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LiHoF4 as a spinhalf nonstandard quantum Ising system
by Tomer Dollberg, Moshe Schechter
Submission summary
Authors (as registered SciPost users):  Tomer Dollberg 
Submission information  

Preprint Link:  scipost_202405_00026v1 (pdf) 
Date submitted:  20240517 10:36 
Submitted by:  Dollberg, Tomer 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
$\mathrm{LiHoF_{4}}$ is a magnetic material known for its Isingtype anisotropy, making it a model system for studying quantum magnetism. However, the theoretical description of $\mathrm{LiHoF_{4}}$ using the quantum Ising model has shown discrepancies in its phase diagram, particularly in the regime dominated by thermal fluctuations. In this study, we investigate the role of offdiagonal dipolar terms in $\mathrm{LiHoF_{4}}$, previously neglected, in determining its properties. We analytically derive the lowenergy effective Hamiltonian of $\mathrm{LiHoF_{4}}$, including the offdiagonal dipolar terms perturbatively, both in the absence and presence of a transverse field. Our results encompass the full $B_{x}T$ phase diagram, confirming the significance of the offdiagonal dipolar terms in reducing the zerofield critical temperature and determining the critical temperature's dependence on the transverse field. We also highlight the sensitivity of this mechanism to the crystal structure by comparing our calculations with the $\mathrm{Fe_{8}}$ system.
Author indications on fulfilling journal expectations
 Provide a novel and synergetic link between different research areas.
 Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work
 Detail a groundbreaking theoretical/experimental/computational discovery
 Present a breakthrough on a previouslyidentified and longstanding research stumbling block
Author comments upon resubmission
We would first like to thank the referees for their very valuable comments. These, we believe, helped us improve the quality and presentation of our work. Here, we will address each comment and point to corresponding amendments made in the revised manuscript.
Referee 1:
 Carefully check the formulas, correcting apparent errors (e.g. the presence of a twobody interaction in H_{3B}) and ambiguities of notation (e.g. the lack of a site index on the crystal field terms and the apparent confusion between H_{full} and H_{micro}).
We added a site index to the crystal field term in Eq. (1) and clarified the caption of Fig. 2, explaining precisely what $H_{micro}$ means. It is $H_{full}$, as is defined in this work, with the addition of a hyperfine term. Other curves also include the hyperfine interaction, but since, in those cases, it is omitted from the Hamiltonian and later accounted for by a temperaturedependent renormalization, we deemed it appropriate to use a different label for when it is included from the start. Regarding the presence of a twobody interaction in $H_{3B}$, we maintain that it is indeed a threebody interaction due to its dependence on the existence of a third spin $i$, as evident in the presence of the index $i$ in, e.g., $V_{ij}^{xz} V_{ik}^{xz}$. In our view, this is an important distinction, as it pertains to the xdependent behavior of the diluted variant $LiHo_{x}Y_{1x}F_4$; namely, it explains the relatively mild reduction of T_{c} with decreasing x, as explored in Ref.
\[30\].
 Expand on the content and justification of the phenomenological scaling and renormalisation procedures, indicating clearly how the results would be different if they were not included (and ideally presenting results without the scalings / renormalisations implemented, so that the reader can see their effects).
The two scaling and renormalization procedures referred to are indeed not optimal. However, we would like to note that both adjustments are not related to the main result of the paper, namely the reduction of $T_c$ at low fields. For this reason, we purposefully chose procedures that were already used in previous papers (particularly Refs.
\[28,37\]), so that comparison is straightforward. To make the manuscript more selfcontained, and in view of the referee's comment, we added Appendices D and E, which discuss the temperaturedependent renormalization that accounts for hyperfine interactions and the rescaling of the longitudinal interaction, respectively. Each appendix includes a new figure (Figs. 6 and 7) that presents the current results with and without these adjustments.
Furthermore, to more directly relate to the main result of the paper, we have included Appendix F, which discusses the analytical behavior of $T_c(B_x)$ at low fields.
 Ensure that sufficient information, including all parameters (e.g. J_{ex}), is provided for the reader to reproduce the results presented. Give references for the numbers quoted, e.g. the values of alpha and rho given below (4).
The omission of the numerical value of $J_{ex}$ is an oversight on our part, which has been amended (below Eq. (2)). We thank the referee for spotting this issue. The other parameters mentioned, $\alpha$, $\rho$, and $\Delta$, are derived in this work by numerical diagonalization of $V_{c}$ and their values are given below Eq. (4). We use the same notations as in Ref.
\[31\], though the values differ slightly because of the use of updated crystalfield parameters. We add a footnote pointing out this difference and its origin.
 Discuss the validity of the truncation of the SchriefferWolff transformation. (This arises especially because of the observation on page 4 that the threebody terms are longrange because of the additional summations. Doesn't that raise the issue that higher orders in the SchriefferWolff transformation might yield four, five, etc.body interactions that are equally important?)
