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Magnetic properties of a capped kagome molecule with 60 quantum spins
by Roman Rausch, Matthias Peschke, Cassian Plorin, Christoph Karrasch
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Submission summary
| Authors (as registered SciPost users): | Christoph Karrasch · Matthias Peschke · Cassian Plorin · Roman Rausch |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202111_00015v2 (pdf) |
| Date accepted: | 2022-03-22 |
| Date submitted: | 2022-03-01 12:52 |
| Submitted by: | Rausch, Roman |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We compute ground-state properties of the isotropic, antiferromagnetic Heisenberg model on the sodalite cage geometry. This is a 60-spin spherical molecule with 24 vertex-sharing tetrahedra which can be regarded as a molecular analogue of a capped kagome lattice and which has been synthesized with high-spin rare-earth atoms. Here, we focus on the $S=1/2$ case where quantum effects are strongest. We employ the SU(2)-symmetric density-matrix renormalization group (DMRG). We find a threefold degenerate ground state that breaks the spatial symmetry and that splits up the molecule into three large parts which are almost decoupled from each other. This stands in sharp contrast to the behaviour of most known spherical molecules. On a methodological level, the disconnection leads to ``glassy dynamics'' within the DMRG that cannot be targeted via standard techniques. In the presence of finite magnetic fields, we find broad magnetization plateaus at 4/5, 3/5, and 1/5 of the saturation, which one can understand in terms of localized magnons, singlets, and doublets which are again nearly decoupled from each other. At the saturation field, the zero-point entropy is $S=\ln(182)\approx 5.2$ in units of the Boltzmann constant.
List of changes
- change of title to "Magnetic properties of a capped kagome molecule with 60 quantum spins"
- shortening of the abstract to 200 words
- incorporation of various new references as suggested by the referees
- correction of S=ln(181) to S=ln(182)
- change of all usages of "spin liquid"
- "magic fraction" removed
- remark that the degeneracy is not a numerical artifact
- remark on symmetry breaking in finite systems
- sketch of the neighbourhood of a tetrahedron in Fig. 3
- more ways to quantitatively characterize the disconnection patterns in Sec. 5
- ball-and-stick drawing of the SOD20 molecule in Fig. 4
- new subsection 5.1: "Nearest-neighbour valence bond picture"
- sketch of the hexagon localization domains in Fig. 5
- discussion of why there are 13 instead of 14 localized magnons in Sec. 6.1
- discussion of the relationship of the localized doublets and singlets to findings from other papers
- conclusion: added Ref. [38] and changed the discussion of the comparison to the pyrochlore lattice
- addition of clarifying remarks regarding the geometry and the computational details
- a number of various small chages: typos, hyphens, word replacements etc. (often as pointed out by the referees)
Published as SciPost Phys. 12, 143 (2022)
Reports on this Submission
Report #3 by Anonymous (Referee 1) on 2022-3-14 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202111_00015v2, delivered 2022-03-14, doi: 10.21468/SciPost.Report.4682
Strengths
see previous report
Weaknesses
see previous report
Report
—— beginning of report ——
I have read the revised draft and the point by point reply of the Authors. The Authors have removed the conflicting term “spin liquid” and have added a discussion on what they mean by “symmetry broken” state for this finite-size system. They have also incorporated other suggestions which have improved the content of the paper.
I would therefore like to recommend the paper for publication.
Further comments/suggestions:
1) On the notion of “symmetry broken” state, the Authors may wish to further clarify sentences such as:
line 8: “We find a threefold degenerate ground state that breaks the spatial symmetry”
lines 95-96: “… making it threefold degenerate and thus in principle symmetry-broken.”
Such statements give the general impression that degeneracy implies symmetry breaking. Perhaps it would help if the Authors clarify that what they mean is that the ground state manifold contains different irreps (angular momentum L=0, 2pi/3 and -2pi/3) of the C3 subgroup, associated with one of the four C3 axes of the molecule. As long as these have different irreps, a linear combination can break the C3 symmetry (in the finite-size sense).
2) The Authors may wish to add a few sentences about the symmetry group and perhaps illustrate one of the three-fold axes in Fig. 1, as this is the symmetry that is broken (again, in the finite-size sense).
3) line 133: “...has the irreducible representations A (1), E (2), T (3)”:
The group Oh has 10 irreps in total: A1g(1), A1u(1), A2g(1), A2u(1), Eg(2),Eu(2),T1g(3) ,T1u(3),T2g(3),T2u(3).
Perhaps the Auhors meant "classes of irreps"?
4) line 166: “...three rotational symmetry axes that pierce the square faces”:
The three-fold axes pierce the hexagon faces (and therefor connect one square face to another) and not the square faces. I guess this is a typo.
5) regarding the NNVB picture and the intuitive explanation of the different strengths of NN spin-spin correlations:
Perhaps the Authors can still add a few sentences and explicitly mention that the spin-spin correlations on squares are much stronger than those on hexagons due to stronger tunneling, but still weaker than some of the outer bonds, which have strong dimerisation due to the apical dangling spins.
