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Fractional magnetization plateaux of a spin-1/2 Heisenberg model on the Shastry-Sutherland lattice: effect of quantum XY interdimer coupling
by T. Verkholyak, J. Strecka
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Submission summary
| Authors (as registered SciPost users): | Jozef Strecka · Taras Verkholyak |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202110_00002v1 (pdf) |
| Date submitted: | 2021-10-05 11:54 |
| Submitted by: | Verkholyak, Taras |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Spin-1/2 Heisenberg model on the Shastry-Sutherland lattice is considered within the many-body perturbation theory developed from the exactly solved spin-1/2 Ising-Heisenberg model with the Heisenberg intradimer and Ising interdimer interactions. The former model is widely used for a description of magnetic properties of the layered compound SrCu$_2$(BO$_3$)$_2$, which exhibits a series of fractional magnetization plateaux at sufficiently low temperatures. Using the novel type of many-body perturbation theory we have found the effective model of interacting triplet excitations with the extended hard-core repulsion, which accurately recovers 1/8, 1/6 and 1/4 magnetization plateaux for moderate values of the interdimer coupling. A possible existence of a striking quantum phase of bound triplons is also revealed at low enough magnetic fields.
Author comments upon resubmission
We are grateful to the Referees for the careful reading and valuable remarks and suggestions, which we have taken into account and addressed in the manuscript. We also submitted our responses to the Referee reports and the list of changes.
We herewith submit a revised version of our manuscript which we hope is suitable for the publication in SciPost Physics.
Yours sincerely,
Taras Verkholyak, Jozef Strečka
List of changes
All implemented changes have been highlighted in blue in the revised manuscript.
Section 1
We extended the introduction.
We added new references suggested by the referees, and also commented about the Dzyaloshinskii-Moriya interaction. (page 2)
Due to the suggestion of Referee 4, we added the schematic phase diagram which highlights the results of the current work. (Fig.1 on page 3 and the text below)
We also made a small correction in the text due to the remarks of all referees.
Section 2
We modified Fig. 3 (Fig.2 in the previous version) to indicate more clearly the hard-core condition. (Fig.3 on page 6)
We extended the explanation regarding the hard-core condition. (see the text after Eq.(4) on page 5)
Section 3
We added the comment about the parameters of the effective model after Eq. (6). (see the text after Eq.(6), pages 7-8)
We changed the paragraph, where the multi-particle interactions are discussed. (page 9)
In the next paragraph we added the clear statement that the hopping terms are ignored in Sec.3 and will be analyzed in Sec. 4. (page 9)
In the subsequent paragraphs we extended the discussion about the origin of the fractional plateaux, the macroscopic degeneracy at the border between different plateux phases. We also provided more details about the 2/15 plateau phase. (pages 9-10)
We changed the paragraph where the relation to SrCu2(BO3)2 is discussed and removed the paragraph about the applicability of the effective model to the low-temperature thermodynamics. (pages 11-12)
We added the missing information to the caption of Fig. 9 (Fig.8 in the previous version). (page 12)
Section 4
We modified Sec.4 for clarity. We pointed out what is the rigorous result there. (pages 13-15)
Section 5
We modified the Conclusions to stress the importance of the achieved results. (page 16)
Appendix B
We added Fig. 12 illustrating virtual processes for the correlated hopping terms and the corresponding text and Eqs. (33), (34) after Eq. (32). (page 24)
New Appendix C is added, where it is described how to prove the ground-state phase diagram for the case of the localized triplons excluded the correlated hopping terms. (pages 25-30)
Current status:
Reports on this Submission
Report #3 by Anonymous (Referee 8) on 2021-11-8 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202110_00002v1, delivered 2021-11-08, doi: 10.21468/SciPost.Report.3812
Strengths
See previous report
Weaknesses
See previous report
Report
Although the authors have made many changes to their manuscript, this referee is disappointed by the results. Despite being given many opportunities for improvements, and in some cases verbatim suggestions, many of the points at issue remain unresolved or very poorly explained. Under these circumstances it is no longer clear that the problems can be blamed only on language, as opposed to lying in the authors' understanding of their subject matter.
