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Topological multiferroic order in twisted transition metal dichalcogenide bilayers
by Mikael Haavisto, J. L. Lado, Adolfo Otero Fumega
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Submission summary
Authors (as registered SciPost users):  Jose Lado · Adolfo Otero Fumega 
Submission information  

Preprint Link:  https://arxiv.org/abs/2204.03360v1 (pdf) 
Date submitted:  20220412 10:14 
Submitted by:  Otero Fumega, Adolfo 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
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Abstract
Layered van der Waals materials have risen as powerful platforms to artificially engineer correlated states of matter. Here we show the emergence of a multiferroic order in a twisted dichalcogenide bilayer superlattice at quarterfilling. We show that the competition between Coulomb interactions leads to the simultaneous emergence of ferrimagnetic and ferroelectric orders. We derive the magnetoelectric coupling for this system, which leads to a direct strong coupling between the charge and spin orders. We show that, due to intrinsic spinorbit coupling effects, the electronic structure shows a nonzero Chern number, thus displaying a topological multiferroic order. We show that this topological state gives rise to interface modes at the different magnetic and ferroelectric domains of the multiferroic. We demonstrate that these topological modes can be tuned with external electric fields as well as triggered by supermoire effects generated by a substrate. Our results put forward twisted van der Waals materials as a potential platform to explore multiferroic symmetry breaking orders and, ultimately, controllable topological excitations in magnetoelectric domains.
Current status:
Reports on this Submission
Anonymous Report 3 on 202268 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2204.03360v1, delivered 20220608, doi: 10.21468/SciPost.Report.5202
Strengths
1 A major and important topic of condensed matter in 2D materials.
2 The numerical method used is adapted and seems to me correctly applied.
3 A thorough, detailed and well explained study.
Weaknesses
1 Is the model used (Wannier Hamiltonian) relevant to study magnetism of twisted transition metal dichalcogenide (TMD) bilayers ? In particular, this approach does not directly take into account the specific roles of the dorbitals (Transition Metal) and porbitals (Chalcogenides) which do not have the same characteristics in terms of magnetization.
2 This study only considers magnetism in the z direction, but the magnetic moment may be in favor of the xy plane rather than the z direction, especially for a system that is not half filled.
Report
This paper presents a detailed numerical study of the magnetism of low angle twisted bilayer TMD, using a Wannier Hamiltonian. It is shown how a topological multiferroic order can emerge in this system at quarterfilling. The competitive effects of onsite and firstneighbor interaction terms are studied in detail. As a result, a multiferroic order displaying simultaneously ferroelectric and ferrimagnetic orders can be obtained. Spinorbit interactions induce a topologically nontrivial multiferroic order, and topological excitations can be control by and external electric fields. The author have also studied the impact of an underlying substrate in the moiré system by considering supermoiré cell, and thus show the emergence of topological excitations. This theoretical work is very rich and brings advances in a complex and new subject the physics of the twisted bilayer semiconductor. It is well presented and I think it deserves to be published in SciPost Physics.
Requested changes
To reinforce the relevance of this work it seems important to me to discuss the following points in more detail.
1 Wannier Hamiltonian model is widely used in the literature to study the electronic properties of the lowenergy states of twisted bilayer. However, its relevance to the study of magnetism is not obvious. The low energy eigenstates, that are well simulated by the Wannier Hamiltonian (without interaction), are complex and localized on a large number of atomic orbitals (mainly dobitals of metal). When the interaction terms are on, the local magnetic order (at the atomic scale) can be complex. Is it well simulated by Wannier states? It would be interesting for the authors to discuss the consequences of their results for the magnetization of the dorbitals of metal.
2 If I am not mistaken, the xy plane magnetization is not include in the calculations. The authors should justify that point, especially for a calculation that is not at halffilling.
3 The authors focused on the quarterfilling case. It would be nice to justify this choice a little more. Is it also possible to discuss, at least qualitatively, the cases of other fillings?
4 The SciPost latex style should be used to produce the manuscript.
Author: Adolfo Otero Fumega on 20220709 [id 2647]
(in reply to Report 3 on 20220608)Please find attached the response to this report.
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Strengths
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Weaknesses
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Report
See attached report
Requested changes
See attached report
Author: Adolfo Otero Fumega on 20220709 [id 2646]
(in reply to Report 2 on 20220602)Please find attached the response to this report.
