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So we start with the definition of grand partition function, $\mathcal{Z} = Tr(e^{-\beta(\hat{H}-\mu\hat{N})})$, with $\beta=1/k_BT$, $\hat{H}$ the Hamiltonian and $\hat{N}$ the operator of total particle number, which commutes with $\hat{H}$. Since you also want to evaluate the average occupation number for each spin, you may add to $\hat{H}$ a source term ...

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