I am given the Ising Hamiltonian \begin{align} H = K \sum_{<ij>}S_i S_j + h \sum_i S_i, \quad K>0 \end{align} to set up a real-space block-spin RG, where the renormalized spins are constructed via majority rule: $S_I ' = \textrm{sgn} (S_{1,I},S_{2,I},S_{3,I})$. Now the first part is to show that the partition function of the blocked spins takes the form \begin{align} Z = \prod_I e^{A + B S_I '} \end{align} and by considering the two possibilities $S_I ' = \pm 1$, we want to find $A(K,h)$ and $B(K,h)$. Doing the calculation, however, I find the 'one-block' partition function \begin{align} Z_1 ^+ = e^{3K+3h}+3 e^{-K + h} \\ Z_1 ^- = e^{3K-3h}+3 e^{-K - h} \end{align} and now I don't see how I would read the $A$ and $B$ from this. Thank you all for your answers already!


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