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REVIEW 2 major objections 4 minor 40 references

Berry Curvature Engineering by Gating Two-Dimensional Antiferromagnets

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Pith's one-line read Electric-field gating drives a 2D antiferromagnet into a Chern insulator with Chern number 3.

desk verdict A specific, internally consistent prediction of a gate-tunable C=3 QAH state in even-layer AFM MnBi2Te4, but the load-bearing AFM stability assumption rests on a figure I cannot see. read the letter →

arxiv 1909.01194 v1 pith:AA3LUBQ5 submitted 2019-09-03 cond-mat.mtrl-sci cond-mat.mes-hall

Shiqiao Du , Peizhe Tang , Jiaheng Li , Zuzhang Lin , Yong Xu , Wenhui Duan , Angel Rubio This is my paper · ORCID
classification cond-mat.mtrl-scicond-mat.mes-hall
keywords BerrycurvatureengineeringMnBi2Te4antiferromagneticspintronicsCherninsulatoranomalousHalleffectdual-gateelectricfieldtopologicalphasetransitionquantum
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper predicts that a simple out-of-plane electric field, of the kind applied by a dual gate, can transform a two-septuple-layer antiferromagnetic MnBi2Te4 film from a trivial insulator into a Chern insulator with Chern number 3. In the pristine film, a combined spatial-inversion and time-reversal symmetry forces the Berry curvature to vanish everywhere, so there is no anomalous Hall signal. The electric field gives the two layers different electrostatic potentials, breaking that symmetry and making the Berry curvature finite and tunable; beyond a critical field the bulk gap closes and reopens with three chiral edge channels, giving a quantized Hall conductance of $3e^2/h$. If the prediction holds, the same material can act as a memory switch whose readout is an electrical Hall signal, with no applied magnetic field, and whose ideal on/off contrast is infinite at zero temperature. The paper checks that the antiferromagnetic order survives the relevant field range, though that check lives in the supplementary material.

What carries the argument

The load-bearing object is $\mathcal{PT}$ symmetry and its controlled removal. In the double-SL film, the absence of separate inversion and time-reversal symmetries combined with the presence of their product forces $\Omega(\mathbf{k})=0$ at every $\mathbf{k}$. An out-of-plane field shifts the electrostatic potential of one septuple layer relative to the other, breaking $\mathcal{PT}$ and allowing nonzero Berry curvature; the field magnitude then acts as a continuous control knob for both the Berry curvature and the gap. The topological transition is carried by the threefold-symmetric band crossings along the $\Gamma$-$K'$ lines, whose closing and reopening produce three chiral edge states and hence Chern number 3.

What would settle it

Apply an out-of-plane electric field to a double-SL MnBi2Te4 device and measure the Hall conductance: the claim requires a quantized plateau at $\sigma_{xy}=3e^2/h$ for gate fields between 0.021 and 0.027 V/Å. If the conductance stays zero or unquantized in that window, or if magnetic characterization shows the antiferromagnetic order cants or becomes ferromagnetic below 0.021 V/Å, the central claim is refuted.

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Extended reading notes

Core claim

The central discovery is that gating can break $\mathcal{PT}$ symmetry in a collinear antiferromagnet without destroying the antiferromagnetic order, and that this is enough to drive a topological phase transition. In the double-SL MnBi2Te4 film the authors identify two critical fields. At $E_{c1}=0.021$ V/Å the gap closes at three equivalent points in the Brillouin zone and reopens, leaving a Chern insulator with $C=3$; at $E_{c2}=0.027$ V/Å a second gap closing turns the film into a trivial antiferromagnetic metal with large but unquantized anomalous Hall response. Reversing the field direction reverses the sign of the Hall signal. The same electric-field mechanism is shown to work in a four-SL film, where the topological window is narrower.

Load-bearing premise

Everything rests on the assumption that the antiferromagnetic order with out-of-plane Mn moments remains the ground state for electric fields up to at least 0.029 V/Å; the paper states this is checked only in the supplementary material, and a field-induced ferromagnetic or canted state would break the PT-symmetry argument and the Chern number classification.

