High temperature series expansions of S = 1/2 Heisenberg spin models: algorithm to include the magnetic field with optimized complexity

1) Detailed description of an algorithm for HTSE of high order. An especially interesting development is the inclusion of other important terms apart from the isotropic Heisenberg term in the expansion (Zeeman term, Dzyaloshinskii–Moriya term, and the possibility of obtaining HTSE for Heisenberg XXZ model)

2) An algorithm is open. The Authors present a library of HTSE coefficients for several models on different lattices.

1) Manuscript is a bit too technical.

2) Some minor grammatical inaccuracies.

As for today, there are two open algorithms for HTSE of S=1/2 Heisenberg models with several nonequivalent interactions [3,12]. Both algorithms allow users to calculate HTSE for \ln Z of 10th order [3] and 12th order for the statical structure factor [12], typically of up to four different exchange interactions. In the submitted manuscript, the Authors improved the order of HTSE over existing ones (for most of the lattices shown in the data repository, improvement is quite significant). A major part of the scientific literature on the HTSE, including many papers by the Authors, was mainly focused on studying the Heisenberg model in zero magnetic field. The presence of a magnetic field in the expansion provides an opportunity to study, for example, the magnetization process. Another interesting result is the possibility of including the Dzyaloshinskii–Moriya term in the expansion (this should be very significant in analyzing the experimental data by the HTSE).

The manuscript is designed as a self-consistent description of the algorithm. Therefore, it is very formal, and sometimes it is hard to follow all the technicalities.

Overall, the manuscript is an important development in the high-temperature studies of spin models, and I support its publication. At the same time, I am asking the Authors to consider improving some points (see below).

1) Title "High temperature series expansions of S = 1/2 Heisenberg spin models: algorithm to include the magnetic field with optimized complexity" --> "High-temperature series expansions of S = 1/2 Heisenberg spin models: an algorithm to include the magnetic field with optimized complexity"

Change high temperature --> high-temperature throughout the manuscript.

2) In Section 2, the Authors mentioned: "... and the interactions are short-range (in practice, first, second, third neighbors)."

Could the Authors be more precise about how many different exchange interactions are possible to get HTSE of a reasonably high order? 

3) In Section 2, the Authors stated: "Nevertheless, B is an experimentally adjustable parameter that has been known to induce various unexpected phenomena such as magnetization plateaus and phase transitions."

Could the Authors elaborate on this statement? If the order of expansion is sufficiently high to get to rather low temperatures (let's say T=0.2J), could one see "melted" by temperature magnetization plateaus?

4) In Appendix 3, in equation (B.3) limits of the sum are missing \sum_{j=1}^{k}.

Ask for minor revision

1) Interesting description of an extension of high-temperature series expansions in the presence of a finite field

1) Article is very technical
2) Introduction is not very convincing
3) References are poorly present

The article entiteled "High temperature series expansions of S = 1/2 Heisenberg spin models: algorithm to include the magnetic field with optimized complexity" by Pierre, Bernu, and Messio describes algorithmic progress of high-temperature expansions in the presence of a finite magnetic field. The described method is interesting and certainly valuable for publication. It has further been already used by the same authors in previous articles. At the same the present article is purely technical and does not contain any new physical results (as also clearly stated by the authors. In my opinion the article is therefore better suited for SciPost Core.

1) The first paragraph about Hubbard models is completely detached from the rest of the paper. Of course, it is one prominent way to obtain effective spin models, but there are many others.
2) There are almost no references in the introduction, e.g. no link is given to existing literature on (numerical/non-perturbative) linked cluster expansions there are many other methods mentioned without refererence.
3) Page 3, "sec." -> "Sec."
4) Page 3, "bidimensional" -> "two-dimensional"
5) Page 3, "don't" -> "do not"
6) Page 4, "(Note..." -> "(note...)"
7) Page 5 (but also everwhere in the article): check "," and "." after equations, e.g. after (8) and (10)
8) Page 5: "measure, (5) and (6)" -> "measure (5) and (6)"
9) Page 6: "multi-graph U" -> "multi-graphs U"
10) Page 7: there are several methods applying non-perturbative linked-cluster expansions, e.g. check the recent work in SciPost

M. Hörmann, K. P. Schmidt
Projective cluster-additive transformation for quantum lattice models
SciPost Physics 15, 097 (2023)

and references therein.

