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Time-reversal-broken Weyl semimetal in the Hofstadter regime
by Faruk Abdulla, Ankur Das, Sumathi Rao, Ganpathy Murthy
This is not the latest submitted version.
This Submission thread is now published as SciPost Phys. Core 5, 014 (2022)
|As Contributors:||Faruk Abdulla · Ankur Das · Ganpathy Murthy · Sumathi Rao|
|Arxiv Link:||https://arxiv.org/abs/2108.03196v2 (pdf)|
|Date submitted:||2021-08-24 08:05|
|Submitted by:||Abdulla, Faruk|
|Submitted to:||SciPost Physics|
We study the phase diagram for a lattice model of a time-reversal-broken three-dimensional Weyl semimetal (WSM) in an orbital magnetic field with a flux of p/q per unit cell. In addition to trivially insulating and layered Chern insulating (LCI) phases, we find new WSM phases with 2q, 4q, 6q, and 8q Weyl nodes. In a robust region of the parameter space (consisting of the ratios of hopping parameters in the different directions) the LCI coexists with the WSM. We analyze the dispersions of the bulk and surface states in the phases and at the phase boundaries, and conclude with the behavior of the spectrum in the limit of small orbital flux.
Submission & Refereeing History
Published as SciPost Phys. Core 5, 014 (2022)
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Reports on this Submission
Anonymous Report 3 on 2021-11-2 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2108.03196v2, delivered 2021-11-02, doi: 10.21468/SciPost.Report.3776
Pedagogical technical discussion
Lacks the vision of broad implications
In this manuscript Abdulla et. al. study a two-band model for Weyl semimetals in the Hofstadter regime by threading a rational p/q flux quanta through each plaquet. This paper has already been reviewed by two other referees, who raised some concerns regarding the novelty, broad implications of this study and the experimental feasibility of the model the authors study. One referee in particular raised some genuine technical questions. Even though I find the responses from the authors to be reasonably convincing, I leave it up to the respective referee to judge the correctness of the responses. Given that two referees asked quite reasonable and probing questions, I will comment on the suitability of this article for publication in SciPost. First of all, I find this work to be convincing and the results to be appealing to a set of audience working on Weyl semimetals, which nowadays is fairly large. Therefore, this paper can be published in SciPost. However, prior to publication, I want the authors to take into account the following suggestions, which in my opinion should increase the impact of this article.
1. I strongly recommend the authors to expand the Abstract of this article. For example, the last sentence from the Abstract is pretty vague and it only mentions what calculations the authors do, while being completely silent about the outcome. I understand that an Abstract cannot contain every technical detail and outcome. But, authors should also bear in mind that nowadays often readers have time to read the Abstract of an article before delving into the paper itself. Therefore, Abstract needs to be punchy and must bear a flavor of broad appeal to attract more audience. Some of the authors are senior and established scientists. So, they do not need my guidance to improve an Abstract. Nonetheless, I strongly urge the authors to extend and polish the Abstract.
2. In the same spirit, I strongly urge the authors to improve the presentation of the Introduction. The first half of the Introduction reads very nicely, providing a broad overview of the field of Weyl semimetals. However, the second half reads very rushed to me. In principle, the authors can divide the whole Introduction into a few parts, for example (a) general Introduction, which is already there, (b) an extensive summary of their results (this is the part I found to be rushed, thus needs more serious work), which should include references to appropriate figures from the main text and (c) broad impact of this work in the general field of Weyl semimetals or even better if the authors can connect it to topology in gapless systems. There are a few sentences in the Introduction which simply do not serve any purpose. For example "Many features of the phase diagrams are universal and independent of p". Statements like this need to be expanded and authors need to state what are the universal features they are referring to, for example.
Therefore, in brief the Abstract and Introduction require serious restructuring. Notice that SciPost is a very good journal and we collectively should maintain its high standard.
