Discrete time crystals enabled by Floquet strong Hilbert space fragmentation

Discrete time crystals (DTCs) are non-equilibrium phases of matter that break the discrete time-translation symmetry and is characterized by a robust subharmonic response in periodically driven quantum systems. Here, we explore the DTC in a disorder-free, periodically kicked XXZ spin chain, which is stabilized by the Floquet strong Hilbert space fragmentation. 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 $π$-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, 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.

💡 Research Summary

This paper establishes “Floquet strong Hilbert space fragmentation (HSF)” as a novel, disorder-free mechanism for stabilizing discrete time crystals (DTCs) in out-of-equilibrium quantum many-body systems. DTCs are non-equilibrium phases that break the discrete time-translation symmetry of a periodic drive, typically characterized by a robust subharmonic response at half the driving frequency. A major challenge has been preventing Floquet systems from heating to a featureless infinite-temperature state, with many-body localization being a prominent but resource-intensive solution. This work explores an alternative route via HSF, a phenomenon where the system’s Hilbert space shatters into exponentially many dynamically disconnected subspaces due to kinetic constraints, previously studied mainly in static systems.

The authors investigate a clean (disorder-free), periodically kicked XXZ spin chain under a two-step Floquet drive: an imperfect π-pulse around the x-axis followed by evolution under an XXZ Hamiltonian. When the ZZ interaction (V) dominates over the XY coupling (J), strong kinetic constraints emerge. Through extensive numerical simulations (exact diagonalization and TEBD) and analytical arguments, they demonstrate the emergence of a DTC phase protected by Floquet strong HSF.

Key findings include:

1. DTC Responses: The system exhibits the hallmark period-doubling (2T) subharmonic response for initial states like the Néel state, stemming from the initial state being predominantly composed of a single dominant “π-pair” in the Floquet spectrum (a pair of eigenstates with a quasi-energy splitting of π). For other initial states, such as domain-wall configurations, a more complex “multiple-period” response with long-period beating dynamics is observed in finite-sized systems. This arises from the coherent interplay of multiple π-pairs, generating additional Fourier peaks at frequencies determined by the quasi-energy differences between these pairs.
2. Lifetime and Scaling: The lifetime (τ) of the DTC order shows three crucial characteristics: (i) independence from the driving frequency (ω), distinguishing it from prethermal DTCs; (ii) a power-law increase with the ZZ interaction strength (V); and most significantly, (iii) an exponential growth with system size (L). This exponential scaling is a definitive signature of “strong” Hilbert space fragmentation, as opposed to “weak” fragmentation where the lifetime scales polynomially.
3. Mechanism - Approximate Conservation: The authors analytically show that in the strong V/J limit, the Floquet operator exhibits an approximate conservation of magnetization and domain-wall number. This leads to the emergent strong HSF, which is corroborated by numerical analysis of the dimensionality ratio of symmetry subspaces, confirming the exponential fragmentation of the Hilbert space.
4. Phase Characterization: The rigidity of the DTC phase against driving imperfections (ϵ) and its phase boundary are identified through finite-size scaling of the Floquet-spectrum-averaged mutual information, as well as via dynamical probes like the subharmonic response amplitude.

In conclusion, this work successfully demonstrates that Floquet strong HSF can act as a potent, disorder-free alternative to many-body localization for stabilizing long-lived DTCs. It opens a new avenue for exploring and stabilizing nontrivial non-equilibrium quantum phases in clean, periodically driven systems, with potential for realization in near-term quantum simulators like superconducting qubits or trapped ions.