Non-Resonant Boundary Time Crystals from Quantum Synchronization Breakdown

Quantum synchronization (QS) in dissipative systems is often inferred from smooth phase locking, leaving open whether its breakdown constitutes a genuine nonequilibrium transition. Here we introduce a Liouvillian framework that classifies driven-dissipative dynamics by the structure of the undriven dissipative background and show that QS breaks down via a Hopf-type dynamical phase transition into a boundary time crystal (BTC). The character of this transition is determined by the background attractor: systems with a self-sustained oscillator (SSO) support robust non-resonant BTCs, whereas those with a polar fixed point (PFP) sustain BTCs only at resonance and lose them under detuning. We identify sharp dynamical and spectral signatures of the QS-BTC transition and thereby establish, within U(1)-symmetric collective-spin Lindbladians driven by a single coherent tone, a background-based allowed/forbidden criterion that unifies QS, its breakdown, and time-crystalline order within a single Liouvillian framework.