Discrete time crystals (DTCs) are nonequilibrium phases of matter that break the discrete time-translation symmetry and are characterized by a robust subharmonic response in periodically driven quantum systems. Here, we explore the DTC, stabilized by the Floquet strong Hilbert space fragmentation, in a disorder-free periodically kicked XXZ spin chain. We numerically show the period-doubling response of the conventional DTC order, and uncover a multiple-period response with beating dynamics due to the coherent interplay of multiple $\pi$ pairs in the Floquet spectrum of small-size systems. The lifetime of the DTC order exhibits independence of the driving frequency and a power-law dependence on the ZZ interaction strength. It also grows exponentially with the system size up to a critical imperfection, as a hallmark of the strong fragmentation inherent to the Floquet model. We analytically reveal the approximate conservation of the magnetization and domain-wall number in the Floquet operator for the emergent strong fragmentation, which is consistent with numerical results of the dimensionality ratio of symmetry subspaces. The rigidity and phase regime of the DTC order are identified through finite-size scaling of the Floquet-spectrum-averaged mutual information, as well as via dynamical probes. Our work establishes the Floquet Hilbert space fragmentation as a disorder-free mechanism for sustaining nontrivial temporal orders in out-of-equilibrium quantum many-body systems.

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• Phase diagrams
• Spin dynamics
• Time crystals
• 1-dimensional spin chains
• Floquet systems
• Strongly correlated systems
• Exact diagonalization
• Renormalization group