  We study the response of a thermal state of an Ising chain to a nonlocal
non-Hermitian perturbation, which coalesces the topological Kramer-like
degeneracy in the ferromagnetic phase. The dynamic responses for initial
thermal states in different quantum phases are distinct. The final state always
approaches its half component with a fixed parity in the ferromagnetic phase
but remains almost unchanged in the paramagnetic phase. This indicates that the
phase diagram at zero temperature is completely preserved at finite
temperatures. Numerical simulations for Loschmidt echoes demonstrate such
dynamical behaviors in finite-size systems. In addition, it provides a clear
manifestation of the bulk-boundary correspondence at nonzero temperatures. This
work presents an alternative approach to understanding the quantum phase
transitions of quantum spin systems at nonzero temperatures.
