# Correlation functions of quantum Ising models

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.