Adiabatic and irreversible classical discrete time crystals

1- Glossy figures
2- Nice videos

Difficult to follow

The authors propose a realization of classical discrete time crystals in a system of paramagnetic colloidal particles that are placed above a magnetic hexagonal pattern and exposed to an external field that changes its direction periodically.

I have to admit that this work is a bit peripheral to my expertise. Thus, although it does seem interesting, I have confess that I have a bit of a hard time following it.

Part of the issue may be related to the figures that are glossy, but nevertheless difficult to understand.

Figure 1 is particularly difficult to understand. If its description would mention specific parts (left, middle, or right panel), this would probably already help. Likewise, section 2 is quite abstract and thus not easy to read, at least not for a non-expert.

Another detail is recurrent terminology such as "north pole". I only found a short explanation on page 5 during a careful second reading. At the same time, the angles $\vartheta$ and $\varphi$ are not properly introduced. Conversely, line 2 of the caption of Fig. 5 mentions a direction that is "normal to the pattern", which might be identical to "north".

The caption of Fig. 7 should mention that this space-time picture is for a chain of honeycomb cells.

The five accompanying video clips are extremely useful to clarify and illustrate the findings of the present work. However, I believe that it would be good to emphasize that these videos are not just a supplementary piece, but actually an important part of this work. At the same time, the source in Ref. [49] needs to be specified more clearly. It may be Ok to use arXiv for them, but since this is not SciPost, this should be stated explicitly.

Content-wise, the middle panel of Fig. 1 does not factorize into a control space ${\cal C}$ and an action space ${\cal A}$, as the notation ${\cal C} \otimes {\cal A}$ seems to suggest.

A further comment on content is the emphasis on the behavior at the edge in section 5.2. After all, a spatial crystal should break translational symmetry spontaneously while an edge breaks it explicitly. I would be grateful if the authors could clarify this point.

Final remark on content: At the very end of the text, the authors say "All integrations were performed numerically". I think that the authors should give a few more details, and in particular specify the integration algorithm.

I would like to see the above points clarified before I commit on a definite recommendation for SciPost Physics.

There are a few further minor issues that I list among the "Requested Changes".

1- Make the description of Fig. 1 more accessible.
2- Try to explain the content of section 2 better to a non-expert.
3- Define the angles $\vartheta$ and $\varphi$ properly and use this opportunity to specify the meaning of the "north pole".
4- Clarify if the "normal to" on line 2 of the caption of Fig. 5 is synonymous to "north".
5- The caption of Fig. 7 should mention that this space-time picture is for a chain of honeycomb cells.
6- Check if the terminology "nuclei" (several) versus "nucleus" (one) is used correctly in the video clip "adFig7.mp4".
7- Emphasize in the text that the "supplementary videos" are an integral part of this work.
8- Specify the source of in Ref. [49] (arXiv?) more clearly.
9- Clarify the notation ${\cal C} \otimes {\cal A}$, or possibly rather the meaning of ${\cal C} $ and ${\cal A}$, as appropriate.
10- Clarify the relevance of the edge in section 5.2 (Fig. 6(b)): in which sense is the resulting state still a crystal?
11- Expand the comment "All integrations were performed numerically" at the end of the Appendix. In particular, specify the integration algorithm.
12- Page 11: eliminate abbreviations that are not used again ("DO TC") or only a second time ("FDO").
13- Page 11: Using angles $\langle \cdot \rangle$ rather than less and bigger signs $<\cdot>$ for expectation values would improve readability.
14- Still page 11: The "timely" should probably rather be a "time" or "time-like".
15- Caption of Fig. 6: I think that "spatial" would be preferred spelling over "spacial".
16- Format Eq. (A3) such that the curly bracket "$\{$" stretches across both alternatives.
17- Last line of text: "effect" $\to$ "affect".
18- Properly upper-case names in the titles of Refs. [16,18,20,27,29,36,38,56].
19- Correct the spelling of the name of the author of Refs. [52,53]: "T. Prosen".

The paper introduces discrete time crystals based on the conformation of colloidal particles above a magnetic hexagonal pattern and subjected to external periodic field. The authors demonstrate complex periodic response of the system and dynamic phase transitions between reversible and non-reversible dynamic particle arrangements defined by the topology of the control loops driving the system. The complexity is further multiplied by the role of long-range multibody interactions and anisotropic filling of the cells.
Colloidal time crystal may introduce a powerful platform for topologically protected transport phenomena in driven colloidal ensembles.

The paper is not easy to understand due to the complexity of the problem at hand. That is not a real weakness though. The authors tried their best to systematically introduce the required terminology and concepts.

The paper addresses an important aspect of topology in space and time to generate a nontrivial dynamic response of the system on external periodic driving. The paper introduces concepts and approaches using the example of a magnetic colloidal system driven above the patterned magnetic substrates to demonstrate a rich variety of topological time crystalline and non-time crystalline phases phenomena available. I was humbled by the complexity of the system.
The paper is well written and suitable for publication in SciPost.

the paper introduces a lot of variables that are needed for the story, but some of them are not defined in time. For instance, on page 8 when defining \phi_A authors use vector a_1 that is not defined right away and only on page 10.