We acknowledge the referee's point regarding higherorder terms in the SchriefferWolff transformation and have given it considerable thought. Indeed, higherorder terms in the SchriefferWolff transformation should yield interactions involving a growing number of spins, e.g., the third order in the SchriefferWolff transformation will yield 4body terms, as well as possibly additional two and threebody terms. The latter are easily discounted, as their relative contribution compared to their respective counterparts from the second SchriefferWolff order is diminished by a factor of $\langle V \rangle / \Delta \sim 1/10$, where $\langle V \rangle$ denotes, symbolically, the energy scale associated with the dipolar interactions.
However, as the referee rightly points out, fourbody interactions cannot be naively discounted based solely on the relative smallness of their coefficient. To emphasize this point, we mention a reference that has recently come to our attention (Rau et al., Phys. Rev. B 93, 184408 (2016)), wherein the authors perform a similar perturbation procedure on the pyrochlore XY antiferromagnet $Er_{2}Ti_{2}O_{7}$, and make the case that due to the combinatoric factor associated with four and sixspin terms, their contribution is comparable to that of lowerorder terms in the perturbation series. The combinatoric factor is the number of ways an open chain of nearestneighbor spins can be chosen in the pyrochlore system (in that work, only nearestneighbor interactions are considered), which gives an increase of two orders of magnitude because of the high coordination number of the pyrochlore system (compared to $LiHoF_4$). In the present case, instead of a combinatoric factor, it is a sum over presumably longrange four,five,etc.body interactions that seemingly may result in a significant enlargement of the otherwise inherently small higherorder terms.
Despite all of the above, we assert that higherorder terms are negligible in this case. First, we make the case that for an orderofmagnitude estimate, it is enough to consider only nearestneighbors, despite the longrange nature of the interaction. Consider, for example, the three spins depicted in Fig. 1 of the manuscript, assuming spin 1 is a nearestneighbor of spin 2 and spin 3 is a (different) nearestneighbor of spin 2. The energy associated with such a triplet from the threebody term is $ \frac{\alpha^2 \rho^2}{\Delta} E_D^2 V_{12}^{xz} V_{23}^{xz} \approx 0.01 K $. Considering that we have 4*3=12 ways to choose such a chain of neighboring spins in the $LiHoF_4$ crystal, we can estimate the energetic contribution to be $ \approx 0.1 K $, which turns out to be a good estimate of the actual MF correction (the last term in Eq. (7)), which is ~0.1 K. In essence, nearest neighbors dominate the total sum despite its longrange nature because the dipolarderived emergent interactions provide both positive and negative contributions that partially offset. This would be the case for fourspin (and higher) interactions as well.
Let us apply the same reasoning to a hypothetical fourspin interaction that emerges from the next order in the SchriefferWolff transformation. The energetic contribution of such a term would be $ 4\times 3 \times 3 \times \alpha^2 \rho^4 E_D^3 [V_{12}^{xz}]^3 / \Delta^2 \approx 5 mK $, so its effect on $T_{c}$ would negligible in the context of the present work.
We add the above explanation as a closing paragraph to Appendix A, where H_{eff} is derived.
Even worse, data with two different values of the longitudinalinteraction scaling factor are presented in the paper.
As an additional point, under "Weaknesses," the referee mentions that two longitudinalinteraction scaling factors are used. This is correct in that Fig. 2 shows results without ODD terms with two rescaling factors (0.785 and 0.805). However, we would like to clarify that the former is just a reproduction of results presented in Ref.
\[37\](specifically, the solid curve labeled "x=1" in Figure 7 therein). To clarify this point, we slightly amended the phrasing of the caption of Fig. 2 to emphasize that the solid line as a whole is taken from Ref.
\[37\]and not just the rescaling factor.
Referee 2:
1) I would have liked a better summary justifying various rescaling factors invoked (with refs. to experimental works and "strong caxis fluctuations"  i do not recall the issue and not inclined to go digging in the literature), esp. considering the paper is all about fixing relatively small discrepancies
1) better justfiication of rescaling factors
As per our response to the first referee's request #2, we have added Appendices D and E that aim to justify the different rescaling and renormalization schemes used and present results without them, showcasing their effect.
2) it might have helped to state the symmetry constraints on the 3body terms, e.g. I assume they are required to be even in \sigma_z operators so as not to break twofold symmetry. These would limit the first of these terms, which apparently only contains 2 Pauli operators (both \sigma_z^s), to be a twospin operator and thus be considered properly as E_D/\Delta order reduction of the bare 2spin dipolar term rather than a 3spin term. if this is the case, the authors should not call it a 3spin term AND also address whether this term by itself (without the true 3spin terms) can reproduced the reduction of the critical temperature or not.
