Generalized Ising model

Question

I am in very trouble with a particular expression. I leave the original pages in order to have everything available and what I am goin to leave are the first pages of nine chapter of Non Perturbative Renormalization by Vieri Mastropietro. My problem is: how can I find the expression (9.48)? I can't remove the sum over σ when I derive with respect of $beta J$ the partition function $Z_I$ in the r.h.s. of (9.48) to get "one" exponential as the l.h.s. .
I skipped the section 9.2 because is off topic.

Abelaer · Answer

So the image resolution is a bit low and I can't see the indices of the $J$ you are deriving to, but I think the point is that you are deriving w.r.t a specific $J_{ij}$, even though they are all evaluated at $J_{ij}=J$.
In the full Hamiltonian, a specific coupling $J_{ij}$ only appears once in the full sum, namely when it couples $sigma_i$ and $sigma_j$. That is why deriving w.r.t. $J_{ij}$ picks out only one term $propto sigma_i sigma_j$ from the sum.

Generalized Ising model

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