Spin Glass Hamiltonian

Question

Why do Edwards and Anderson use the hamiltonian $$ H = \sum_{i,j} J_{ij} \mathbf{s}_i \cdot \mathbf{s}_j $$ to describe the interactions in a spin glass?

Naively I would think that from the interaction energy $U = - \mathbf{m} \cdot \mathbf{B}$ for a dipole $\mathbf{m}$ in a magnetic field $\mathbf{B}$, and the formula $$ \mathbf{B}(\mathbf{r}) \propto \frac{3 \mathbf{\hat{r}}(\mathbf{m}\cdot \mathbf{\hat{r}}) - 5 \mathbf{m}}{r^3} $$

for the magnetic field generated at a displacement $\mathbf{r}$ from a dipole, that the Edwards Anderson hamiltonian amounts to a neglecting of the $3 \mathbf{\hat{r}}(\mathbf{m}\cdot \mathbf{\hat{r}}$) term. What is the purpose of this neglect?

Answer 1

Not a dipole interaction
One reason is that actual interactions between spins are not necessarily dipole interactions. E.g., RKKY interaction is often present and significantly stronger than the dipole interactions.

Model Hamiltonians and critical behavior
Another reason is that Anderson-Edwards Hamiltonian, like the other similar Hamiltonians inspired by the Ising model, aims not at modeling the details of the spin interaction, by the global thermodynamic/critical behavior - notably at capturing different phases of matter (paramagnetic, ferromagnetic, spin glass, etc.) Although originally such model Hamiltonians were suggested ad-hoc, the development of the renormalization group approach has shown that the critical behavior is independent on the particularities of interaction - even more, the system of different physical nature may be in the same criticality class!