Editorial extensions

If this is right

  • An electric field alone can switch a collinear antiferromagnet between zero and quantized Hall response, so a memory bit would need no magnetic field and no current-driven spin torque.
  • The on/off ratio of the proposed device is infinite in the ideal zero-temperature limit and remains $10^6$ to $10^{14}$ at practical temperatures, far above existing AFM memory cells.
  • The sign of the Hall signal is set by the direction of the gate field while the magnetic order is unchanged, giving a natural two-state electrical readout.
  • The mechanism extends to even-SL MnBi2Te4 films generally; the four-SL case shows the same field-driven transition with smaller critical fields and a narrower topological window.
  • Finite temperature degrades the plateau gradually, so quantization should survive up to a substantial fraction of the Néel temperature.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: the essential ingredient is layer-resolved electrostatic asymmetry, so other even-layer antiferromagnets with $\mathcal{PT}$ symmetry and similarly segregated orbitals may show the same electric-field-driven Hall switching, not only MnBi2Te4.
  • Inference: the large field-tunable Berry curvature near the conduction-band anticrossing could be used separately from the quantized plateau, for example to steer spin-polarized currents or build a Berry-curvature diode in the trivial phase.
  • Inference: because the critical fields are tens of mV/Å, testing the $3e^2/h$ plateau in exfoliated films with dual gates should be within reach of current experiments; a null result there would separate the quantitative DFT prediction from the generic symmetry mechanism.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The manuscript uses first-principles DFT (GGA+U, U=4 eV) to study even-septuple-layer MnBi2Te4 thin films, focusing on the double-SL film. It shows that an out-of-plane electric field, as produced by dual gates, breaks the PT symmetry of the collinear AFM ground state, thereby generating nonzero Berry curvature and a large anomalous Hall signal. The authors report a critical field Ec1 = 0.021 V/Å at which the band gap closes, followed by a Chern insulator phase with Chern number 3 and quantized anomalous Hall conductance 3e^2/h, and a second critical field Ec2 = 0.027 V/Å above which the film becomes a trivial AFM metal. They corroborate the Chern number with edge-state calculations and Kubo-Greenwood conductance, and propose a dual-gate AFM memory device with the AH signal as readout. The paper also claims generalization to other even-SL MnBi2Te4 films and presents a four-SL case in the supplementary material.

Significance. If the predictions hold, the paper offers a concrete voltage-controlled topological switch in a realistic antiferromagnetic thin film, with a quantized high Chern number and a large on/off Hall ratio. The central C=3 phase is supported by converging evidence: the gap closure at Ec1, the extended Berry curvature distribution near the three crossing points, three chiral edge states in the semi-infinite tight-binding model, and a quantized AH conductance from the Kubo-Greenwood formula. The device proposal is concrete, and the generalization to four-SL films adds scope. The main strength is that the topological characterization is internally consistent and uses standard ab initio and Wannier-based methods with no parameter refitting of the target result.

