  The existence of topological zero modes in nontrivial phase of quantum Ising
chain results in not only the Kramers-like degeneracy spectrum, but also
dynamic response for non-Hermitian perturbation in the ordered phase (2021
Phys. Rev. Lett. 126 116401). In this work, we investigate the possible
response of the degeneracy spectrum for Hermitian perturbations. We provide a
single-particle description of the model in the ordered phase, associating with
an internal degree of freedom characterized as a pseudospin. The effective
magnetic field, arising from both local and nonlocal perturbations in terms of
string operators, acts on the pseudospin. We show that the action of string
operator can be realized via a quench under the local perturbations. As an
application, any ground states and excited states for the Hamiltonian with
perturbation can be selected to identify the quantum phase, by adding the other
perturbations to trigger a quench and measuring the Loschmidt echo.