11) Page 11: I find the logic a bit strange that one states that (22) is now proven, but then continues with proving (26) which is given half a page later. Maybe one can (26) a bit close to this statement.

12) Page 13: "anti-ferromagnetic" -> "antiferromagnetic"

Accept in alternative Journal (see Report)

1) Comprehensive introduction to the method discussing all relevant aspects

2) Discussion of a relevant idea that can improve the well established method of HTSE

3) Completely self-contained article. All relevant proofs and definitions can be found in the appendix.

1) Many technical details are only discussed in the specialized notation. Examples and more illustrations would help to better communicate the information.

2) The introduction including the Hubbard model is somehow detached from the main body of the article. A focus on the relevance of the HTSE method and some remarkable results obtained using the method would be preferable.

3) There are many technical details that require to jump between the main body of the manuscript and the appendix (and vice versa).

4) There are several small issues regarding the grammar.

5) Regarding the introduction of many mathematical symbols and abbreviations the formatting of the text, equations and definitions is not ideal.

The manuscript describes an idea to improve graph-based high-temperature series expansions (HTSE) for magnetic spin Hamiltonians in a magnetic field.
The main result of the work is a scheme to deduce the contribution for bridged and tree graphs with an improved algorithmic complexity of \(n^2\).
These graphs are required for the HTSE of Heisenberg Hamiltonians in a magnetic field.
With the presented scheme and its improvements the authors claim a benefit of one additional order in the series expansion.

The manuscript is a technical description of a method, therefore there are no "physical examples" discussed.

The manuscript falls under the "Expectations" acceptance criteria 3. and 4. as one can argue that it provides a theoretical/computational advantage to improve existing HTSE by an order.
Since graph-based series expansions are a hard problem this improvement is significant.

Nevertheless, the manuscript still requires some minor changes in order to meet the "general" acceptance criteria of SciPost Physics.

General points:

1) Could you address the points raised in the weaknesses section? These points refer to the entire manuscript.

2) To reach a "bigger audience", it would be very useful to visualize many of the graph-related concepts in figures.

3) It is completely acceptable that there is no big emphasis on physical results in the manuscript. Nevertheless, it would be beneficial to have some connection points to "relevant physics" in the introduction and the summary/outlook.

Specific points:

1) Introduction: Regarding the wording: "we still do not get a solvable model in presence of frustration (competing interactions)". Many non-frustrated models are also not solvable (Heisenberg model on the cubic lattice). Could you please clarify the statement?

2) Introduction: Regarding the sentence: "Frustrated spin models are realized in numerous new materials and exhibit various unconventional phases". Could you please provide some (overview) references to give interested reader an easy access to this statement.

3) Introduction: Regarding the sentence: "Understanding these systems requires increasingly sophisticated methods, including variational methods, mean-field methods, tensor-product numerical methods, and renormalization group methods, among others". Could you please provide some (overview) references to give the interested reader easy access to these methods.

4) Introduction: Regarding the sentence: "Furthermore, extrapolation techniques have been developed to extend the analysis to lower temperatures, necessitating the inclusion of the largest possible number of coefficients in the series". Could you please provide the relevant references.

5) Sec. 2: Regarding the sentence: "2-spin or multispin interactions are possible". Please explain how multispin interactions are incorporated in the graph expansion scheme?

5) Sec. 4: Add a more descriptive caption to Figure 2.

6) Sec. 4: At the end of the paragraph "of the usual method" is mentioned. Please add more description and references to this statement.

7) Appendix C.3: "We now explain a better criterium (C.5), and give an algorithm to compute it.". A better criterium for what? Plase clarify the sentence.

8) Appendix C.3: "Condition “if one of them . . . to an odd islet” is important". Please clarify the condition.