Primary issues:
1) The explanation of the fact that the treatment is perturbative in J'_xy remains totally inadequate. It is completely confusing that J' is fixed in Eq. (2) and then reused in Eq. (5) when in fact the correct statement of this prefactor is that in Eq. (21); it is then further confusing that a clear statement is presented concerning J' possibly needing to be very small, but nothing is stated about J'_xy and the authors persist in showing results only for J'_xy = J'_{Eq. (2)}.
2) The figures, state degeneracies and the authors' claims concerning the plateau phases remain confusing. The text states "highly" and in places "macroscopically degenerate," but the authors show figures with lines of dimers (in general, too few dimers are shown to judge whether a triplon arrangement is non-repeating) and use text such as "vertical and horizontal ordering," both of which imply that there are not so many possible states. The fact that the main-text figures for the 1/8 states show highly ordered configurations while the appendix figures are less (but not clearly dis)ordered feeds this confusion.
3) The bound-triplon state remains incomprehensible, because of a complete failure to answer the explicit question of whether this is an isolated two-triplon object or a sliding crystal of two-triplon objects. Is Fig. 11 trying to show one two-triplon excitation or a crystal of bitriplons ? The sentence in the summary "stripe-like order of delocalised bound triplons" appears to be self-contradictory: are they delocalised or ordered ? Physically, do these objects have to propagate to gain energy (the authors' use of "free-wave state") ? How can they propagate if many other triplets are occupied ? What is the meaning of the red arrows in Fig. 11 ? [See the confusion voiced in New Report 1.] Separately, do these bitriplon objects have any connection to the excited states known to form at zero field, or are they only a property of the perturbative theory at the 1/3-plateau boundary ?
Secondary issues:
a) Indeed the authors added words about the Dzyaloshinskii-Moriya interactions in SCBO in 2 places, but only as a disconnected observation with no comment (even a qualitative one) on how these interactions might affect their results.
b) The words about the 2/15 state appear to remain a meaningless "conjecture" if the authors do not present an argument for why these specific configurations might be favoured among the macroscopically degenerate possibilities.
c) Other referees have commented on the enduring problems in explaining the correlated hopping processes; perhaps the authors could explain whether the situation at the 1/3-plateau edge where they work is different from the conventional discussion of correlated hopping terms.
Detail issues:
The authors' attention to detail is poor: they cannot even make suggested changes to the English, resulting in enduring confusion about "sufficiently low" (the 1/3-plateau boundary is absolutely not a low field) and "significantly low/small" (it is necessary to use a word other than "significantly" -- see also New Report 1); there is poor linking of text and figures (Eq. (4) is certainly not shown in Fig. 3); the authors' hard-core condition remain only an assertion, such that the manuscript is not self-contained on such an important conceptual issue; it seems that problems with paper titles were already raised at the first round of refereeing.
In summary, the manuscript remains unsuitable for publication despite the extensive input provided in the refereeing process, and both authors and editors should consider whether the potential exists to rectify this situation.
Report #1 by Anonymous (Referee 7) on 2021-10-18 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202110_00002v1, delivered 2021-10-18, doi: 10.21468/SciPost.Report.3625
Report
The work is a new pathway to addressing the in-field phase diagram of the Shastry-Sutherland model. This is a fundamental problem in frustrated magnetism and is also connected to important experiments on SrCu2(BO3)2. The authors have demonstrated that key features of the phase diagram can be captured from an effective Hamiltonian obtained by perturbing around the Ising-Heisenberg limit. The paper, in addition, meets the general acceptance criteria for publication.
The authors have addressed the very extensive remarks from the reviewers and the revised version is now significantly clearer. I recommend publication in Scipost following a couple of minor clarifications.