major comments (2)
  1. [Results, paragraph after Fig. 2] The sentence 'its magnetism keeps the AFM order as the ground state in a finite field range (see Fig. S2 in Supplementary Materials)' is the sole support for the persistence of the collinear out-of-plane AFM order up to and beyond Ec1 and Ec2. This assumption is load-bearing: the PT-breaking mechanism, the gap evolution in Fig. 2, and the C=3 classification in Fig. 3 all require that the two Mn sublattices remain exactly opposite and out-of-plane. The referenced Fig. S2 is not included in the version under review, and no total-energy comparison, canting angle, or ferromagnetic/canted phase check is provided in the main text. Please include in the manuscript or in an accessible supplement the total energies of AFM, ferromagnetic, and canted configurations as a function of electric field over at least 0-0.03 V/Å, and state the dependence of the stable magnetic order on U and on spin-orbit coupling.
  2. [Methods and Fig. 2] The electronic structure and Wannier functions are computed with GGA+U using U=4 eV on Mn 3d, but no sensitivity analysis of the topological phase boundaries to U is reported. The QAH window between Ec1=0.021 V/Å and Ec2=0.027 V/Å is only 0.006 V/Å wide, and since the gap and the critical fields for MnBi2Te4 are known to depend on the Hubbard U, a modest change in U could shift or close this window. Please provide Ec1, Ec2, and the Chern number for at least U=3, 4, and 5 eV, and ideally show the HSE06 band gap at zero field and near Ec1, since the HSE06 check is mentioned but no results are shown.
minor comments (4)
  1. [Abstract and Introduction] The phrase 'high Chern number of 3' is used repeatedly; since the paper claims this is the first AFM-material QAH proposal with high Chern number, please add a brief comparison with existing high-Chern QAH predictions in non-AFM or engineered systems to support the novelty statement.
  2. [Throughout] There are several typographical and grammatical errors, including 'doublely degenerated', 'valance' for 'valence', and 'antiferromagent' in the Introduction. A careful proofreading pass is needed.
  3. [Fig. 2 caption] The caption notes that in panel (E) the black dashed line overlaps the red one; this makes it difficult to locate the Fermi level in that panel. Please adjust the color or line style so that the Fermi level is clearly distinguishable.
  4. [Device proposal section] The on/off ratio of 10^6-10^14 quoted in the text is an enormous range; the manuscript would benefit from a sentence specifying the main sources of the temperature and dissipation dependence that produce this spread.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Chern number, critical fields, and AH conductance are emergent outputs of ab initio DFT/Wannier calculations, not refits of the target result.

full rationale

The paper's central predictions—the electric-field-driven transition into a Chern insulator with C=3 and the quantized AH conductance of 3 e^2/h—are obtained from first-principles DFT band structures, Wannier-interpolated Berry curvature, and Kubo-Greenwood conductance calculations. The critical fields (Ec1 = 0.021 V/Å, Ec2 = 0.027 V/Å) are read off from computed band-gap closings, and the Chern number is determined from the Berry curvature distribution and edge-state counting; none of these quantities is inserted as an input or fitted to reproduce the claimed phase. The magnetic ground state (out-of-plane AFM order) is re-derived in the present DFT relaxation, with the citation to prior work [26, 27] given only as consistency confirmation, not as the load-bearing evidence. The symmetry argument that PT symmetry forces zero Berry curvature and that an electric field breaks PT is a standard, externally established result, not a self-citation. The paper's reliance on Fig. S2 for the persistence of AFM order under field is a verification/completeness concern, not a circularity: even if that check were missing, the logic of the computation would not reduce the prediction to its inputs. No equation in the text equates a fitted parameter with the predicted Chern number or Hall conductance, and no load-bearing claim is justified solely by a same-author citation. The derivation chain is therefore self-contained with respect to the claimed results.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard DFT approximations, the experimental magnetic structure, and the validity of the intrinsic AHE picture. The only manually chosen numeric parameter is the Hubbard U. No new physical entities are introduced.

free parameters (1)
  • Hubbard U for Mn 3d = 4 eV
    Chosen for the GGA+U calculation of Mn 3d electrons; not derived from the target result, but the band gap and topological phase boundaries depend on this value.
assumptions (4)
  • domain assumption The PBE exchange-correlation functional with GGA+U approximates the ground-state electronic structure of MnBi2Te4 accurately enough for topological classification.
    Computational results rely on this; HSE06 is used only to check the band gap, not the topology.
  • domain assumption Each septuple layer is ferromagnetically ordered with out-of-plane moments, and adjacent layers couple antiferromagnetically, giving the PT-symmetric AFM ground state.
    Supported by previous calculations and experiments cited in refs [26-29] and by the authors' own DFT; it is the starting point for the PT symmetry breaking scenario.
  • domain assumption The intrinsic Berry curvature contribution dominates the anomalous Hall effect; extrinsic mechanisms such as skew scattering are negligible.
    Invoked in the discussion of the proposed device; the paper argues high crystal quality justifies it.
  • domain assumption The AFM order is maintained under the applied electric field up to the studied range.
    Confirmed in the supplementary but not visible in the main text; if the field changed the magnetic order to ferromagnetic or canted, the PT-breaking mechanism and C=3 phase would be modified.

