Quantum Phase Transition in a Quantum Ising Chain at Nonzero Temperatures

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.