Honeycomb rare-earth magnets with anisotropic exchange interactions

Q: What is specific to rare earths? The model of Eq. (1) has already been studied in the context of iridates, right? Is it just the fact that this model is expected to be a more accurate approximation than for iridates because one can neglect further neighbour interactions in the case of rare earths? When I saw the title of the paper, I was expecting some derivation (or at least some discussion) of the form of Eq. (1) for rare earths, and to get a feeling for which parameters can be expected to dominate, but I did not find anything of that sort.

A: Yes, one can neglect further neighbor interactions in the case of rare earths. As for the dominating parameters for general rare-earth magnets, due to the low angular momentum of these materials, the interactions between the rare-earth ions are generically quantum, with all the (symmetry-allowed) exchanges being potentially significant.
A detailed derivation of the exchange parameters requires the knowledge of the environment and chemical information for specific materials. Different rare-earth ions would be different from each other. Instead, we take a more phenomenological view and rely more on the experimental input to extract the parameters. For example, using our calculations and the experimental data of susceptibilities in Ref[51], we find that $J_{zz}\sim 8K$ and $J_{\pm}\sim 6K$. We have added this into the discussion section.

Q: I think it would be nice to comment on the qualitative implications of Eq. (9). Usually, torque magnetometry relies on the anisotropy of the g-tensor to measure the magnetization. Here, the torque response will be different because of the other sources of anisotropy.

A: Thanks for pointing that out. We have added some remarks after equation (9).

Q: ESR linewidth: What is the take-home message? By measuring the line width, one can check the parameters of the model with the help of Eqs. (13) and (17)? This should be specified.

A: Yes, thanks. We have expanded the discussions in the caption of figure 3 and below eqn. (17).

Q: Polarized phase with a strong field normal to honeycomb phase: What is new with respect to Ref. 39? Maybe I missed something, but the result looks very similar to that of Ref. 39, with just another parametrisation.

A: This is easy to be achieved in rare-earth magnets and is an indispensable experiment to be carried out for these materials. Although it is only one small section in our paper, We feel it is needed to have a full story for the understanding and the exploration of the experimental consequences from the anisotropic interactions between rare-earth local moments.

1) Q: What is precisely the physical quantity shown in Fig. 3? Is it the one defined in Eq. (13)? In the text it is hard to make a connection.

A: Yes, it is the one in eqn. (13). We have made this point more explicit in the caption of figure 3.

2) Q: In connection to the question above, I do not quite understand the sentence "the ESR linewidth for a Lorentzian-shaped spectrum is ... "

A: We added more explanations in the new version.

3) Q: In the set of Eqs. (17), $\Gamma_{zz}$ is not given. Should I infer $\Gamma_{zz}=-\Gamma_{xx}-\Gamma_{yy}$ from the traceless properties of the tensor $\Gamma_{ij}$?

A: Yes, we have only listed the independent components. But to avoid confusion, we have added the zz component into the equation.

4) Q: I find interesting the result shown in Fig. 5. In connection to it, I have some questions:

a) Being this band structure topologically non-trivial, are the gapless states protected by some symmetry? which one?

b) Would be worthwhile to investigate the symmetry class of the effective Hamiltonian in this regime, if it has not been done in some reference I'm no aware of?

A: It is BdG Hamiltonian without effective time reversal symmetry (the $B(k)$ terms break it), so the system is in symmetry class D, and the gapless states are protected by the effective charge conjugation symmetry $U^\dagger H^* U=-H$.

Minor points:

1) Q: In the introduction, the authors state "Due to the possible proximity to Kitaev physics, these materials were referred as Kitaev materials." The word proximity here causes confusion because of the known term "proximity effect" typically found in topological materials. I would suggest perhaps replacing proximity by similarity.

A: Indeed the word "proximity" can cause confusion. "Similarity" should work, but we tended to mean something more like "relevance". We have changed it in the new version.

