnetic nearest neighbour Ising coupling is explored by Monte Carlo simulations.     

The density dependence of the diffusion constant in interacting lattice gases: Application to surface diffusion of 0 onW(110)

The influence of interactions between adsorbed particles on their diffusion constant is investigated by kinetic Ising models with independent nearest neighbour hops. This leads to expressions for the ratio of the diffusion constant at arbitrary coverage to its value at =0 as a function of the interaction energies relative to temperature. It is shown that under certain conditions this quantity obeys a particle-hole symmetry. Exact results in the whole range of densities are given in one dimension for nearest neighbour interaction. They already yield a qualitative agreement with experimental results and are also compared to corresponding numerical simulations. The introduction of a next nearest neighbour interaction is shown to produce drastic changes in the density dependence of the diffusion constant in some of the cases. A generalized quasichemical approximation and a virial expansion are made in two dimensions, leading to a better agreement with the measurements.     

Diagrammatic decompositions on lattice for correlation functions of Ising spin models by iterative method

In the present paper the iterative method enables us to calculate correlation functions of Ising spin models by two approximate ways. The equations resulting from two models A and B, based on different physical considerations concerning correlations in surroundings of the reference spin, are solved for one, two, and three-dimensional Ising models with nearest neighbour interactions. Except of anomalies occuring in low-dimensional systems the model A leads to the critical pointA _c=0.191 for the three-dimensional cubic Ising model.