  A cornerstone assumption that most literature on discrete time crystals has
relied on is that homogeneous Floquet systems generally heat to a featureless
infinite temperature state, an expectation that motivated researchers in the
field to mostly focus on many-body localized systems. Some works have however
shown that the standard diagnostics for time crystallinity apply equally well
to clean settings without disorder. This fact raises the question whether an
homogeneous discrete time crystal is possible in which the originally expected
heating is evaded. Studying both a localized and an homogeneous model with
short-range interactions, we clarify this issue showing explicitly the key
differences between the two cases. On the one hand, our careful scaling
analysis confirms that, in the thermodynamic limit and in contrast to localized
discrete time crystals, homogeneous systems indeed heat. On the other hand, we
show that, thanks to a mechanism reminiscent of quantum scars, finite-size
homogeneous systems can still exhibit very crisp signatures of time
crystallinity. A subharmonic response can in fact persist over timescales that
are much larger than those set by the integrability-breaking terms, with
thermalization possibly occurring only at very large system sizes (e.g., of
hundreds of spins). Beyond clarifying the emergence of heating in disorder-free
systems, our work casts a spotlight on finite-size homogeneous systems as prime
candidates for the experimental implementation of nontrivial out-of-equilibrium
physics.
