Projective cluster-additive transformation for quantum lattice models

1- New development that improves previous calculations with similar methods.
2- Thorough review of the pre-existent bibliography.
3- Benchmarking through a clear example of application.

1- Unnecessary technical introduction which is hard to follow for non-experts.
2- Sentences are often hard to read.
3- Lack of explicit specification of limits of validity and exactness of the methods themselves.

The authors present a projective cluster-additive transformation that solves the problem of previous linked-cluster expansion methods for excited states. The existing problem consisted in non-block-diagonal elements that gave rise to hopping elements between disconnected clusters. The authors introduce a way to cure such a problem for single- and multi-particle excitations, which further offers improved efficiency. The new method is exemplified by the low-field expansion of the transverse-field Ising model on a square lattice. For this model, the authors calculate expansions around h=0 of the spin-flip and bound-state gaps up to much higher orders than previously existing series. The results are then analyzed and compared against other methods.

Overall, I think that the method presented is sufficiently new and relevant to gran publication in SciPost Physics. However, I feel that all (the authors, the article, and the community) would benefit from a careful revision of the manuscript. I mention some key points below:
- Even though the introduction is lengthy and very complete in a matter of references, many times concepts and quantities appear out of the blue. For example, the first time that "transformation" appears in the main text is on Page 3 in the sentence: "However, not every transformation yields an effective Hamiltonian that allows a decomposition of the form (3)". This is also the first time the concept of "effective Hamiltonian" appears. At this point, what kind of effective Hamiltonians are we talking about and what kind of transformations?

- Sentences are often a little confusing. For example, on Page 3: "In this paper we will introduce an optimal transformation: It shares the efficiency of the projective method, can also be applied non-perturbatively using the exact lowest eigenvectors and energies, but also allows for a cluster expansions with linked clusters only". What does "can $\textit{also}$ be applied" refer to if no previous application is mentioned? What contrast or addition is "but also" indicating at the end?

- Also, generally speaking, many methods are introduced and/or presented but usually, the discussion tends to be too technical. To anchor the methods for the non-expert reader, comments on the scope of validity of the methods are necessary. When are these methods exact? When do they give good qualitative or quantitative results? In which limit do they fail? Which model Hamiltonians can be solved? Which can not? In which dimensions? In which cases they are better than other standard methods? In which cases they are not?

Moving on with questions on the results:
- I have worked before with Padé Approximants and have never observed spurious poles to be a source of big problems. Normally, spurious poles in the numerator and denominator cancel each other and the functions themselves behave continuously at such points. Small divergencies are observed only when the PAs are numerically evaluated and poles and zeroes do not match exactly. In your article, however, the coefficients are not exact and are only obtained up to a certain digit. The question: do the spurious poles arise because of the inexactitude of the coefficients? Or are they inherent in this kind of problem?

- Once the series is known, the Dlog Padé method can be used to obtain critical points and exponents. One calculates the logarithmic derivative of the series and then the PAs of the result. These PAs have poles and residues that give the values of the critical points and exponents. Typically, one can calculate all poles and corresponding residues for all the PAs [m,d] with m+d=n, and for several n. These points tend to form a curve of critical exponent vs critical point and are more concentrated around the real physical singularity. Have you made this analysis? Does it work in this case? Why is the PA [10,12] the only one mentioned on Page 18?

A little typo: at some point, the article reads "$\textit{lwo}$-field ordered phase"

Some little things like the name of Section 4 need to be corrected: "Low-field expansion for transverse-field Ising model on square lattice". Things are missing. Should be something like "Low-field expansion for the transverse-field Ising model on the/a square lattice".

I would like to re-emphasize that these are not major corrections, but minor ones. All in all, the method itself is well introduced and its advantages are clearly shown.

1- Carefully review the introduction to ensure that it is self-contained and accessible to the readers.
2- Carefully review the introduction to make sentences clearer. For example: avoid unnecessary passive voice, add commas, periods, etc.
3- Correction of typos and grammar throughout the article.
4- Add clear specifications of the advantages and limitations of these methods.

1) Very interesting problem : treating interacting Hamiltonian with quasi-particles
2) Technical high-level new method
3) Illustrative example of application of the new method.

1) Very technical introduction, which would require to be more pedagogic and self-contained
2) Notations not all very precise

In this article, the author present a variant of the CUT method.
The gCUT (graph continuous unitary transformation) method (developped previously by one of the authors) allows a linked cluster expansion of an effective Hamiltonian, in the CUT formalism, but is heavy, whereas the CORE method (a projective method) is numerically faster, but does not verify the linked-cluster property. The authors have derived a method combining the advantages of these two.
They first review the linked-cluster expansions and projective transformation. Then, they explain the difficulty to combine both approaches and describe in detail their method, that they finally apply to the transverse field Ising model (TFIM), both perturbatively and non perturbatively.
This article is very technical, and would need to be more self-contained. It attempts an exhaustive bibliography and discussion of the methods related to the presented one. However, their elliptic description makes the article hard to follow. A re-reading of the introduction with a more detailed/pedagocic description of the other methods would be useful. Some example of difficulties are listed below, mainly due to definition of the notation that are missing.

- If the dimension of the Hilbert space is N (p4), the sum must go from 0 to N-1, or from 1 to N in Eqs. 4, 5, 6, ...
- If the dimension of the Hilbert space is really N (p4), then the eigenspaces H_O^n all have dimension 1, and H_0 is not only block-diagonal, but diagonal ?
- H^1_eff is the effective Hamitonian in the one particule space (p5). But according the the definition p.4, it is the block of the Hamilotian in the eigenspace of energy e1. Does it mean that all quasi particules have the same energy ? It's generally not the case (dispersion relation). Is it only the case for the unperturbed Hamiltonian H_0 (then it has to appear clearly) ?
- With all these unprecisions, it's hard to understand the meaning of Eq. 11, where two notations are simultaneously introduced. The |_1, and the overline.
- Between Eq. 16 and 17, what is X ? (from the following, it seems to be a vector, but become clear only pages later)
- There is a typo in Eq. 50 (H_A->H_B)

Apart from this pedagogically difficult approach, this article is a very interesting and serious work and deserve publication in SciPost.