  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, 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 average level spacing ratio and find that the
interval of parameters where the system does not thermalize and persistent
period-doubling is possible corresponds to the one where the Floquet spectrum
shows time-crystal properties.
