Emergence of Light Cones in Long-range Interacting Spin Chains Is Due to Destructive Interference

The authors aim to elucidate the origin of a linear light cone in models with power-law interactions. Overall I thought that it was an interesting read and the counter-intuitive ideas in the figure in the main text were very intriguing. Like the other referee, I did not find it so obvious what \mathcal{Q} was probing and why it was important -- to me it seems like an unusual way to probe locality. But assuming that the authors can suitably clarify that issue, I would support publication of the paper, after the authors address the following comments as well.

The authors state multiple times that destructive interference is necessary for the emergence of a light cone. It was not clear what mathematically the authors meant by "light cone", but taken at face value the statement seems false. For example even if I have only ZZ type interactions, which one might say in the Schrodinger picture cause only constructive interference, there is still a light cone if p>3. If the authors instead mean that the series in Eq. (7) has many cancelling phase factors, then I would agree. However, this latter result is crucial for all the Lieb-Robinson bounds that have been derived for power-law interacting systems also -- without these phase cancellations indeed the combinatorial series that just count paths from one point to the next (a la Hastings) grow very fast. Therefore from this Heisenberg picture, the idea that there are many phase cancellations was understood before.

In Eq. (8) is there no O(1) factor for velocity, e.g. a factor of 2 coming from the binomial coefficient? That is surprising to me. Also, at small times t, Eq. (8) decays much faster than Eq. (9), although maybe the change in subscript (which I didn't see explained until the supplement) is meant to be important. Please clarify.

The scaling in Eq. (13) actually looks rather weak. If I resum the series don't I get an exponential growth in time, with only polynomial decay in q? So I would expect there is a lot more destructive interference than the theorem guarantees term by term in the series. Unlike in the NN case, here the series in Eq. (7) does not begin at O(q).

The authors should simply make the supplementary material into appendices for SciPost Physics. Also, the way that citations are done in the supplement is different from the main text and this should be more standardized, which would be natural if they simply become appendices.

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The manuscript by Azodi and Rabitz reports results supposedly related to the emergence of lightcones in long-range interacting spin chains. I write "supposedly" because I wasn't able to establish the connection of their results to the emergence of lightcones. I wasn't even sure lightcones of what. (Expectation values? Entanglement? All of these? It is known that in general these cones don't coincide in long-range systems.). Also, since I wasn't able to follow some of the initial considerations in the paper, I wasn't in the position to work through the later parts, and this referee report is only a partial assessment of the manuscript. However, the results, if correct, would be of interest, and I wouldn't mind reviewing a thoroughly revised and extended version of the paper.

The main problem I have with the paper is the claim that \mathcal{Q}_q(t) "serves as an upper bound for variations in the expectation value of O, evaluated with respect to an arbitrary wavefunction". I tried to verify this statement, but without success. I request the authors of the manuscript to include a derivation of this upper bound in the paper. (I am not even sure I understand what is meant by "variations in the expectation value of O [...] when transported to distance q under the unitary evolution", which is the quantity to be bounded by \mathcal{Q}_q(t).) How do you manage that the (-1)^{r_1} term in Eq. (7) survives when deriving an upper bound? (When deriving Lieb-Robinson bounds, which try to achieve something similar, the interference effects are easily lost because absolute values have to be taken in some way or the other.) These points need to be clarified for me to be able to continue assessing the content of the paper.

In addition to this main point of my criticism, I have to say that the splitting of the paper into a "narrated" main part and a more mathematical supplemental part doesn't work well, at least in its present form. The main part doesn't mention Holstein-Primakoff, but at times talks about "hopping events" etc. Also, Holstein-Primakoff will, to my understanding, imply certain limitations on the validity of the results, which in turn need to be clearly specified also in the main text. Section III of the Supplemental Material seems to be, in my understanding, entirely on the level of linear spin wave theory, but the theorem is stated as if it were general. Please clarify thoroughly!

Many other parts of the main text also remain way too vague without mathematical specifications, and therefore become incomprehensible. An example being the "definitions" of type 1 and type 2 functions in Section IV.A.

(a) At the end of Section II, the reader is referred to Ref. [46] for a discussion of the fact that the series expansion (7) is restricted to even powers of t. At least with moderate effort I couldn't find this discussion in that reference, so I suggest it be included into the manuscript for convenience.

(b) Paragraph 2 of the introduction: The exponential decay outside the lightcone is discussed as a "tunnelling effect". I believe this is misleading. No barrier to tunnel through is present to my understanding, and the occurrence of exponential tails is not enough to identify a tunneling process.

(c) Eq. (9): The symbol \mathcal{M} is used without explanation. (It will be defined in the Supplemental Material, but needs to be clarified where it first occurs.)

(d) Below Eq. (9): $\alpha_r\approx0$ is so vague a statement that ii is essentially meaningless. Please clarify.

(e) Section IV.C: It is a good idea to phrase the main technical result of the paper in the form of a theorem. But this should be a clear, self-contained statement which includes all relevant conditions, assumptions, etc.

(f) Section V.A: When presenting numerical evidence, it would be extremely important to show not only data for \mathcal{Q}_q(t), but also compare with data for the actual physical quantity that is supposedly bounded by \mathcal{Q}_q(t).

Ask for major revision