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Groundstate and thermodynamic properties of the spin$\frac{1}{2}$ Heisenberg model on the triangular lattice
by Matías G. Gonzalez, Bernard Bernu, Laurent Pierre, Laura Messio
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Submission summary
Authors (as registered SciPost users):  Matías Gonzalez · Laura Messio 
Submission information  

Preprint Link:  https://arxiv.org/abs/2112.08128v1 (pdf) 
Date submitted:  20211216 15:14 
Submitted by:  Gonzalez, Matías 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
The spin$\frac12$ triangular lattice antiferromagnetic Heisenberg model has been for a long time the prototypical model of magnetic frustration. However, only very recently this model has been proposed to be realized in the Ba$_8$CoNb$_6$O$_{24}$ compound. The groundstate and thermodynamic properties are evaluated from a hightemperature series expansions interpolation method, called "entropy method", and compared to experiments. We find a groundstate energy $e_0 = 0.5445(2)$ in perfect agreement with exact diagonalization results. We also calculate the specific heat and entropy at all temperatures, finding a good agreement with the latest experiments, and evaluate which further interactions could improve the comparison. We explore the spatially anisotropic triangular lattice and provide evidence that supports the existence of a gapped spin liquid between the square and triangular lattices.
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Reports on this Submission
Anonymous Report 2 on 2022117 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2112.08128v1, delivered 20220117, doi: 10.21468/SciPost.Report.4194
Strengths
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Weaknesses
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Report
The present work deals with applications of the entropy method to calculate the
ground state energy and specific heat c(T) for spatially anisotropic triangular spin
lattices. The results are also compared with measurements [37,38] on the perovskite
Ba8CoNb6O24, which is considered to be a good realization of the isotropic triangular
lattice.
The entropy method [41] interpolates between the hightemperature behavior (given
by the power series expansion in the inverse temperature) and the lowtemperature
behavior of the specific heat, which must usually be assumed to be known. However,
the latter is often a problem if, for example, the ground state energy is only known
imprecisely. In this situation, the authors offer a captivating solution approach, which
has also been used in previous work [4446], but is the focus of the present paper:
The ground state energy chosen is the one for which the largest majority of consistent
Padé approximants of different order (m,n) for c(T) can be found. This approach is
compared with the known values for certain limiting cases (isotropic triangular
lattice, square lattice, system of disconnected chains) and proves to be surprisingly
accurate.
After calculating c(T) based on these results, a comparison with the experimental data
yields the somewhat sobering result that the interpolation does not provide a
significant improvement over the usual Padé approximations published in [7,37,38].
These approximations already reproduce the broad maximum of c(T), which agrees
with the experiments. For low temperatures the interpolated c(T) correctly goes to 0,
as expected based on the approach, but deviates from the experimental results.
Analogous limitations arise for the spatially anisotropic lattice in the transition ranges
between J=J'=1 and J'→0 on the one hand and J'→ꝏ on the other: in the intermediate
range, the present method fails at certain points for reasons that do not become
entirely clear, and one can only say, "At least the method is able to detect something
unusual."
Nevertheless, this is a beautiful work that seems worthy of publication in SciPost
Physics. After all, it is questionable to appreciate only positive results. The
presentation of the method and the discussion of the state of research and the physical
significance of the results are very well done. Before the final decision, I still
recommend the following minor points to be considered in a possibly revised
manuscript:
Requested changes
1. In the Introduction it is not quite clear what TLHAF means: the general
anisotropic or the isotropic triangular lattice. It should also be made clear that
Ba8CoNb6O24 is always understood as an isotropic triangular lattice,
notwithstanding the remarks in section 4.4.
2. The last sentence before (5) “When the ground state is a gapped spin liquid, we
have c_ν(T)~T^2 exp(T_0/T)...” is insofar misleading as the restriction to α=2 of
the general power term T^α is only due to the entropy method, not to the
physics.
3. In Section 4.1, one would like to find reasons why groups of 3 orders are
formed for the pCPA. Obviously, this leads to a smoothing and to an
improvement of the approximation. Why?
4. Padé approximants are rational functions in T which may have poles on the
positive T axis. Usually, these cases will have to be discarded (but see Figure 3
in [7]). This problem is not mentioned in the paper. Do no poles occur, or are
these cases discarded and simply do not belong to the majority of consistent
approximants?
Author: Matías Gonzalez on 20220207 [id 2167]
(in reply to Report 2 on 20220117)We provide the response to the report in the attached file.
Attachment:
Anonymous on 20220214 [id 2201]
(in reply to Matías Gonzalez on 20220207 [id 2167])This is the response to Report 1 and has been uploaded a second time in response to that report.
Anonymous Report 1 on 202215 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2112.08128v1, delivered 20220105, doi: 10.21468/SciPost.Report.4144
Report
The article entiteled "Groundstate and thermodynamic properties of the spin1/2 Heisenberg model on the triangular lattice" by Gonzalez and collaborators uses hightemperature series expansions in combination with the entropy method, which is a specific extrapolation tool, to calculate and analyze the groundstate energy and the specific heat of the Heisenberg model on the anisotropic triangular lattice. The latter includes as limiting cases the 1d chain, the square lattice, and the isotropic triangular lattice, which are all well known from previous analytical and numerical studies. In addition, extensions of the model including spatial anisotropies, nextnearest neighbor Heisenberg interactions, and nonmagnetic impurities are also considered which are in particular motivated by experimental findings on the frustrated quantum magnet Ba_8CoNb_6O_24. In my opinion the article is well written and the topic is interesting. Globally, I therefore recommend publication in SciPost. I nevertheless have some points, which the authors should address to further improve their manuscript.
Requested changes
List of points:
 Title: I wondered whether one should put "anisotropic triangular lattice" in the title which expresses better the focus of the article
 Figure 3, page 7: Is there a microscopic reason why a "1/n^2" scaling is used? If yes, maybe one can specify the argument. If not, one should mention it.
 Page 8: There are two equations in the text which are too wide.
 Page 9/10: Is there are a (physical) reason why the quality of the extrapolation seems worse on the square lattice compared to the triangular and chain limit? Naively I would have expected that the square lattice is the simplest system.
 Page 13: Is there a simple argument why the specific heat does not (almost does not) depend on the spin anisotropy?
 Page 13: Another possibility/option could be the presence of multispin interactions. Can the relevance of such interactions be excluded for the considered material?
 Page 15: Title of reference [18] seems to be spoiled.
Author: Matías Gonzalez on 20220214 [id 2200]
(in reply to Report 1 on 20220105)We provide the response to the report in the attached file.
Author: Matías Gonzalez on 20220207 [id 2168]
(in reply to Report 2 on 20220117)We provide the response to the report in the attached file.
Attachment:
response2.pdf
Anonymous on 20220209 [id 2178]
(in reply to Matías Gonzalez on 20220207 [id 2168])Based on the authors' responses and the changes made, I have no objection to publication of this article, which represents a valuable contribution to a current area of research.