- #1
Danny Boy
- 49
- 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 following 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.
Thanks for any assistance.
$$\mathcal{H} = \sum_{j < k} J_{jk} \hat{\sigma}_{j}^{z}\hat{\sigma}_{k}^{z}$$
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 following 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.
Thanks for any assistance.