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A brief note on the G$_2$ Affleck-Kennedy-Lieb-Tasaki chain
by Hosho Katsura, Dirk Schuricht
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Dirk Schuricht |
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| Preprint Link: | https://arxiv.org/abs/2503.18885v2 (pdf) |
| Date submitted: | March 28, 2025, 5:59 a.m. |
| Submitted by: | Dirk Schuricht |
| Submitted to: | SciPost Physics Core |
| for consideration in Collection: |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We consider the valence bond solid (VBS) state built of singlet pairs of fundamental representations and projected onto adjoint representations of the exceptional Lie group G$_2$. The two-point correlation function in the VBS state is non-vanishing only for nearest neighbours, but possesses finite string order. We construct a parent Hamiltonian for the VBS state, which constitutes the G$_2$ analog of the famous AKLT chain.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2025-7-1 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2503.18885v2, delivered 2025-07-01, doi: 10.21468/SciPost.Report.11495
Strengths
2-The presentation is very detailed and easy to follow
3-The state appears to exhibit some non-standard features such as a vanishing two-point correlation function beyond nearest neighbors
Weaknesses
Report
I checked various of the technical statements in detail and could not find a mistake. Still, I would recommend to comment more clearly on the role of the two different bases $|\psi_a\rangle$ and $|a\rangle$. Specifically, it should be mentioned that the $\Lambda^a$ in the adjoint representation as defined at the end of Appendix A act on the $|\psi_a\rangle$. This is sort of clear given that $\Lambda^3$ and $\Lambda^8$ are not diagonal in the adjoint representation (if one works out the matrices) [which is the reason to introduce the $|a\rangle$] and also from footnote 7 where the matrix elements appear and are (implicitly) identified with the structure constants to get the antisymmetry property. However, I started a computer implementation based on the information in the appendices and got quite confused initially.
Requested changes
There are a few typos and I leave it to the discretion of the authors to address these
1-"periondic" above Eq (2)
2-In footnote 7 there are two incarnations of $g_2$ instead of $G_2$
3-In contrast to the other referee I would suggest to stick to "parent Hamiltonian"
4-Please clarify the role of the two different bases in the adjoint representation (see above)
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report #1 by Anonymous (Referee 1) on 2025-4-30 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2503.18885v2, delivered 2025-04-30, doi: 10.21468/SciPost.Report.11116
Strengths
Weaknesses
Report
Referee Report
The manuscript entitled “A brief note on the G2 Affleck–Kennedy–Lieb–Tasaki chain” is an excellent short paper on the valence bond solid (VBS) with G2 symmetry. The authors provide a concise and well-rounded introduction and review of the construction by Affleck–Kennedy–Lieb–Tasaki.
The fundamental representation of G2 is 7-dimensional. On each lattice site, two such representations are introduced. In the decomposition of their tensor product, the representations 1 ⊕ 7 ⊕ 14 ⊕ 27 appear, of which all except the 14-dimensional adjoint representation are projected out.
The resulting rank-3 tensor serves as the basic building block for the multi-spin state composed of 14-dimensional adjoint representations: neighbouring tensors are connected by contracting their adjacent 7-dimensional components into singlets. Locally, the product of two adjoint (14-dimensional) representations may contain 1 ⊕ 14 ⊕ 27 ⊕ 77 ⊕ 77′. However, due to the specific construction of the VBS state — starting from four 7-dimensional representations of which two are contracted to a singlet — at most 1 ⊕ 7 ⊕ 14 ⊕ 27 can appear, resulting in 1 ⊕ 14 ⊕ 27.
The parental Hamiltonian, designed to take this constructed VBS state as its ground state, is chosen to annihilate the 1 ⊕ 14 ⊕ 27 components while assigning finite positive energy to the 77 ⊕ 77′ components. The authors investigate the two-point correlation function via a transfer matrix method and find it to be non-vanishing only for nearest neighbours. However, a string order parameter can be defined, which turns out to be finite.
The paper is well written, very accessible, and the results are, to my knowledge, new. The main body is presented with minimal technical detail — the authors rely primarily on fundamental properties of tensor product decompositions and the eigenvalues of the quadratic Casimir operator. The actual computations begin with the evaluation of the correlation function and the string order parameter, with technical details deferred to the appendices.
I would, however, suggest that the authors comment on the uniqueness of the ground state (not just of the VBS state itself). Typically, one would need to prove that the excited states remain gapped, even in the thermodynamic limit, to confirm uniqueness — a technically demanding part of the original AKLT construction. It would be valuable to clarify whether the G2 symmetry affects this aspect or simplifies the situation in any way.
I recommend this manuscript for publication in SciPost.
discovered "typos":
can be regarded as the G2 generalisation of the famous AKLT construction. -> can be regarded as the G2 version/analog of the famous AKLT construction.
ie -> i.e.
parent Hamiltonian -> parental Hamiltonian
hidden antiferro-magnetic order exist. -> hidden antiferro-magnetic order exists.
subindex -> index or subscript
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)