Non-Resonant Boundary Time Crystals from Quantum Synchronization Breakdown

Abstract

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.

Figures (4)

• Figure 1: (a) The Liouvillian of a driven--dissipative spin system is decomposed into $\mathcal{L}_0$, which governs the U(1)-symmetric undriven dissipative background, and $\mathcal{L}_1$, which represents the symmetry-breaking drive. (b,c) Schematic phase diagrams in the rotating frame for cases where the background steady state set by $\mathcal{L}_0$ is a polar fixed point (b) or a self-sustained oscillator (c). For a PFP background, only a resonant BTC appears when drive strength $\epsilon>\epsilon_c$ and collapses under detuning $\Delta$, whereas an SSO background supports a non-resonant BTC over a finite detuning window. PFP behaves as a heavily damped pendulum, whereas SSO responds in a gyroscope-like manner, sustaining rotation.
• Figure 2: Dynamics of the minimal collective-spin model under the driving protocol shown on top: (a) magnetization $m_z(t)$, comparing a PFP-only background ($\Gamma_-=1$) with an SSO-hosting background ($\Gamma_-=2$); (b) relative phase $\delta\phi(t)=\phi(t)-[\omega_d t]_{2\pi}$ for two initial conditions on the SSO manifold, diagnosing phase locking and its breakdown. Parameters: $\omega_0=20\pi$, $\Gamma_+=1$.
• Figure 3: Verification of the QS-BTC dynamical phase transition under resonant driving. (a–b) Time evolution of the magnetization $m_z(t)$ for various system sizes $N$, initialized at $(m_x, m_y, m_z) = (1, 0, 0)$, under (a) weak driving $\epsilon = 1$ and (b) strong driving $\epsilon = 4$. (c) The nonanalytic behavior of the dynamical order parameters—the time-averaged magnetization $\overline{m}_z$ and its variance $\text{Var}[m_z]$—as the system size increases, indicates a DPT from QS to BTC. (d) Universal data collapse in the finite-size scaling of $\overline{m}_z$ confirms a DPT. Other parameters: $\omega_0=\omega_d=20\pi,~\Gamma_{+} = 1,~\Gamma_{-} = 2$.
• Figure 4: (a) Arnold tongue phase diagram showing the boundary between two phase-locked Quantum synchronization (QS) regimes and BTC regimes. (b) Bloch trajectories and Liouvillian spectra for the three representative points marked in Fig. \ref{['fig3']}(a): square (BTC), star (QS-I), and triangle (QS-II), with $N = 200$. Each thick trajectory is initialized at the north pole to highlight their distinct dynamical behaviors: persistent oscillations in BTC, exponential relaxation in QS-I, and oscillatory decay in QS-II. (d-e) Finite-size scaling of the Liouvillian gap $\alpha=\alpha_r \pm i\alpha_i$ for the three cases in (b). Dashed lines are guides to the eye. Other parameters: $\omega_0=20\pi,~ \Gamma_+ = 1$, $\Gamma_- = 2$.