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Soviet Journal of Low Temperature Physics

Phase transition in a layered Ising model

A. I. Banshchikov, Yu. N. Mitsaı̆, N. V. Shakhova

Published Apr 1, 1981• 0 citations
Publisher

Abstract

The magnetization and magnetic susceptibility are calculated for a layered system in which it is assumed that the interaction between spins in a plane is long-range and only neighboring planes interact. The calculations are carried out to first order in the square of the inverse interaction radius in a plane.
DOI
10.1063/10.0030500

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The position of the phase transition in the two dimensional Ising model were determined byusing Monte Carlo simulation in a quadratic for area of variable length with external magnetic fieldswitched off (B = 0). The magnetization (M) per site (µ), magnetic susceptibility (x) of aferromagnetic and paramagnetic materials were calculated as a function of temperature T for(20×20,40×40,60×60), (80×80,120×120,200×200) spin lattice interactions. Nearest neighborinteraction is assumed (i.e. each spin has 4 neighbors); uses periodic boundary conditions. The Curietemperature (Tc = 2.27 J/kB ) is determined by measuring the magnetic susceptibility at which theferromagnetic and paramagnetic undergoes a phase change from order to disorder. There is thus aphase transition defined by the Curie temperature. The Monte Carlo method were used to check theseresults and to confirm the phase transition. The data are analyzed using the Curie-Weiss law whichcontains the Curie temperature as a parameter.
Thermodynamics of a layered Ising model with infinite-range ferromagnetic intralayer interaction and random nearest-neighbor interlayer coupling is considered. A detailed analysis of the model with vanishing average interlayer coupling is presented. The Gibbs free energy is found in the critical region, and the existence of many metastable states is demonstrated. Thermodynamic parameters of the system are found for periodic states. As the mean square interlayer coupling increases, the equilibrium state of the system undergoes an infinite sequence of first-order phase transitions, the number of magnetic planes and the distance between them change discontinuously, and so do both bulk magnetization and magnetic susceptibility.
A bilayer Ising model consisting of two Bethe lattices, each of which is coupled with crystal fields of different strengths and each with a branching ratio of q Ising spins with one of the layers having only spin-1 and the other having only spin-3/2, is laid over the top of the other and the two layers are tied together via an interaction between the vertically aligned spins. After obtaining the ground-state (GS) phase diagrams on different possible planes depending on the given system parameters, the changes in the order-parameters and the free energy are investigated by use of the exact recursion relations in a pairwise approach to calculate the phase diagrams of the model. The ferromagnetic ordering in each of the layers and ferromagnetic or antiferromagnetic ordering of the adjacent nearest-neighbor (NN) spins of the layers are considered. The system presents both second- and first-order phase transitions. The lines of the first-order phase transitions end on either the stable or unstable tricritical points or at the isolated critical points. The model also displays one or two compensation temperatures when the bilinear interaction of the upper layer with spin-1 can compete with that of the lower layers with spin-3/2.
The phase diagram of the Ising square lattice with first- and second-nearest-neighbor antiferromagnetic interactions (J1{\mathit{J}}_{1}J1​ and J2{\mathit{J}}_{2}J2​) is calculated within the framework of the cluster-variation method. The Hamiltonian is solved within the nine-point-cluster approximation. In contrast to previous studies we find that the phase transition from the superantiferromagnetic to the paramagnetic state is of first order in the range 0.5R=J1{\mathit{J}}_{1}J1​/${\mathit{J}}_{2}$1.144. The non-Ising critical behavior observed in that range of interactions might be due to that phenomenon.
The surface scaling theory previously presented by the authors is developed further, and derived heuristically from a cluster model. Monte Carlo calculations are carried out to obtain the spatial and temperature dependence of the magnetization in Ising and Heisenberg systems with free surfaces. The exponent ${\ensuremath{\beta}}_{1}$ of the (surface) layer magnetization is shown to agree with the scaling value (${\ensuremath{\beta}}_{1}\ensuremath{\approx}\frac{2}{3}$) previously derived. In the Heisenberg system, the results at low temperature agree with a spin-wave calculation by Mills and Maradudin. Ising models with modified exchange ${J}_{s}=J(1+\ensuremath{\Delta})\ensuremath{\ne}J$ on the surface are considered, both in mean-field theory and by means of high-temperature-series expansions. The critical value ${\ensuremath{\Delta}}_{c}$ for surface ordering is found from the series to be 0.6, compared to the mean-field value of 0.25. For $\ensuremath{\Delta}>{\ensuremath{\Delta}}_{c}$ there is a temperature region in which the surface behaves like a bulk two-dimensional Ising model near its phase transition. The critical exponents experience a crossover at $\ensuremath{\Delta}={\ensuremath{\Delta}}_{c}$, which is reflected in poorly behaved series, and effective exponents differing from the true ones for $\ensuremath{\Delta}\ensuremath{\lesssim}{\ensuremath{\Delta}}_{c}$. In the case of weakened surface exchange (0<Js<J0<{J}_{s}<J0<Js​<J), the layer magnetization is shown to fit a linear temperature dependence over a large temperature range below Tc{T}_{c}Tc​, thus providing a possible explanation for previous experiments. For sufficiently strong negative Js{J}_{s}Js​, mean-field theory predicts that the surface will order antiferromagnetically while the bulk is ferromagnetic.
The critical behaviour of the square-lattice Ising model, with nearest and next-nearest neighbour interactions of either sign, has been investigated by means of high-temperature series. lhe location of the critical lines in the coupling constant plane has been accurately determined. Along the critical line which corresponds to transitions to the layered or superantiferromagnetic state a breakdown of universality is observed and explicit numerical estimates obtained for the exponent of the ordering susceptibility.