A high temperature expansion is obtained for the zero field susceptibility of a generalized Ising model, which can be used to describe the critical behaviour of various multicomponent systems. The first seven terms of the series are obtained for the face centred cubic lattice. Estimates of the critical temperature obtained from this series are considerably more precise than those obtained previously from the zero field free energy. Estimates are also obtained for the critical exponents of the susceptibility and the derivatives of the zero field free energy. The results are compared with the predictions of mean field theory.
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<J), the layer magnetization is shown to fit a linear temperature dependence over a large temperature range below
Tc, thus providing a possible explanation for previous experiments. For sufficiently strong negative
Js, mean-field theory predicts that the surface will order antiferromagnetically while the bulk is ferromagnetic.
In this study, we explored the magnetic coupling in a multilayer system consisting of thin layers separated by a distance
d. We have employed Monte Carlo (MC) simulations to calculate the thermodynamic quantities such as the magnetization per spin
mLμ, magnetic susceptibility
χLμ, and the reduced fourth-order Binder cumulant
ULμ as a function of temperature
T and for several values of lattice size
L. These quantities were obtained for each layer (
μ=l), for the bulk system (
μ=b), as a function of interlayer distance
d and the parameter
α that defines the scale length of the exponential decay of the interaction. Furthermore, we applied the finite-size scaling theory to calculate the critical exponents. Our results reveal that the system exhibits 2D Ising exponents when the layers are sufficiently separated. On the other hand, in the compact limit, where
d equals the distance between two adjacent sites within the same layer, our bulk results show that the system exhibits 3D Ising critical exponents, provided that the number of layers
Lz increases proportionally to the layer sizes. Even with the separation increasing up to
d=2.0, the layers are so correlated that the set of critical exponents retains the values of 3D critical exponents. However, when the number of layers
Lz remains fixed, even in the compact limit with periodic boundary conditions, only the exponent
β aligns closely with the predicted literature values, on the other hand, the other exponents show significant deviations.
We investigate the static and dynamic critical behaviour of a uniformly driven bilayer Ising lattice gas at half-filling. Depending on the strength of the inter-layer coupling J , phase separation occurs across or within the two layers. The former transitions are controlled by the universality class of model A (corresponding to an Ising model with Glauber dynamics), with upper critical dimension d c = 4. The latter transitions are dominated by the universality class of the standard (single-layer) driven Ising lattice gas, with d c = 5 and a nonclassical anisotropy exponent. These two distinct critical lines meet at a nonequilibrium bicritical point which also falls into the driven Ising class. At all transitions, novel couplings and dangerous irrelevant operators determine corrections to scaling.