3

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.