3. Fig. 2 for example the authors show the energy spectra for the surface states, which I believe correspond to the Fermi arcs. Instead of only showing the energy spectra, the authors should also display the localization of the Fermi arcs, especially how they become delocalized as the Weyl nodes are approached, as shown in PRB 96, 201401 (2017).
4. The distribution of the topological charges to the Weyl nodes and their connectivity via Fermi arcs, shown in Fig. 6, closely resemble intertwined Weyl phases, discussed in arXiv:2105.08443. I recommend the authors to consult this preprint and make connections, if they exist.
6. In the second last paragraph of the "Summary and Outlook" authors mention effects of electronic interactions in Weyl systems. But, I find it to be quite surprising that the entire paragraph does not contain a single reference, while ample efforts have been invested to investigate interaction effects in Weyl materials. For example, see PRB 87, 161107 (2013), PRB 90, 035126 (2014), PRB 95, 201102 (2017). The authors should do literature surveys and find other important works in this direction. But, this paragraph definitely demands at least a few key references.
7. The discussion on the disorder effects on Weyl semimetal is somewhat biased toward rare region effects and the authors cite multiple papers from a single group of researchers, which I find to be somewhat unfair. Let me first comment on physics. Authors should note that numerical evidence for the rare region effect is strictly slim, as rare states have only been observed in the very close vicinity of the WSM-metal critical point, roughly about 10-15% of the critical disorder strength. Even though it has been claimed that those rare states persist all the way down to infinitesimal disorder, there exists no real trend or direct evidence supporting such a claim. Given that authors have not studied the disorder effects by themselves, I suggest that they only state the current status of this problem, as the jury is still out. Also it is not clear yet whether those rare states really convert the system into a metal or not. Only if those rare rares lead to finite dc conductivity at infinitesimal disorder and at zero temperature the phase can be called a metal, according to the definition by Nevil Mott. Also they need to acknowledge efforts from other research groups on this problem by referring to the following articles PRL 112, 016402 (2014), PRL 113, 026602 (2014), PRB 93, 210302 (2016), PRB 94, 220201 (2016), ... I urge authors to expand the reference listing on this topic to honor the effort invested by different research groups.
8. A minor point: Ref. 28 and Ref. 36 are identical. Please fix it.
To summarize, I think this paper can be published in SciPost once the authors fix the issues I pointed out, and answer the questions two other referees asked. As far as the experimental relevance of this model is concerned, I suggest authors add more concrete discussion rather than saying "it can be realized in cold atomic setups".
PS: I want to say sorry to the authors for the delayed report. There were some personal issues, which I had to deal with and which I could not avoid.
Anonymous Report 2 on 2021-10-15 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2108.03196v2, delivered 2021-10-15, doi: 10.21468/SciPost.Report.3685
In this paper, the authors do a detailed analysis of the phase diagram of a Weyl semimetal in the presence of large magnetic field. They find a handful of new features, the most notable of which in my mind is the phase they label W2’, which corresponds to coexisting Weyl nodes and layered Chern insulator. I think the work is correct so merits publication, but does not currently rise to the level of importance that Scipost is targeting.
With papers such as this, there are a few questions that I ask to determine whether the paper has novel results, namely:
1. Are there fundamentally new phases of matter? If so, is there any meaningful discussion of the physical import of the new phase of matter (experimental detectability, topological response, etc.)
2. Is there a proposal for how this model Hamiltonian may be realized experimentally?
3. Is there a meaningful discussion of stability to, e.g., disorder or interactions?
Of these three topics, I only see discussion of the third. Without some reason to believe this is more interesting than stated (via criteria 1 and 2), I recommend against publication in SciPost.
