  We study parity-time-symmetric non-Hermitian quantum systems at finite
temperature, where the Boltzmann distribution law fails to hold. To
characterize their abnormal physical properties, a new quantum statistics
theory (the so-called quantum Liouvillian statistics theory) was developed, in
which the Boltzmann distribution law was replaced by the Liouvillian-Boltzmann
distribution law. Using it, we derived analytical results of thermodynamic
properties for thermal PT systems and found that a "continuous" thermodynamic
phase transition occurs at the exceptional point, where a zero-temperature
anomaly exists.
