Microscopic origin of Ising model

Question

A typical Hamiltonian for Ising model is $$ H=-\sum_{i,j} J_{ij}S_iS_j - K \sum_i S_i.$$ In many references we can find exact solutions for special cases, mean-field approach, phase transition, and many others. However, I cannot find how valid is this model in real physical materials, and what is the microscopic origin of the phenomenological constants $J$ and $K$.

1. What is the microscopic origin of this Hamiltonian? Can this Hamiltonian be obtained by a more systematic approximation of a more general Hamiltonian that describes magnetic systems?

2. For what type of material this model is valid? Since there are enormous types of interactions that can occur in solids and Ising model features only some of them, in what cases the "Ising-interaction" is dominant than others?

Answer 1

Although it is often introduced as a model of ferromagnetism, the Ising model is a pretty poor model of this phenomenon, due to the many severe approximations made.

Its main value, however, lies elsewhere: it is a very simple (in a sense, the simplest) model of cooperative phenomena, which makes it ideally suited for testing all kinds of approximation techniques, as well as deriving a detailed description of its properties without making additional assumptions. Its role in the development of the theory of critical phenomena was crucial, for example, and it is the topic of countless papers in mathematical physics, thanks to its relative tractability.

That being said, there are magnetic systems that are well described by an Ising model; see this paper for several examples.

In addition, it is a reasonable model of the liquid/gas phase transition (for example, the critical exponents of the 3d model are quite close to those associated to such a phase transition). It is an even better model for adsorption, as well as for binary mixtures.

The fact that it provides an (albeit crude) description of many different physical systems (as well as systems away from physics) are an additional benefit of its simplicity.