# Quantum Ising model correlation function query

In this paper on quantum Ising model dynamics, they consider the Hamiltonian $$\mathcal{H} = \sum_{j < k} J_{jk} \hat{\sigma}_{j}^{z}\hat{\sigma}_{k}^{z}$$ and the correlation function $$\mathcal{G} = \langle \mathcal{T}_C(\hat{\sigma}^{a_n}_{j_n}(t_n^*)\cdot\cdot\cdot \hat{\sigma}^{a_1}_{j_1}(t_1^*)\hat{\sigma}^{b_m}_{k_m}(t_m) \cdot\cdot\cdot \hat{\sigma}^{b_1}_{k_1}(t_1)) \rangle$$ where $$a,b= \pm$$ and the time dependence of the Heisenberg picture $$\hat{\sigma}_{j}^{a}(t) = e^{it\mathcal{H}}\hat{\sigma}_{j}^{a}e^{-it\mathcal{H}}$$ where the time ordering operator $$\mathcal{T_C}$$ orders operators along a closed path $$\mathcal{C}$$.

Question: Can anyone see the reasoning behind the subsequent statement on page 6:

If an operator $$\hat{\sigma}^{a=\pm}_{j}$$ occurs in $$\mathcal{G}$$ one or more times, the operator $$\hat{\sigma}^{z}_{j}$$ (appearing in the time evolution operator) is forced to take on a well defined value $$\sigma_{j}^{z}(t)$$ at all points in time.

I understand why it would be well-defined for all times after $$t_1$$ (denoting the first occurrence of the operator $$\hat{\sigma}^{a=\pm}_{j}$$) since this results in the state on site $$j$$ being an eigenstate of $$\hat{\sigma}_{j}^{z}$$, hence it commutes with the Ising Hamiltonian above. But surely before $$t_1$$ the state on site $$j$$ could be in any spin orientation and hence would be affected by the Ising interaction?

Thanks for any assistance.