The Hamiltonian for this system is given by
\begin{equation}
\mathcal{H} \{S\} = -H\sum_i S_i – \frac{J_0}{2} \sum_{ij} S_i S_j,
\end{equation}
where $H$ is the external magnetic field and there is no restriction to nearest neighbour interaction. The spin at each site, $S_i$, may take value $+1$ or $-1$.
Now my question is:
Why does this model only make sense if $J_0 = J/N $, where $N$ is the number of spins in the system?
Best Answer
The energy and the particle number should be both extensive (i.e. $E/N\to\rm{const}$ in the thermodynamic limit, $N\to\infty$). If you calculate the energy of the above system for $H=0$ and, for example, all spins aligned, you get something like $E \propto J_0 N$. Thus, $J_0$ must be $\propto 1/N$.