Stroboscopic aliasing in long-range interacting quantum systems

1)Topical subject (dynamical stabilisation, time crystal phases in periodically driven systems)
2)Interesting results (subharmonic response with arbitrary multiple of the drive frequency)
3)Intuitive and qualitative picture to understand the main results (discussion in Sec. 2.2 for example, analogy with aliasing)

1)The main claims of the paper seem to concern generic (non-integrable) systems while most of the results concern the fully connected limit, which is a special limit
1)The discussion of the role of quantum fluctuations due to finite-range interactions is limited to Sec 3.2 and would probably benefit from some revision.
2)Same for the discussion on the role of dissipation

The article by Kelly et al concerns the physics of a long-ranged periodically driven quantum system and the dynamical stabilisation of an oscillatory regime at a frequency given by an arbitrary multiple of the drive one. As such this work provides an interesting addition to the family of discrete time crystal phases.
Furthermore the authors provide a nice interpretation of this phenomenon in terms of stroboscopic aliasing.

In practice the authors focus on a long-ranged Ising chain with a transverse field, with the strength of the Ising interaction which is modulated, alternating two values within one single period. The authors mostly discuss the fully connected case (alpha=0) for which one can reduce the dynamics to classical equations of motion. In this limit the authors provide a nice and transparent physical interpretation for the origin of the aliasing. In Sections 3.2 and 3.3 they discuss the role of quantum fluctuations due to finite-range interactions and dissipation.
Both these sections would require in my opinion some change both in terms of content and of presentation of the results(see below)

Overall, I think the paper meets the criteria for publication in Scipost and I would be happy to recommend publication provided the authors address the points below.

Sec 3.2

I am puzzled by the results in Fig.4 and by the conclusions the authors draw from them. In particular the fact that at finite alpha it seems that increasing N the stability of the n=2 aliasing increases, leading to the conclusion in Sec. 3.2 that:

"The stroboscopic aliasing is therefore not a fine tuned point in parameter space, rather, it shows robustness to the inclusion of long-range spin-spin interactions....In this respect, it survives a purely mean-field description of dynamics."

I would have expected that for alpha\neq0, when the model is non-integrable, the system would heat up towards infinite temperature in the thermodynamic limit as expected from a periodically driven system.
Could the authors comment on this expectation and whether it should be met by their model at finite alpha?Can they quantify this heating?

In this light, the fact that increasing the system size makes the aliasing more and more stable, as shown in Fig.5 is, according to me, a demonstration of the fact that the method chosen by the author (DTWA) is not able to actually capture thermalization. It seems instead that the fluctuations included by DTWA become less and less important in the thermodynamic limit. Can the authors comment on this point?

Overall I am rather skeptical about the stability of the aliasing to many-body quantum fluctuations since, as the authors state at the beginning of Sec.3 .3, it requires the existence of a well defined single-frequency collective mode, which is probably not the case in a generic many-body setting. Can the authors comment on this point?

Sec. 3.3

The authors could improve the presentation of this section, including (i)clarifying which case they are considering in Fig 5 (fully connected alpha=0 or not?Finite N or large N), (ii) writing down the Lindblad master equation they consider and (iii) mention in few words how they actually solve it.


Minor Issue:
References - some of the articles seem misplaced or not properly cited. For example Ref 39 has, from what I can see, nothing to do with Discrete Time Crystals and should not be cited together with 37-38.