# From where does the Ising Hamiltonian come?

So in my Stat Mech course, we were introduced to the classical Ising Model: $$H = -J\Sigma _{}S_iS_j - K\Sigma_i S_i$$ But from where does this come from? Is there any rationale behind this?
I want to know the origins of especially the first term (with J).
I guess since this has magnetic moments in the formula, is it some kind of dipole-dipole interaction?

A short answer is: Ising model comes from Ising. I realize however that the question has deeper meaning in it. There are two ways to look at it:

1. Thinking of the Ising model as a model approximating some real-world materials (this is suggested by the question). To this I have to say that it does not model dipole-dipole interactions, which rarely lead to (anti-)ferromagnetism, but rather the exchange interactions, such as RKKY.
2. Thinking of the Ising model as a model Hamiltonian which actually describes many critical phenomena in the same universality class, which may not have anything to do with (anti-)ferromagnetism, e.g., phase transitions in liquids. It is a very special and stunning feature of the critical phenomena, that close to a critical point they can be described disregarding many particularities of their macroscopic structure and microscopic interactions. This however goes well beyond the first course in statistical physics.

Finally, the Ising model has important historical and theoretical role as the first model capable of describing a phase transition, and as a model that can be solved exactly in one and two dimensions (there is no finite temperature phase transition in one dimension, whereas Lars Onsager was awarded a Nobel prize for solving the Ising model in 2D).

Update
@YvanVelenik has correctly pointed that Onsager was awarded the Nobel Prize for the Onsager identities, rather than for solving the Ising model.

• – Yvan Velenik Sep 10 '20 at 9:43

The microscopic Hamiltonians are (usually) postulated on the basis of intuition and general requirements. For example, The $$J$$ term can be regarded as the easiest way of including interactions that have symmetry under the rotation of all the spins.

It is possible to add more contributions that could be more representative of a particular physical system. However, the J-term is the minimum "ingredient" needed to explore the bast phenomenology of interacting systems with critical points (divergent correlation lengths, symmetry breaking, universality,...).