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Anonymous Report 1 on 202261 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2204.03360v1, delivered 20220601, doi: 10.21468/SciPost.Report.5162
Report
The authors study topological multiferroic order in twisted transition dichalcogenides bilayers (TMDs) in the presence of spinorbit interactions and applied electrical field. Starting from a model Hamiltonian that captures nearestneighbor hopping between the AB/BA sites forming an effective honeycomb lattice, the authors analyze spin/charge excitations (SDW/CDW) that emerge due to onsite and nearestneighbor interactions within a selfconsistent meanfield theory. They demonstrate that spinorbit interactions render this model a topological multiferroic with nonvanishing Chern Number whose magnetoelectrical coupling can further be tuned by the application of electrical fields. The topological character manifests in the existence of interface modes between either ferrimagnetic (FM), ferroelectric (FE) or multiferroic (FEM) domains. These findings are argued to exist in supermoire potentials arising from twisted TMDs encapsulated in substrates, e.g. hBN, with slightly different lattice constant. The authors demonstrate that the additional moiré potential leads to the formation of different topological regions of the multiferroic and hence naturally to the existence of topological interface modes.
The realization of topological states in twisted TMDs with strong intrinsic spinorbit coupling represents an uprising field of research beyond the physics of twisted multilayer graphene and aligns with recent theoretical and experimental efforts. The discussion of topological multiferroic order is novel and adds to the toolbox of correlated topological phases that can be realized via moiré engineering. The connection to the supermoiré potential as possible candidate to host topological interface modes displays an interesting avenue, in particular because such effects have been observed experimentally in twisted trilayer graphene (TTG) to have major impact on correlated quantum states. The manuscript is soundly written and easy to follow for the reader. I therefore recommend publication as long as the following (minor) points are addressed properly:
Some technical aspects could be described in more detail to provide further clarification or to eventually allow for easier reproduction of the results, especially since there is no Appendix/Supplementary Material. Some aspects I came across are the following:
1.) Concerning the TMD model Hamiltonian: The nearestneighbor honeycomb model has been widely studied in the literature within, e.g. meanfield approximation, functional renormalization group studies, quantum Monte Carlo etc. Most of these studies focus to halffilling/filling in the vicinity to the vanHove singularity. Could the authors clarify how quarterfilling in their manuscript is related to the position of the vHS? This would make it easier to compare with existing results.
2.) The authors write: „The interacting model is solved using a selfconsistent meanfield procedure including all the Wick contractions, including magnetic symmetry breaking, hopping renormalization, and charge order.“ I would prefer to avoid the statement „all Wick contractions“ as e.g. superconducting order is not accounted for as another possible Wick contraction or do the authors also account for this kind of instability, even though it may be irrelevant for purely repulsive interactions? Furthermore, what are the technical details of the meanfield analysis? How many moiré unit cells were taken into account (or what momentum resolution was used to sample the Brillouin zone) for the selfconsistent procedure? Were the calculations performed in real or momentumspace? Did the initial guess for the selfconsistent procedure already include multiferroic order or was the initial state chosen randomly?
3.) Concerning the results of the meanfield analysis: It is interesting that for U/t<4 and up to V/t<2 no chargedensity wave is present, but the latter only emerges for U/t>4 in the multiferroic phase. Is there an intuitive picture to understand why the formation of CDW order is diminished at U/t=0?
4.) The authors write: „Since the origin of the interface modes is topological, small reconstructions at the interface will not impact the boundary modes“. For simple graphene, the edge termination (zigzag/armchair) is crucial for the existence of edge states. Have the authors tried different edge terminations for the L/R domains and found invariant results?
5.) How is the momentum dependence of the interface spectral function A(w, k_x) related to the sketch in Fig. 5 a) and what is the length of the domains in numerical simulations? From the sketch, I would intuitively expect that the system is periodic in ydirection and the interface separates the junction into the left/right domain along x. Then the momentum k_y would describe the bulk dependence. Maybe it would be helpful to add a small coordinate system in Fig. 5a) or to add a more detailed instruction why/how the spectral function A(w, k_x) is computed.
6.) Concerning the supermoiré potential in the last section: Can the authors give the functional form of the moiré potential and how it couples to the Wannier moiré orbitals? The potential looks to preserve the D_6h symmetry of the original honeycomb lattice, though the LDOS in the multiferroic state only seems to preserve D_3h. Is that due to the inversionbreaking Rashba SOC term? Furthermore, how are the Chern Numbers calculated for different regions of the supermoiré? Was the set of parameters for the meanfield Hamiltonian (m, Delta_z, lambda_R) the same as in the section before?
I hope the authors appreciate the comments given above, which should not distract from the high quality of the manuscript.
Typos:
1.) Introduction: „which may drive a nontrivial topological character in a the moire system[16].“
2.) Manuscript: „moire“. In the literature (and for all cited References) „moiré“ seems to be the usual term, but I leave this point up to the authors.
3.) Page 5: Up to this point, we have been based all the analyses on a full selfconsistent solution of the interacting Hamiltonian of eq. (1).
4.) Page 6: and acts as an independent topological source.
Author: Adolfo Otero Fumega on 20220709 [id 2645]
(in reply to Report 1 on 20220601)Please, find attached the response to this report.
Anonymous on 20220602 [id 2552]
Editorial Comment: This is the attachment to Anonymous Report 2 on 202262.
Attachment:
sci_post_06_2022.pdf