Typos, Style and Grammar:

These are some typos / sentences that caught the eye during the examination of the manuscript. They can be used to improve the manuscript. The authors can adjust their manuscript, but do not need to address these suggestions "point-by-point".

- General: Often a sentence structure like "We [verb] here ..." is chosen. It would improve the clarity of the sentences if "Here, we [verb] ..." would be used instead.

- General: Sometimes required articles (a, an, the) are missing in sentences.

- Title: "algorithm" -> "an algorithm"

- Introduction: "limited to spin" -> "limited to its spin"

- Sec. 2.2: "Series expansions of the previous subsection" -> "The series expansions described in the previous section"

- Sec. 2.2: "is a sum over connected multi-graph" -> "is a sum over connected multi-graphs"

- Sec. 2.2: "it possesses a well defined HTSE in the thermodynamic limit" -> "it results in a well defined HTSE in the thermodynamic limit"

- Sec. 2.2: Eq.~(14) -> added "." at the end

- Sec. 2.2: "To simplify the notations in this presentation" -> "To simplify the notations in this manuscript"

- Sec. 2.3: "coefficient of" -> "the coefficient of"

- Sec. 2.3: "are not equal, denominator of coefficient" -> "are not equal, the denominator of the coefficient"

- Sec. 2.3: "To store in a uniform way the series" -> "To store the series in a uniform way"

- Sec. 2.3: "and publicly available" -> "and publicly available results"

- Sec. 2.4: "that do the job" -> "that does the job"

- Sec. 2.4: "if we keep or not a vertex" -> "if we keep a vertex or not"

- Sec. 2.4: "It uses the McKay’s algorithm" -> "Canonical labels can be calculated using McKay’s algorithm"

- Sec. 2.4: "They consist roughly in first generating" -> "They consist roughly of a first step generating"

- Sec. 2.5: Maybe replace "In the sequel" by "in the following"

- Sec. 4: The hyperref to Eq.~(34a) and Eq.~(34b) does not work.

- Appendix A: The listing of definitions might be better as a bulletpoint structure.

- Appendix B: "but that what follows" -> "but what follows"

- Appendix B.5: "without isolated site" -> "without an isolated site"

- Appendix B.5: "Their exponentials too." is not a sentence.

- Appendix B.6: "Then with Eq.~(B.13) we see that cumulant too." is not a proper sentence.

- Appendix B.7: "According to induction hypothesis" -> "According to the induction hypothesis"

- Appendix B.7: "Only remains what we want to prove." is not a proper sentence.

- Appendix C: There is an unclear structure around Eq.~(C1). Should it be a "," or a "." in the equation?

- Appendix C.3: "Multigraph U is graph G where some links are doubled". Please clarify the sentence.

- Appendix D.1: "But there is simpler way" -> "But there is a simpler way"

- Appendix D.1: The formatting style around the C Code example is not ideal to follow the argument.

- Appendix D.1: "We can still save half computational time" -> "We can still save half the computational time"

- Appendix D.2: "since first multiplicand is allways" -> "since the first multiplicand is always"

- Appendix D.2: "and coefficient" -> "and the coefficient"

Ask for minor revision

1) Detailed, step by step presentation of a new algorithm for calculation of the high-temperature series for the general spin-1/2 exchange Hamiltonian with the Zeeman term

1) introduction is too short and technical
2) no actual results are presented

This is an important theoretical work which extends capability of the high-temperature expansion technique for quantum spin models by including the effect of strong magnetic field. Still a couple of improvements on the manuscript can be made that will increase visibility of the current work within quantum magnetism community. The introduction is extremely short and a bit out of point. Instead of a somewhat ambiguous statement about general validity of the Hubbard model (not studied in this work) authors may give a brief historical overview of the high-temperature expansion methods. For deeper appreciation of their results, authors may also include explicit results for one or two simple spin models. For example, the series obtained for square and kagome lattice antiferromagnets can be used to compute the uniform magnetization M(T,h) for a few values of h ~ J.

Ask for minor revision