Anonymous Report 1 on 2021-10-1 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2108.03196v2, delivered 2021-10-01, doi: 10.21468/SciPost.Report.3599
1. The paper is coherently written
2. The subject is timely
3. Results and discussions are overall sound
1. The manuscript does not report much novelty
2. Taking into account the above not much beyond the phase diagram is addressed; the discussion of the phases is relatively shallow. This is also reflected in the motivation of the work
3. There is not much new physics reported, in particular no new observable effects or connection to materials
In the submission by Abdulla et al. the authors consider a Weyl semi-metal [WSM] in presence of magnetic field. To this end they consider a rather well-known toy model for a WSM, eq 3. Adding the p/q rational flux induces a magnetic unit cell and a richer phase diagram with new phases dubbed W2' and I' for example. The phases and boundary states are considered numerically and, exploiting the simplicity of the model, also analytically in some cases.
The manuscript is coherently written and the question of considering topological phases in the presence of magnetism is timely. Although the interesting aspect of topological materials is that they can occur without external field, considering the effect of such a readily available experimental probe is interesting in its own right and the subject is increasingly receiving attention.
Nonetheless, I also have some concerns that I would like to discuss with the authors. Most importantly, there is the issue of novelty. There has been work on considering WSMs in magnetic fields that range from the weak field limit to strong field limit, see also ref 23 and quite some others. The authors say they go beyond these works by considering multiple Weyl nodes and different orientations of the magnetic field relative to the nodes. The point however is that having more Weyl nodes does not significantly change the physics of the nodes or total phase in itself. One of the earlier papers proposing WSMs [PRB 83, 205101 (2011)] in fact has many nodes, while the important local features per node [resulting in anomaly, stability of each node and their flux] can be effectively addressed by a “two-node” model akin to the one used. Possibly, the extra nodes can lead to more effects beyond just observing intermediate phases with more nodes, but then one really has to consider crystalline symmetries and their role to signify the physics. The authors neither give such crystalline interpretations nor motivate why their analysis involving extra nodes is important or worthwhile.
This rather shallow perspective is also illustrated in the superficial discussion of the I' phase in Sec 3.3.4; the authors merely state that the phase diagram features an insulating phase with counter propagating states. They rightfully connect this to surviving crystalline symmetries, but make no deeper analysis. Hence, not much is gained other then just identifying the phase in the phase diagram. Evidently, the model has an inversion symmetry and generalized mirrors that persist and should be taken into account. Such a broader analysis that includes the role of e.g. the inversion should be made. Similarly this deeper analysis should be related to the particle hole symmetry of the spectrum mentioned in section 3.1.
Overall, given the fact that this work is not radically new, at least its role as extension on previous results should be enforced by exploring all details in depth. Hence, these aspects should be improved.
Some other concrete points that I like to raise are
1. The authors say above Eq. 5 “...Note that we consider the orbital
effect only. The reason is the following: The orbital coupling, since it couples to the charge degree of freedom, is universal. On the other hand, the pseudospin label is a k-dependent linear combination of spin and orbital labels, which does depend sensitively on the microscopic material parameters...” I am confused by this. I agree that pseudospin label is k-dependent but the orientation can be important, although being model dependent. Indeed, consider for example the edge states. If it is spin polarised, meaning it is in the direction of one Pauli matrix, then anti-commuting Zeeman terms directly induce a gap, whereas commuting ones do not. Hence, there is an effect. This is rather similar to how the orbital parts also depends on specific model settings. This lack of broader view also ties to the motivation aspect discussed above.
2. In the discussion it is left implicit that when edge states are not mentioned periodic boundary conditions are used.
3. The authors say “...We find that the topological phase diagram, in general, depends only on three of the hoppings (tx,t(1)y ,t(1)z ) and the onsite mass parameter M...” Given that the phase diagram is determined essentially by the tuning of the mass it seems that one can scale out t_x and consider (1, t_y/t_x, t_z/t_x, M/t_x) as is usual for such kinds of topological toy models.
Finally some minor points
-The authors comment on the chiral anomaly in the second paragraph. The transport properties they describe semi-classically leads to the chiral anomaly, hence I would not use “due to” rather it is the other way around; the anomaly arises due to non-conservation of the density of states at each Weyl node.
-Ref 25 studies Chern and fragile insulators rather than Weyl or double Weyl. The se systems are then linked to higher order topology in contrast to ref 24.