Requested changes
1. There are many objects in figure 11 that are not clearly defined: dimers shaded or not, circles that are empty, red, black, red-black-horizontal or vertical, black-white horizontal and black and red arrows. All this on top of the hard core condition illustrated in Fig 3. At the moment it is hard for me to see that a reader could find this illuminating. It would be helpful to explain this figure in much more detail (maybe in a reasonably self-contained way as far as possible) and tie it to equations in the text as well as to Fig. 3.
2. Pg. 12. "..significantly small interplane [58] couplings" The sentence is not completely clear. Also, the cited work is at high pressure. Is there ambient pressure evidence for inter-layer couplings? If not, perhaps soften the statement to "possible small interplane couplings."
Author: Taras Verkholyak on 2021-11-15 [id 1942]
(in reply to Report 1 on 2021-10-18)
We are grateful to the referee for the positive value of our manuscript and the remarks. We changed the manuscript according to them.
Requested changes
Referee says:
-
There are many objects in figure 11 that are not clearly defined: dimers shaded or not, circles that are empty, red, black, red-black-horizontal or vertical, black-white horizontal and black and red arrows. All this on top of the hard core condition illustrated in Fig 3. At the moment it is hard for me to see that a reader could find this illuminating. It would be helpful to explain this figure in much more detail (maybe in a reasonably self-contained way as far as possible) and tie it to equations in the text as well as to Fig. 3.
Our response: In a new version of Fig. 11, horizontal and vertical ellipses representing the bound state of two triplons were missed. We changed the figure to restore the correct picture, and to show the possible configuration for the stripe-like phase of bound triplons. We added the description to the figure caption and extended the text after Eq. (14).
Referee says: 2. Pg. 12. "..significantly small interplane [58] couplings" The sentence is not completely clear. Also, the cited work is at high pressure. Is there ambient pressure evidence for inter-layer couplings? If not, perhaps soften the statement to "possible small interplane couplings."
Our response: We are not aware of any paper where the interplane coupling in SrCu2(BO3)2 has been determined or measured. In Ref. [58] (Ref.[59] in the present version) a general assumption for such a coupling has been made irrelevant to the applied pressure. Therefore, we changed the text due to referee's suggestion. (see the last sentence of Sec. 3).
Report #2 by Anonymous (Referee 6) on 2021-10-12 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202110_00002v1, delivered 2021-10-12, doi: 10.21468/SciPost.Report.3655
Report
The revised version of the paper has a better focus of what
has been done and what has been achieved.
The authors have implemented quite a number
of changes to follow the Referees' comprehensive suggestions
and to improve the manuscript thereby.
There are a few points which (still) puzzle me:
1)
The correlated hopping is an essential process in the
effective model. I understand that its influence is
reduced in certain crystallized triplon solids.
Still, it appears in order J'^2/J^2 and is thus
potentially large. In the reply to Referee 1 it is argued
that it is order six. This is not true, see for instance
Knetter et al. PRL 85,3958 (2000) and
Knetter et al. PRL 92, 027204 (2004).
2)
In Knetter et al. PRL 92, 027204 (2004) it appears
that the two-triplon bound state disperses quite sizably
along the horizontal/vertical axes as well as
along the diagonals. So it appears appropriate to
discuss why this does not show up in the effective
model investigated where only horizontal/vertical
motion is found. Is it because the manuscript
uses second order perturbation theory?
3)
Your approach breaks the isotropic spin symmetry.
Does this constitute a problem in the sense that
you get unphysical admixtures?
4)
The titles in the references are screwed up with lots
of small letters where capital letters should appear as
ising -> Ising and so on.
Once the above points are accounted for I
recommend publication.
Author: Taras Verkholyak on 2021-10-13 [id 1845]
(in reply to Report 2 on 2021-10-12)
We are grateful to the referee for the remarks and positive judgment of the revised version of our manuscript.
Please find our response to the remarks below.
Referee says:
1)
The correlated hopping is an essential process in the
effective model. I understand that its influence is
reduced in certain crystallized triplon solids.
