Intensive numerical studies of exact ground states of the two-dimensional ferromagnetic random field Ising model at
T=0, with a Gaussian distribution of fields, are presented. Standard finite size scaling analysis of the data suggests the existence of a transition at ${\ensuremath{\sigma}}_{c}=0.64\ifmmode\pm\else\textpm\fi{}0.08.$ Results are compared with existing theories and with the study of metastable avalanches in the same model.
It is often assumed that for treating numerical (or experimental) data on continuous transitions the formal analysis derived from the renormalization-group theory can only be applied over a narrow temperature range, the ``critical region''; outside this region correction terms proliferate rendering attempts to apply the formalism hopeless. This pessimistic conclusion follows largely from a choice of scaling variables and scaling expressions, which is traditional but very inefficient for data covering wide temperature ranges. An alternative ``extended scaling'' approach can be made where the choice of scaling variables and scaling expressions is rationalized in the light of well established high-temperature series expansion developments. We present the extended scaling approach in detail, and outline the numerical technique used to study the three-dimensional (
3D) Ising model. After a discussion of the exact expressions for the historic
1D Ising spin chain model as an illustration, an exhaustive analysis of high quality numerical data on the canonical simple cubic lattice
3D Ising model is given. It is shown that in both models, with appropriate scaling variables and scaling expressions (in which leading correction terms are taken into account where necessary), critical behavior extends from
Tc up to infinite temperature.
In this paper, we theoretically study the critical properties of the classical spin-1 Ising model using two approaches: 1) the analytical low-temperature series expansion and 2) the numerical Metropolis Monte Carlo technique. Within this analysis, we discuss the critical behavior of one-, two- and three-dimensional systems modeled by the first-neighbor spin-1 Ising model for different types of exchange interactions. The comparison of the results obtained according the Metropolis Monte Carlo simulations allows us to highlight the limits of the widely used mean-field theory approach. We also show, via a simple transformation, that for the special case where the bilinear and bicubic terms are set equal to zero in the Hamiltonian the partition function of the spin-1 Ising model can be reduced to that of the spin-1/2 Ising model with temperature dependent external field and temperature independent exchange interaction times an exponential factor depending on the other terms of the Hamiltonian and confirm this result numerically by using the Metropolis Monte Carlo simulation. Finally, we investigate the dependence of the critical temperature on the strength of long-range interactions included in the Ising Hamiltonian comparing it with that of the first-neighbor spin-1/2 Ising model.