# Fractional magnetization plateaux of a spin-1/2 Heisenberg model on the Shastry-Sutherland lattice: effect of quantum XY interdimer coupling

### Submission summary

 As Contributors: Taras Verkholyak Preprint link: scipost_202110_00002v1 Date submitted: 2021-10-05 11:54 Submitted by: Verkholyak, Taras Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory 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.

###### Current status:
Editor-in-charge assigned

### Author comments upon resubmission

Dear Editor,

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)

### Submission & Refereeing History

Resubmission scipost_202110_00002v1 on 5 October 2021
Submission 2104.05797v2 on 16 April 2021

## Reports on this Submission

### 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."

• validity: -
• significance: good
• originality: good
• clarity: good
• formatting: good
• grammar: good

### 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.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Author:  Taras Verkholyak  on 2021-10-13  [id 1845]

(in reply to Report 2 on 2021-10-12)
Category:
remark
correction
pointer to related literature

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.