— end of report —
Requested changes
Further comments/suggestions:
1) On the notion of “symmetry broken” state, the Authors may wish to further clarify sentences such as:
line 8: “We find a threefold degenerate ground state that breaks the spatial symmetry”
lines 95-96: “… making it threefold degenerate and thus in principle symmetry-broken.”
Such statements give the general impression that degeneracy implies symmetry breaking. Perhaps it would help if the Authors clarify that what they mean is that the ground state manifold contains different irreps (angular momentum L=0, 2pi/3 and -2pi/3) of the C3 subgroup, associated with one of the four C3 axes of the molecule. As long as these have different irreps, a linear combination can break the C3 symmetry (in the finite-size sense).
2) The Authors may wish to add a few sentences about the symmetry group and perhaps illustrate one of the three-fold axes in Fig. 1, as this is the symmetry that is broken (again, in the finite-size sense).
3) line 133: “...has the irreducible representations A (1), E (2), T (3)”:
The group Oh has 10 irreps in total: A1g(1), A1u(1), A2g(1), A2u(1), Eg(2),Eu(2),T1g(3) ,T1u(3),T2g(3),T2u(3).
Perhaps the Auhors meant "classes of irreps"?
4) line 166: “...three rotational symmetry axes that pierce the square faces”:
The three-fold axes pierce the hexagon faces (and therefor connect one square face to another) and not the square faces. I guess this is a typo.
5) regarding the NNVB picture and the intuitive explanation of the different strengths of NN spin-spin correlations:
Perhaps the Authors can still add a few sentences and explicitly mention that the spin-spin correlations on squares are much stronger than those on hexagons due to stronger tunneling, but still weaker than some of the outer bonds, which have strong dimerisation due to the apical dangling spins.
Report
The authors have taken into consideration all suggestions and remarks from my previous reviewer report. Accordingly, the manuscript has been substantially improved and I therefore recommend the present version of the manuscript for publication.
Author: Roman Rausch on 2022-05-06 [id 2447]
(in reply to Report 3 on 2022-03-14)>On the notion of “symmetry broken” state, the Authors may wish to further clarify sentences such as:
>
>line 8: “We find a threefold degenerate ground state that breaks the spatial symmetry”
>lines 95-96: “… making it threefold degenerate and thus in principle symmetry-broken.”
>
>Such statements give the general impression that degeneracy implies symmetry breaking. Perhaps it would help if the Authors clarify that what they mean is that the ground state manifold contains different irreps (angular momentum L=0, 2pi/3 and -2pi/3) of the C3 subgroup, associated with one of the four C3 axes of the molecule. As long as these have different irreps, a linear combination can break the C3 symmetry (in the finite-size sense).
We would say that symmetry breaking implies degeneracy, but not vice versa (topological degeneracy is a prime counterexample), so we agree that the statement is somewhat misleading and we have revised it in the final version.
However, the ground-state manifold does not contain several irreps. Rather, we have a nontrivial, 3-dimensional irrep.
>2) The Authors may wish to add a few sentences about the symmetry group and perhaps illustrate one of the three-fold axes in Fig. 1, as this is the symmetry that is broken (again, in the finite-size sense).
We have illustrated the C4 and C3 symmetry axes in Fig. 1 of the final manuscript.
>3) line 133: “...has the irreducible representations A (1), E (2), T (3)”:
>The group Oh has 10 irreps in total: A1g(1), A1u(1), A2g(1), A2u(1), Eg(2),Eu(2),T1g(3) ,T1u(3),T2g(3),T2u(3).
>Perhaps the Auhors meant "classes of irreps"?
This has been improved in the final version.
>4) line 166: “...three rotational symmetry axes that pierce the square faces”:
>The three-fold axes pierce the hexagon faces (and therefor connect one square face to another) and not the square faces. I guess this is a typo.
This is not a typo: There are both 3-fold axes and 4-fold axes. The 4-fold axes are easier to recognize on the square projection, so we were mainly thinking and writing in terms of them.
To be precise, the symmetry is broken such that two of the three 4-fold axes become 2-fold axes, while the 3-fold axes cease to be symmetry axes altogether. We have included this explanation into the final manuscript.
>5) regarding the NNVB picture and the intuitive explanation of the different strengths of NN spin-spin correlations:
>Perhaps the Authors can still add a few sentences and explicitly mention that the spin-spin correlations on squares are much stronger than those on hexagons due to stronger tunneling, but still weaker than some of the outer bonds, which have strong dimerisation due to the apical dangling spins.
We believe that our statement expresses exactly this:
"In the case of SOD60, parallel bonds are found on the square plaquettes (blue-blue) and this may explain their relatively large correlations [...] at the expense of the red-blue and green-blue ones. This leaves the red (apex) spins to couple more strongly with the green spins."
Therefore, we have chosen not to do further changes in this case.
We would like to once more thank the referee for a thorough reading of our work and for these final comments.