Many-body localized discrete time crystals are globally delocalized

The many-body localized discrete time crystals (MBL-DTC) are the first and most-studied models in discrete time crystals. Their stability and non-ergodicity are believed to originate from the many-body localization. However, in this work, we show that they are actually globally delocalized in the thermodynamic limit when any non-vanishing small extensive local perturbations are added. We use the prototypical kicked Ising DTC model as an example. By perturbative analysis, we show that the perturbed eigenstates of the Floquet operator are globally delocalized with a length O(λL), in the presence of a strength-λ local perturbation. When λL≪1, the system remains localized and shares identical properties with MBL systems. However, the condition is unrealistic in the thermodynamic limit. For λL≫1 with small λ, the system will quickly delocalize with length O(λL), but will remain there and exhibit robust subharmonic oscillations for an exponentially long time. We also confirm the above arguments with numerics. Finally, we offer a new proof, with a reasonable conjecture, for the stability of disorder DTCs without invoking MBL. The proof is based on proving the robustness of the π quasi-spectral gap via perturbative analysis to the O(L)-th order.