Still, it appears in order J'^2/J^2 and is thus
potentially large. In the reply to Referee 1 it is argued
that it is order six. This is not true, see for instance
Knetter et al. PRL 85,3958 (2000) and
Knetter et al. PRL 92, 027204 (2004).
Our response:
Thanks to the current report, we realized that our response to Referee 1 was not comprehensive enough on this point. Unfortunately, in the previous response we accidentally confused the term pair hopping (which means the correlated hopping) with the simple hopping term (which is proportional to (J'/J)^6). Surely, we agree with the Referee that the correlated hopping is of order J'^2/J^2. This is also clear from Eq. (35) of our manuscript. The results presented in Fig. 7(a) bear evidence that the magnitude of the correlated hopping term is of the same order as the effective pair interaction between triplons. We should stress that in the manuscript we never claimed that the correlated hopping term is proportional to (J'/J)^6.
We hope that the explanation above is emending our previous response to Referee 1 as well.
Referee says:
2)
In Knetter et al. PRL 92, 027204 (2004) it appears
that the two-triplon bound state disperses quite sizably
along the horizontal/vertical axes as well as
along the diagonals. So it appears appropriate to
discuss why this does not show up in the effective
model investigated where only horizontal/vertical
motion is found. Is it because the manuscript
uses second order perturbation theory?
Our response:
Within the perturbative theories in [Knetter et al. PRL 85,3958 (2000); Knetter et al. PRL 92, 027204 (2004); Momoi and Totsuka PRB 62, 15067 (2000)] the correlated hopping terms include also configurations with triplons on the nearest horizontal and vertical dimers. As a result, the quantum state can be extended onto the whole lattice. In our theory the states with the triplons on the nearest horizontal and vertical states are forbidden due to the hard-core condition shown in Fig.3. Therefore, the aforementioned correlated hopping terms will not appear even in the higher order perturbation of our approach.
Thank you for the remark. We will add the corresponding note to the next revision of our manuscript.
Referee says:
3)
Your approach breaks the isotropic spin symmetry.
Does this constitute a problem in the sense that
you get unphysical admixtures?
Our response:
Our approach is consistent with the symmetry of the system which already looses its isotropic character in the presence of the magnetic field. As far as we can see, the effective model does not manifest any unphysical admixture within the second-order perturbation theory.
Referee says:
4)
The titles in the references are screwed up with lots
of small letters where capital letters should appear as
ising -> Ising and so on.
Our response:
Thank you for the remark. We will correct the references in the bibliography accordingly.
Author: Taras Verkholyak on 2021-11-15 [id 1943]
(in reply to Report 3 on 2021-11-08)Primary issues:
1) The explanation of the fact that the treatment is perturbative in J'xy remains totally inadequate. It is completely confusing that J' is fixed in Eq. (2) and then reused in Eq. (5) when in fact the correct statement of this prefactor is that in Eq. (21); it is then further confusing that a clear statement is presented concerning J' possibly needing to be very small, but nothing is stated about J'_xy and the authors persist in showing results only for J'_xy = J'.
We do not agree with referee's assessment. We think that our treatment is consistent. From the very beginning we consider the isotropic Heisenberg model on the Shastry-Sutherland lattice. Therefore, the selection of the perturbative part is rather straightforward. Probably, the main concern of the referee is that the formal procedure for the perturbation uses the expansion with respect to a small parameter, which finally can be set to 1. The latter can be used to indicate the order of the perturbative expansion. To avoid the further confusion, we changed Eq. (5) and text below it.
2) The figures, state degeneracies and the authors' claims concerning the plateau phases remain confusing. The text states "highly" and in places "macroscopically degenerate," but the authors show figures with lines of dimers (in general, too few dimers are shown to judge whether a triplon arrangement is non-repeating) and use text such as "vertical and horizontal ordering," both of which imply that there are not so many possible states. The fact that the main-text figures for the 1/8 states show highly ordered configurations while the appendix figures are less (but not clearly dis)ordered feeds this confusion.
