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I have a single technical question regarding a statement on page 7 of the paper "Dynamical quantum correlations of Ising models on an arbitrary lattice and their resilience to decoherence". The paper up until page 7 defines a general correlation function $\mathcal{G}$ of a basic quantum Ising model (with only an interaction term in the Hamiltonian). The correlation function $\mathcal{G}$ up to page 7 deals only with raising and lowering operators of the form $\sigma^{\pm}_{j}$ on sites $j$. To insert operators of the form $\hat{\sigma}_{j}^{z}$ into the correlation function $\mathcal{G}$ the following is stated:

The insertion of an operator $\hat{\sigma}_{j}^{z}$ inside a correlation function $\mathcal{G}$, which we denote by writing $\mathcal{G} \mapsto \mathcal{G}^{z}_{j}$, is relatively straightforward. If $\alpha_j = 0$, then clearly the substitution $\hat{\alpha}_{j}^{z} \mapsto \alpha_{j}^{z}(t)$ does the trick. If $\alpha_j = 1$, $\hat{\alpha}_{j}^{z}$ can be inserted by recognizing that the variable $\phi_j$ couples to $\hat{\alpha}_{j}^{z}$ as a source term, and thus the insertion of $\hat{\alpha}_{j}^{z}(t)$ is equivalent to applying $i \frac{\partial}{\partial \phi_j}$ to $\mathcal{G}$.

Can anyone see the reasoning behind the last sentence:

If $\alpha_j = 1$, $\hat{\alpha}_{j}^{z}$ can be inserted by recognizing that the variable $\phi_j$ couples to $\hat{\alpha}_{j}^{z}$ as a source term, and thus the insertion of $\hat{\alpha}_{j}^{z}(t)$ is equivalent to applying $i \frac{\partial}{\partial \phi_j}$ to $\mathcal{G}$.

Thanks for any assistance.

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