2) symmetry constraints on and possibly correction/reclassification of 3spin terms
The comment regarding symmetry constraints on emergent terms is, of course, correct. We partially address this fact in Appendix A, right before Eqs. (10)(13). In the revised manuscript, we rephrase that statement to make that point more explicitly and add a similar statement right after Eq. (6).
As for the reclassification of the first term in $H_{3B}$, as we mention in our response to comment #1 by the first referee, we reaffirm that its classification as a threebody interaction is justified. The interaction fundamentally involves three spins, with the third spin $i$ mediating the emergent interaction—an observation that is crucial to explaining the behavior of the system under mild dilution, as considered in Ref.
\[30\].
As to whether this term alone can reproduce the reduction in the critical temperature—that is precisely what we find in this work. Namely, that this term dominates the reduction, at least in MF at zerofield, where the other two terms in $H_{3B}$, which we denote "quantum terms," completely vanish due to lattice symmetry. To emphasize this point, we slightly change the wording following Eq. (7).
Not only is the term in question dominant at zerofield, but we also show in a new appendix (Appendix F) that its behavior under nonzero transverse field is inline with experimental observations.
Outside of MF, we believe these terms may contribute to the reduction, albeit in a subdominant manner, possibly accounting for the remaining discrepancy compared to the experimental $T_{c}$. We raise this possibility in the Conclution section.
3) it seems the 3spin terms in the Hamiltonian imply the existence of nonclassical expectation values induced by classical 2point correlators, i.e. finite values of \<X> and \<Z> (possibly with shapedependent spatial profiles?). If true, the authors may want to discuss these as possible predictions for future experiments.
Offdiagonal dipolar terms indeed lead to a nontrivial distribution of internal magnetic fields, which subsequently induce finite local expectation values \<X> and \<Y>. This phenomenon was partly addressed in Ref.
\[30\]by examining the local xdirection fields in a Monte Carlo simulation (see Fig. 4 in that reference). While a similar analysis is possible within the analytical lowenergy effective Hamiltonian approach undertaken in the current paper, it does not follow trivially from the threespin terms in $H_{3B}$ and is thus outside the scope of the present work.
List of changes
 Added missing summation sum in Eqs. (14)(15).
 New appendix: D The hyperfine interaction
 includes new Figure 5
 New appendix: E Rescaling the longitudinal interaction
 includes new Figure 6
 New appendix: F Extended analytical characterization of $T_{c}$ in nonzero $B_x$
 includes new Figure 7
 Previous appendices "D Simulation details" and "E Effective lowenergy description of the Fe8 molecular magnet" moved to G and H, respectively
 Minor correction: in the paragraph preceding Figure 3, removed redundant k index.
 Added an index to the first term in Eq. (1)
 Amended the paragraph following Eq. (2) to include the value of $J_{ex}$
 Added a footnote on page 4 remarking on the values of the parameters $\rho$, $\alpha$, and $\Delta$
 Added a footnote on page 4 explaining the presence of the first term in $H_{3B}$
 Added a sentence mentioning the symmetry requirements on $H_{eff}$ in the paragraph following Eq. (6)
 Added a clarification at the end of the same paragraph (the one following Eq. (6)) regarding the naming distinction between the first term and last two terms in $H_{3B}$
 Changed "the result of the threebody interaction" to "the result of the first of the three threebody terms in $H_{3B}$" after Eq.(7)
 Added brief explanation and reference to Appendix F at the end of Section 4
 Exlicitly refer to the absence of terms with an odd number of Pauli z operators right before Eq. (10)
 slightly amended the phrasing of the caption of Fig. 2 to emphasize that the solid line as a whole is taken from Ref. \[37\]
 Refined the definition in Eq. (25) and modified the surrounding text (Appendix C) accordingly, emphasizing that $J^{\mu}_{\text{eff}}$ is defined within the lowenergy subspace. The referring text (first paragraph of Section 4) has been slightly changed to accordingly
 Two paragraphs added to the end of Appendix A justifying the cutoff of $H_{eff}$ at secondorder in $H_{T}$
Current status:
Reports on this Submission
Report
I thank the authors for the time they have taken to consider the recommendations in my first report, and to reply to them both via written comments in their resubmission and via extensive changes to the manuscript.
These changes, which include the addition of several appendices, have fully addressed my concerns. In particular, the additional argument at the end of appendix A regarding the negligibility of higherorder terms in the SchriefferWolff expansion is persuasive, and the new appendices D and E  and especially their figures  clarify the rescaling procedures used.
My former conclusion, viz. that one of the SciPost 'expectations' criteria is met, still stands; so, given that the authors have now also satisfied the 'general acceptance' criteria  including those that I felt were not satisfied by the original submission  I recommend acceptance for publication without further review.
Recommendation
Publish (meets expectations and criteria for this Journal)