Information scrambling and entanglement dynamics in Floquet time crystals

We study the dynamics of out-of-time-ordered correlators (OTOCs) and the entanglement of entropy as quantitative measures of information propagation in disordered many-body systems exhibiting Floquet time-crystal (FTC) phases. We find that an OTOC spreads in the FTC with different characteristic timescales due to the existence of a preferred ``quasiprotected'' direction, denoted as the $\ensuremath{\ell}$-bit direction, along which the spins stabilize their period-doubling magnetization for exponentially long times. While orthogonal to this direction the OTOC thermalizes as an usual many-body localization time-independent system (at stroboscopic times), along the $\ensuremath{\ell}$-bit direction the system features a more complex structure. The scrambling appears as a combination of an initially frozen dynamics (while in the stable period-doubling magnetization time window) and a later logarithmic slow growth (over its decoherence regime) until full thermalization. Interestingly, in the late-time regime, since the wave front propagation of correlations has already settled through the whole chain, scrambling occurs at the same rate regardless of the distance between the spins, thus resulting in an overall envelopelike structure of all OTOCs, independent of their distance, merging into a single growth. In addition, the entanglement entropy shows a logarithmic growth over all time, reflecting the slow dynamics up to a thermal volume-law saturation.