Lindbladian PT phase transitions

Abstract A parity-time (PT) transition is a spectral transition characteristic of non-Hermitian generators; it typically occurs at an exceptional point, where multiple eigenvectors coalesce. The concept of a PT transition has been extended to Markovian open quantum systems, which are described by the Gorini–Kossakowski–Sudarshan–Lindblad equation. Interestingly, the PT transition in many-body Markovian open quantum systems, the so-called Lindbladian PT (L- PT ) phase transition , is closely related to two classes of exotic nonequilibrium many-body phenomena: continuous-time crystals and non-reciprocal phase transitions . In this review, we describe the recent advances in the study of L- PT phase transitions. First, we define PT symmetry in three distinct contexts: non-Hermitian systems, nonlinear dynamical systems, and Markovian open quantum systems, highlighting the interconnections between these frameworks. Second, we develop mean-field theories of L- PT phase transitions for collective-spin systems and for bipartite bosonic systems with particle-number conservation. Within these classes of models, we show that L- PT symmetry can induce a breaking of continuous time-translation symmetry down to a discrete one, leading to persistent periodic dynamics. We further demonstrate that the L- PT phase transition point is typically a critical exceptional point , where multiple collective excitation modes with zero excitation spectrum coalesce. These findings establish an explicit connection to continuous-time crystals and non-reciprocal phase transitions. Third, going beyond the mean-field theory, we analyze statistical and quantum properties, such as purity and quantum entanglement indicators of time-independent steady states for several specific models with the L- PT symmetry. Finally, we discuss future research directions for L- PT phase transitions.