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Lowenergy effective theory and anomalous Hall effect in monolayer $\mathrm{WTe}_2$
by S. Nandy, D. A. Pesin
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Submission summary
Authors (as registered SciPost users):  Snehasish Nandy 
Submission information  

Preprint Link:  https://arxiv.org/abs/2110.14586v1 (pdf) 
Date submitted:  20211108 18:17 
Submitted by:  Nandy, Snehasish 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We develop a symmetrybased lowenergy theory for monolayer $\mathrm{WTe}_2$ in its 1T$^{\prime}$ phase, which includes eight bands (four orbitals, two spins). This model reduces to the conventional fourband spindegenerate Dirac model near the Dirac points of the material. We show that measurements of the spin susceptibility, and of the magnitude and time dependence of the anomalous Hall conductivity induced by injected or equilibrium spin polarization can be used to determine the magnitude and form of the spinorbit coupling Hamiltonian, as well as the dimensionless tilt of the Dirac bands.
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Reports on this Submission
Report #2 by Anonymous (Referee 2) on 202222 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2110.14586v1, delivered 20220202, doi: 10.21468/SciPost.Report.4305
Strengths
1 It provides a comprehensive discussion of models for WTe2 monolayer. The discussion includes a description of differences of previous models in the literature which will be a useful starting point for other researchers interested in this system.
2 It makes an effort to connect detail parameters of the model with various experimental probes. This effort might help experimentalist in interpretation of results and it may also help or provide ideas for other compounds.
Weaknesses
1 The presentation (good overall) might be improved in some parts.
Report
The authors present a lowenergy model for monolayer WTe2 in the inversion symmetric phase. They provide insightful discussions into the possible variants that can describe the lowenergy electronic structure of this system and also presents certain phenomena that could experimentally reveal information about specific parameters of such models.
I find the paper wellwritten and the discussions of interest. I recommend publication together with a revision of the points made below.
Requested changes
1  When the authors say "there is a little mismatch between the dispersions close to the Dirac nodes", it would be nice to be more precise. Is the mismatch in the meV range? Is it in the 10meV range? Is it larger?
I think it would be best to show the band structure of the different models so that readers can jugde by themselves.
2 I am sorry I was not able to follow the sentence 'Since the tightbinding model cannot not reproduce exactly the energy dispersion obtained from the DFT calculations anyway, we ignore such terms from now on'. Why the TB model cannot not reproduce the DFT results?
3 I am sorry I was not able to certainly understand this sentence: "the leading kindependent spinorbit coupling gives rise to the conservation of the spin projection on a particular axis near the Γpoint, which is consistent with the sample, edge orientation, and gatevoltage independent spin quantization axis on edges of a sample". What is meant by "consistent with the sample, edgeorientation"? Does it means that it is at fixed angle away from the edge? Perhaps this sentence could be broken into peaces that makes its comprehension easier.
4 It could be useful to clarify why the Berry curvature appears in the fieldinduced Hall effect. Most readers will understand why it usually appears in the anomalous Hall effect (zerofield, by definition a different experiment).
5 I don't understand why smaller spinorbit coupling leads to larger Hall effect in Eq. 25. At least in the conventional AHC, in linear response, the AHC vanishes without SOC.
Report #1 by Anonymous (Referee 1) on 20211222 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2110.14586v1, delivered 20211222, doi: 10.21468/SciPost.Report.4083
Strengths
1 Derivation from theory of invariants of a k.p lowenergy model for the 1T' phase of WTe2 both in the presence and in the absence of spinorbit coupling
2 Interesting prediction on the appearance of an anomalous Hall effect with applied inplane magnetic fields
Weaknesses
1  The analysis based on the theory of invariants does not bring result different from the paper by Muechler and collaborators [PRX 6, 041069 (2016)] in the absence of spinorbit coupling.
2 The manuscript lacks some quantitative estimations. For instance, it is not immediately obvious if the anomalous Hall effect in the presence of inplane magnetic fields can be observed in experiments.
Report
In "Lowenergy effective theory and anomalous Hall effect in monolayer WTe2", the authors develop a k.p lowenergy model for the 1T' phase of WTe2. The model is first derived in the absence of spinorbit coupling, in which case the continuum k.p model reproduces the electronic properties of the tightbinding model originally developed by Muechler and coworkers in PRX 6, 041069 (2016). In a second step, spinorbit coupling is explicitly introduced. Here the authors restrict to momentumindependent terms of spinorbit coupling. However, a discussion of other symmetryallowed spinorbit coupling terms would have been beneficial. Using an additional SchriefferWolf transformation, the authors then derive an effective fourband model for tilted Dirac cones that are gapped by spinorbit coupling. In my opinion, the most interesting aspect of this manuscript is the discussion concerning the occurrence of an anomalous Hall effect triggered via Zeeman coupling by a planar magnetic field. However, I would like to point to the authors that precisely the same effect has been originally proposed by Battilomo and collaborators in Physical Review Research 3, L01006 (2021), and dubbed Anomalous planar Hall effect. Later, the same effect has been discussed by Cullen and collaborators in Physical Review Letters 126, 256601 (2021).
Requested changes
1 When deriving the spinorbit coupling Hamiltonian in Sec.IIA, the authors should discuss if spinorbit coupling term that are linear in k are symmetry allowed. In principle those term, if appearing, should be explicitly taken into account considering that the spinorbit free Hamiltonian is expanded up to linear order in momentum
2 In the same subsection, the authors also mention that there is conservation of the spinprojection in their effective Hamiltonian. Is this enforced by a crystalline symmetry, as it happens for instance in graphene when preserving the Mz horizontal mirror symmetry? Or this is an accidental property due to the assumptions made on the spinorbit coupling terms?
3  A minor remark concerns the statement in Sec.IIB "There are several notable features of Hamiltonian. Each band that it describes is double degenerate at every kpoint, as appropriate for a system with both timereversal and inversion symmetry". I would not consider "notable" the double degeneracy of the bands in an inversion and timereversal symmetric system.
4 I am confused by the expression for the anomalous Hall conductance due to planar magnetic field, that seems to be inversely proportional to the spinorbit coupling strength (see Eq.25).The Berry curvature indeed grows proportionally with the SOC. Could the authors clarify the scaling and what happens to the transverse conductance in the spinorbit free limit?
5 It would be also beneficial to estimate the size of the Hall conductance for moderate magnetic fields and estimate if the effect can be in principle observed in experiments.
Author: Snehasish Nandy on 20220217 [id 2214]
(in reply to Report 1 on 20211222)
We thank the referee for their report and helpful comments and suggestions. We also thank the referee for bringing these useful references (Physical Review Letters 126, 256601 (2021) and Physical Review Research 3, L012006 (2021) ) to our attention. We have now cited these references (Refs.~30 and 31) in the revised version of the manuscript. (First Paragraph of SectionIIIC, Page 8).
Please find our response to the questions raised by the referee in the attached file.
Author: Snehasish Nandy on 20220217 [id 2215]
(in reply to Report 2 on 20220202)We are grateful to the Referee for their report and positive assessment of our work. Please find our response to the comments from the Referee in the attached file.
Attachment:
Response_to_Report_2.pdf