  It presents a significant challenge to elucidate the relationship between the
phases of open quantum many-body systems and the spectral structure of their
governing Liouvillian, which determines how the density matrix evolves.
Previous studies have focused on the Liouvillian gap, defined as the decay rate
of the most slowly-decaying mode, as a key indicator of dissipative phase
transition, noting its closure in symmetry-broken phases and opening in
disordered phases. In this work, we propose an additional spectral gap, termed
the oscillating-mode gap, defined as the decay rate of the most slowly-decaying
oscillating mode. Through the analysis of a prototype dissipative boson system,
we demonstrate the necessity of both the Liouvillian gap and the
oscillating-mode gap for the comprehensive characterization of the system's
phases and the transitions between them.
