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?


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 $\propto 1/N$.

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