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Competition between frustration and spin dimensionality in the classical antifer romagnetic $n$-vector model with arbitrary $n$
by N. P. Konstantinidis
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Submission summary
| Authors (as registered SciPost users): | Nikolaos Konstantinidis |
| Submission information | |
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| Preprint Link: | scipost_202208_00004v2 (pdf) |
| Date submitted: | 2022-09-17 20:10 |
| Submitted by: | Konstantinidis, Nikolaos |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
A new method to characterize the strength of magnetic frustration is proposed by calculating the minimum dimensionality of the absolute ground state of the classical nearest-neighbor antiferromagnetic $n$-vector model with arbitrary $n$. Platonic solids in three and four dimensions and Archimedean solids have lowest-energy configurations in a number of spin dimensions equal to their real-space dimensionality. Fullerene molecules and geodesic icosahedra can produce ground states in as many as five spin dimensions. Frustration is also characterized by the maximum value of the ground-state energy when the exchange interactions are allowed to vary.
List of changes
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Reports on this Submission
Report #1 by Anonymous (Referee 4) on 2022-9-20 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202208_00004v2, delivered 2022-09-20, doi: 10.21468/SciPost.Report.5726
Report
According to the abstract, the paper proposes a "new method to characterize the strength of magnetic frustration". I understand this as a broad statement about the geometry that should have implications for both the quantum and the classical case. The more narrow scope would be an investigation of the n-vector model on various frustrated geometries.
It seems to me that the author is inconsistent on this point: In the reply, he cites the excited states of C60, which corroborate his approach; but dodges the question on the mismatch for other fullerenes as "incomplete comparison" because the quantum case has only three spin components.
If the author makes an earnest effort to discuss the implications of his frustration measure for the quantum case (where it works and where it doesn't, how it compares to other measures etc.), I can accept the premise of the abstract and recommend it for a publication in SciPost. If the author wants to make a more narrow statement on the n-vector model, I would recommend a publication in SciPost Core.
Since no one has come up with a perfect measure of frustration yet, I realize that it makes no sense to be very strict here; but I think that at least an effort should be made.
Requested changes
- a discussion of the implications of the proposed frustration measure for the quantum case
- I think an inclusion of the icosidodecahedron, the truncated tetrahedron, and the truncated icosahedron in Tab. 2 is still helpful to have a comprehensive list (even if the results have been published before), but this is up to the author
Author: Nikolaos Konstantinidis on 2022-10-26 [id 2955]
(in reply to Report 1 on 2022-09-20)In order to perform the calculations in the paper, a method of classical minimization had to be used in more that three spin-space dimensions for the first time, and the method can be used up to arbitrary spin dimensions, limited only by computational resources. Consequently the calculations were very time consuming, especially since there is typically a number of symmetry-independent unique exchange interactions. Still, a relatively large number of molecules has been considered (to quote one of the remarks of the Referee from the first round of reviewing: "brings some classification into the zoo of molecules".)
To set up the analogous quantum-mechanical calculation would be very tedious, since one has to do it in more than three spin-space dimensions, and this calculation would have to be set up. To make matters worse, even to examine a small fraction of the molecules considered in the paper would take a very long time, since the quantum-mechanical calculations are significantly more time consuming. It must also be remembered that, unlike classical calculations, the calculation of the lowest-energy state for low-energy sectors of large molecules is not possible. In fact, calculating the low-energy spectrum of C60, reference 62 in the text, requires the use of DMRG and ended up as a separate publication in the current journal, SciPost. The corresponding quantum-mechanical project would be very time consuming, even in comparison with the current classical one.
With respect to the reference to C60 (reference 62), according to the text fullerenes of its symmetry have classical three-dimensional ground states, so a comparison with the quantum states (of three dimensions at least) is possible. This comparison is far from complete, but having the quantum-mechanical low-lying states for C60 is very challenging even for equal exchange interactions (and this is a paper published in SciPost). On the other hand, molecules like the fullerene with 28 vertices have more than three-dimensional classical ground states, and a comparison with the corresponding quantum states of reference 61 is not possible. This leaves as the only possible candidates with three-dimensional classical states the 24 and 30 sites fullerenes, since the other molecules with three-dimensional ground states are too big. As was stated in the previous paragraph, a formal calculation would first treat the quantum mechanical problem in arbitrary dimensions, which would be quite tedious. Even in three dimensions the time required would be significant. In reference 61 the quantum model was considered only for spatially uniform exchange interactions, unlike the classical case presented here, and those calculatons required significant time. It was also not possible to find the excited states of the 36 cite cluster in reference 59, due to computational limitations that still exist. Finding only the lowest-energy state in a specific irreducible representation of a specific Sz sector is technically easier than calculating more low-lying states in the same sector.
To summarize, it is highly challenging and not only time-wise to scrutinize the current method for the quantum case, to compare it with other measures etc. Only the corresponding calculations for the systems in this paper and only up to three dimensions, classical or quantum, have spanned many papers. Therefore I do not understand the purpose served by words such as "dodges" and "earnest".
With respect to the second requested change, the calculations in this paper have shown that for the truncated tetrahedron and the truncated icosahedron the lowest-energy classical ground states are three-dimensional and have been already found in the literature, so there is nothing new to be added. For the icosidodecahedron it was already known that the lowest-possible classical ground state is two-dimensional. Apart from the molecules associated with new results, I prefer to only include the odd polygons in the Tables since they are fundamental to the concept of frustration.
Anonymous on 2022-11-10 [id 3003]
(in reply to Nikolaos Konstantinidis on 2022-10-26 [id 2955])This is clearly a misunderstanding, I don't expect the author to make any new calculations that are beyond the scope of the paper, but merely to compare with existing data.
Let me try to reformulate: One common measure of frustration is given by the degeneracy of the classical Heisenberg model on a particular geometry. For example, the kagome lattice is infinitely degenerate, while the triangular lattice is not, and is in this sense stronger frustrated. This is a statement on the geometry. The strong frustration that manifests itself in the degenerate classical state will manifest itself differently in the quantum case; but generally, we can use the classical result as a guide to identify interesting geometries and parameter regimes to look at in the quantum case.
As I initially understood the paper, it introduces another measure of frustration given by the dimension n_g. So if we have n_g=4 for C28, it similarly seems to imply that this particular geometry is stronger frustrated than others, which should manifest itself in some manner in the quantum case in the more physical 3-dimensional case.
If it does, then we have very interesting implications. For example, it would reveal the most interesting molecular geometries (such as the geodesic polyhedra with n_g=5) out of a large zoo of possibilities, where one would expect interesting states for other models as well. It would furthermore motivate to look for these specific molecules and states in the experiment.
(Please note that I'm trying to help to improve the paper.)
But such implications are not discussed and the author seems to maintain that the n-vector model can only be compared to the n-dimensional Heisenberg model. In this case, the scope of the paper is more narrow than I understood initially and I would tend to recommend a publication in SciPost Core.