# Physical origins of the Heisenberg model of ferromagnetism

I am trying to understand physical intuition behind the Ising and Heisenberg models (thus I am not sure if my question is appropriate for this mostly mathematical site). I will concentrate of the Heisenberg model.

In the Heisenberg model of ferromagnetism the atoms are assumed to be arranged in a lattice. To the $i$-th atom corresponds a spin operator $\vec S_i$ (here $i$ belongs to the lattice). The Hamiltonian is given by $H=-J\sum_{<ij>}\vec S_i\cdot\vec S_j$ where the sum runs over all pairs of atoms connected by an edge, $J>0$ is a constant.

As far as I understand from the literature that this form of the Hamiltonian comes from the exchange interaction which, in turn, comes from the fact that each (neutral) atom is a boson (see e.g. vol. IX of Landau-Lifshitz, paragraph 72).

QUESTION. In addition to exchange interaction there exists a different spin-spin interaction which has a different form. Why it was neglected?

The spin-spin interaction between two atoms with magnetic moments $\vec\mu_1,\vec\mu_2$ and large separation $\vec R$ is equal to $$\frac{\vec\mu_1\cdot\vec\mu_2}{R^3}-3\frac{(\vec\mu_1\cdot\vec R)(\vec\mu_2\cdot\vec R)}{R^5}.$$ After quantization, $\vec \mu_i$ is replaced by $\vec S_i$ times a constant.

As it is claimed in "Statistical mechanics" by R. Feynman, Section 7.1, the spin-spin interaction is very small in comparison to the exchange interaction. However this seems to be true only on short distances since the exchange interaction decays exponentially with the distance $R$ between the atoms, while the spin-spin decays as $1/R^3$. Hence when we are on 3d-lattice the sum over the lattice of terms of order $1/R^3$ diverges logarithmically. This cannot be neglected from the experimental point of view too. When the ferromagnetic material is magnetized, it creates a magnetic field, which will interact with another magnet or moving charge. This interaction has already nothing to do with the exchange interaction, as far as I understand.