Physics of PT-symmetric quantum systems at finite temperatures

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 quantum statistical theory (the so-called quantum Liouvillian statistical theory) is developed, in which the Boltzmann distribution law is replaced by the Liouvillian-Boltzmann distribution law. Using the theory, we derive analytical results of thermodynamic properties for parity-time-symmetric non-Hermitian quantum systems and find that a ``continuous'' thermodynamic phase transition occurs at the exceptional point, where a zero-temperature anomaly exists.