We do not want to overload the main text by the extensive graphic material. Therefore, we put the details for the 1/8 plateau phase to Appendix C, and added the reference in the main text. We hope that this concern of the referee is not so crucial, since other reviewers did not claim that there could be any confusion in the nature of the presented ground states.
3) The bound-triplon state remains incomprehensible, because of a complete failure to answer the explicit question of whether this is an isolated two-triplon object or a sliding crystal of two-triplon objects. Is Fig. 11 trying to show one two-triplon excitation or a crystal of bitriplons ? The sentence in the summary "stripe-like order of delocalised bound triplons" appears to be self-contradictory: are they delocalised or ordered ? Physically, do these objects have to propagate to gain energy (the authors' use of "free-wave state") ? How can they propagate if many other triplets are occupied ? What is the meaning of the red arrows in Fig. 11 ? [See the confusion voiced in New Report 1.] Separately, do these bitriplon objects have any connection to the excited states known to form at zero field, or are they only a property of the perturbative theory at the 1/3-plateau boundary ?
We changed the text below Eq. (14). We also changed Fig. 11 and its caption, to demonstrate more clearly the possible stripe-like phase of delocalized bound triplons. On this schematic picture we draw the pairs of triplons on particular position of the dimer lattice, however it is evident that the quantum hopping inherent for these pairs leads to the quantum phase along the hopping direction. The description of the stripe phase is given in the last paragraph of Sec. 4. The referee also states: "The sentence in the summary "stripe-like order of delocalised bound triplons" appears to be self-contradictory: are they delocalised or ordered ?"
In fact, our statement is not self-contradictory. The bound triplons state is delocalized in one dimension. The corresponding explanation has been given in the last paragraph of Sec.4.
Secondary issues:
a) Indeed the authors added words about the Dzyaloshinskii-Moriya interactions in SCBO in 2 places, but only as a disconnected observation with no comment (even a qualitative one) on how these interactions might affect their results.
The aim of our work was to study the pure Shastry-Sutherland model. The Dzyaloshinskii-Moriya interaction in SrCu2(BO3)2 has rather complicated form. Although its effect can be studied within the suggested method, we did not touch this point in our work and, therefore, it cannot be commented extensively here.
b) The words about the 2/15 state appear to remain a meaningless "conjecture" if the authors do not present an argument for why these specific configurations might be favoured among the macroscopically degenerate possibilities.
The 2/15 plateau has been conjectured in a number of previous theoretical and experimental papers. In our work we would like to stress that such a plateau, although rather tiny, might appear in the Shastry-Sutherland model. The same statement holds also for other intermediate plateaus below the 1/6 plateau phase.
c) Other referees have commented on the enduring problems in explaining the correlated hopping processes; perhaps the authors could explain whether the situation at the 1/3-plateau edge where they work is different from the conventional discussion of correlated hopping terms.
We added the corresponding comment on page 9 (please see also our response to referee 2).
Detail issues:
The authors' attention to detail is poor: they cannot even make suggested changes to the English, resulting in enduring confusion about "sufficiently low" (the 1/3-plateau boundary is absolutely not a low field) and "significantly low/small" (it is necessary to use a word other than "significantly" -- see also New Report 1); there is poor linking of text and figures (Eq. (4) is certainly not shown in Fig. 3); the authors' hard-core condition remain only an assertion, such that the manuscript is not self-contained on such an important conceptual issue; it seems that problems with paper titles were already raised at the first round of refereeing.
We generally do not agree with this criticism. In particular, what concern some technical remarks, "significantly small" appeared in the revised version, and we have corrected it due to the suggestion of referee 1 in the present revision. The referee has the remark that "Eq. (4) is certainly not shown in Fig. 3", but there is no such statement in our manuscript. As far as the title of our paper is concerned we believe that it exactly matches the results presented in this work. For instance in Fig. 9 we directly showed the effect of quantum XY part of the interdimer coupling, when comparing the exact results for the Ising-Heisenberg model without this term with the fully isotropic Heisenberg model.