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We study a model of a quantum collective spin weakly coupled to aspin-polarized Markovian environment and find that the spectrum is divided intotwo regions that we name normal and exceptional Liouvillian spectral phases. Inthe thermodynamic limit, the exceptional spectral phase displays the uniqueproperty of being made up exclusively of second order exceptional points. As aconsequence, the evolution of any initial density matrix populating this regionis slowed down and cannot be described by a linear combination of exponentialdecays. This phase is separated from the normal one by a critical line in whichthe density of Liouvillian eigenvalues diverges, a phenomenon analogous to thatof excited-state quantum phase transitions observed in some closed quantumsystems. In the limit of no bath polarization, this criticality is transferredonto the steady state, implying a dissipative quantum phase transition and theformation of a boundary time crystal.

Exceptional Spectral Phase in a Dissipative Collective Spin Model