Time crystals are a peculiar state of matter. Their emergence hinges on ergodicity breaking, which typically originates from many-body localization or Floquet prethermalization. Here we propose a novel scheme for devising robust dissipative time crystals where the ergodicity is broken through symmetry-induced fragmentation. Building upon a $U\left(1\right)$-symmetry-induced Liouville-space fragmentation, we first propose a generic Liouvillian with long-time oscillations typical of time crystals. We then show that, even when the $U\left(1\right)$ symmetry is broken, a prethermal time-crystal behavior survives, with distinct oscillation frequencies at different times of the steady-state approaching dynamics. Intriguingly, the stagewise prethermal dynamics derive from Fermi statistics and the Liouvillian skin effect of our model—as the excitations above the boundary-localized dark states can be mapped to the irreducible representations of the permutation group, the branching rules of the permutation group ensure the robustness of the prethermal time crystal. Our work paves the way for devising time crystals through Hilbert-space fragmentation. It also sheds light on the dynamic effects of non-Hermitian physics in many-body quantum open systems.

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• Symmetry breaking states
• Time crystals
• Lattices
• Non-Hermitian systems
• Ergodic theory
• Lindblad equation
• Perturbative methods
• U(N) symmetry