Parity-time (PT) non-Hermitian (NH) system has significant effects on observable in a great variety of physical phenomena in NH physics. However, the PT-symmetric NH quantum system at finite temperature (the so-called thermal PT system) has never been addressed. In this letter, based on a controlled open quantum system coupling two separated environments, we proposed a design to realize a thermal PT system. After solving quantum master equation in the Gorini-KossakowskiSudarshan-Lindblad form, the unexpected, abnormal, universal properties of NH thermal states (the unique final states under time evolution) are explored, for example, the non-Boltzmann/Gibbs distribution, high-temperature non-thermalization effect, etc. To understand the anomalous behaviours in thermal PT system, we developed the quantum Liouvillian statistical theory-the generalization of usual quantum statistical theory to finite-temperature NH systems. With its help, we derived the analytical results of thermodynamic properties. In addition, we found that at exceptional point (EP) a continuous thermodynamic phase transition occurs, of which there exists zero temperature anomaly. This discovery will open a door to novel physics for NH systems at finite temperature.
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