Physics of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">PT</mml:mi></mml:math> -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 statistical theory (the so-called Liouvillian theory) is developed, in which replaced by Liouvillian-Boltzmann law. Using theory, we derive analytical results of thermodynamic properties for and find that ``continuous'' phase transition occurs exceptional point, zero-temperature anomaly exists.