Swapping Floquet time crystal

We propose a Floquet period-doubling time-crystal model based on a disordered interacting long-range spin chain where the periodic swapping of nearby spin couples is applied. This protocol can be applied to systems with any local spin magnitude $s$ and in principle also to systems with nonspin (fermionic or bosonic) local Hilbert space. We explicitly consider the cases $s=1/2$ and $s=1$, using analytical and numerical methods to show that the time-crystal behavior appears in a range of parameters. In particular, we study the persistence of period-doubling oscillations in time and the time-crystal properties of the Floquet spectrum (quasienergy $\pi$-spectral pairing and long-range correlations of the Floquet states), and introduce a quantity (the local imbalance) to assess what initial states give rise to a period-doubling dynamics. We also consider the quasienergy average level-spacing ratio, and find that the interval of parameters where the system dose not thermalize and persistent period doubling is possible corresponds to the one where the Floquet spectrum shows time-crystal properties.