2) Q: At the beginning of Sec. III, the author could indicate Appendix B where the calculation of thermodynamics is shown with some detail.

A: Revised as suggested, thanks.

1) Q: In the last paragraph of page 1, the authors argue that a small magnetic field is enough for getting a fully polarized state, but then they consider only strong magnetic fields in their calculations. Would the author care elaborate on what is “small” and what is “strong”, by providing experimental orders of magnitudes in order to make these points clear?

A: "A small magnetic field" means the absolute magnitude of the field is small, while the ``strong magnetic fields'' in the calculation means that compared with the exchange energy scale of the system, the field is strong. See also the reply to question 6. The exchange energy scale for rare-earth magnets is usually much smaller than $d$ electrons, thus a conventionally small magnetic field would be sufficient to generate a large effect.

2) Q: In Section II, after the parametrization for the Hamiltonian is done (Eq. 2), please briefly explain what the terms J, K, and $\Gamma$ are. Especially considering that one of the main findings is that the rare-earth magnets can be treated as Kitaev models, but the Kitaev term is not introduced to the reader.

A: Added explanations below equations (1) and (3).

3) Q: For the reader’s benefit, please after Eqs. 5 and 7 define what $k_B$, $\mu_0$, and $\mu_B$ are. Also, around Eq. 7 define that h’s are the components of the external magnetic field.

A: Done as suggested.

4) Q: In the caption of Figure 2, write what the dashed-blue and solid-black lines are.

A: Done as suggested.

5) Q: At the very end of Section IV, when discussing ESR shown in Fig. 3 please provide a longer discussion on what is shown in Fig. 3, and highlight the difference among the panels rather than saying that is ESR is anisotropic-coupling dependent.

A: We have expanded the discussions in the caption of figure 3 as well as below eqn. (17).

6) Q: First paragraph of Section V.A, the line “…energy scales for them are usually rather small.” should have a reference at the end. Please provide it.

A: The exchange interactions in rare-earth materials are typically two orders of magnitude smaller than in transition metal magnets with similar inter-atomic distances. For example, those of the 5d material Na2IrO3 [Phys. Rev. Lett. 113, 107201 (2014)] and 4d material RuCl_3 [Scientific Reports volume 6, 37925 (2016)] are O(1) meV, while those of YbMgGaO4 [Phys. Rev. Lett. 115, 167203, (2015)] and Ba3Yb2Zn5O11 [Phys.Rev.B, 98:054408, (2018)] are O(10^{-2}) meV. We have added the general comment to the beginning of section V. There is no satisfactory comprehensive literature that compares the 4d/5d versus rare-earth honeycomb materials (the latter are not widely studied at all), so we prefer not to cite the separate references, which may be distractive. However, we are able to combine our calculations and experimental data of YbCl3 [arXiv: 1903.03615] and find its exchange to be O(10^{-2}) meV. We have added this to the discussion section.

7) Q: Title of Section V.B should read “Strong field along the honeycomb plane”, since “in” could be misinterpreted.

A: Revised as suggested. Thanks.

8) Typos: - Page 3, left column, last paragraph: “there is” should read as “this is”. – Page 3, right column, last line: “a traceless” should read as “is a traceless”. – Page 7, first paragraph: “spin-obital” should read as “spin-orbital”.

A: Revised as suggested. Thanks.

9) Q: After Eq. A1, please define what D(Szi) is.

A: $D(S_i^z)^2$ is the single-ion anisotropy as explained in the first sentence after equation (A1). We edited the text to make it more explicit.

10) Q: After Eq. B5, please define what this $\beta$ is, since in Eq. 2 $\beta$ was something else.

A: Done. Thanks for catching that.

11) Q: Why the perpendicular susceptibility of Eq. B9 only contains Sx terms and not Sy? Please elaborate.

A: Revised as suggested. The $\langle Sy Sy\rangle$ terms give the same result as the $\langle Sx Sx\rangle$ terms, and the cross-terms don't contribute.