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Cite this review

Pith. "Pith review of Berry Curvature Engineering by Gating Two-Dimensional Antiferromagnets." pith.science (2026). https://pith.science/paper/AA3LUBQ5

@misc{pith2026190901194,
  author       = {Pith},
  title        = {Pith review of: Berry Curvature Engineering by Gating Two-Dimensional Antiferromagnets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AA3LUBQ5}},
  note         = {Machine review of arXiv:1909.01194}
}
abstract

Recent advances in tuning electronic, magnetic, and topological properties of two-dimensional (2D) magnets have opened a new frontier in the study of quantum physics and promised exciting possibilities for future quantum technologies. In this study, we find that the dual-gate technology can well tune the electronic and topological properties of antiferromagnetic (AFM) even septuple-layer (SL) MnBi$_2$Te$_4$ thin films. Under an out-of-plane electric field that breaks $\mathcal{PT}$ symmetry, the Berry curvature of the thin film could be engineered efficiently, resulting in a huge change of anomalous Hall (AH) signal. Beyond the critical electric field, the double-SL MnBi$_2$Te$_4$ thin film becomes a Chern insulator with a high Chern number of 3. We further demonstrate that such 2D material can be used as an AFM switch via electric-field control of the AH signal. These discoveries inspire the design of low-power memory prototype for future AFM spintronic applications.

Figures

Figures reproduced from arXiv: 1909.01194 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Works this paper leans on

40 extracted references · 29 canonical work pages

  1. [1]

    Baltz, A

    V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Rev. Mod. Phys.90, 015005 (2018)

    2018

  2. [2]

    Gomonay, T

    O. Gomonay, T. Jungwirth, and J. Sinova, Phys. Status Solidi RRL 11, 1700022 (2017)

    work page 2017

  3. [3]

    Jungwirth, X

    T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nat. Nanotechnol. 11, 231 (2016)

    2016

  4. [4]

    A. H. MacDonald and M. Tsoi, Phil. Trans. R. Soc. A 369, 3098 (2011)

    work page 2011

  5. [5]

    ˇZelezn´ y, H

    J. ˇZelezn´ y, H. Gao, K. V´ yborn´ y, J. Zemen, J. Maˇ sek, A. Manchon, J. Wunderlich, J. Sinova, and T. Jung- wirth, Phys. Rev. Lett. 113, 157201 (2014)

    work page 2014

  6. [6]

    X. F. Zhou, J. Zhang, F. Li, X. Z. Chen, G. Y. Shi, Y. Z. Tan, Y. D. Gu, M. S. Saleem, H. Q. Wu, F. Pan, and C. Song, Phys. Rev. Appl. 9, 054028 (2018)

    work page 2018

  7. [7]

    Marti, I

    X. Marti, I. Fina, C. Frontera, J. Liu, P. Wadley, Q. He, R. J. Paull, J. D. Clarkson, J. Kudrnovsk´ y, I. Turek, J. Kuneˇ s, D. Yi, J.-H. Chu, C. T. Nelson, L. You, E. Arenholz, S. Salahuddin, J. Fontcuberta, T. Jung- wirth, and R. Ramesh, Nat. Mater. 13, 367 (2014)

    work page 2014

  8. [8]

    X. Z. Chen, R. Zarzuela, J. Zhang, C. Song, X. F. Zhou, G. Y. Shi, F. Li, H. A. Zhou, W. J. Jiang, F. Pan, and Y. Tserkovnyak, Phys. Rev. Lett. 120, 207204 (2018)

    work page 2018

Show all 40 references
  1. [9]

    M. Wang, W. Cai, D. Zhu, Z. Wang, J. Kan, Z. Zhao, K. Cao, Z. Wang, Y. Zhang, T. Zhang, C. Park, J.-P. Wang, A. Fert, and W. Zhao, Nat. Electron. 1, 582 (2018)

    2018

  2. [10]

    Moriyama, W

    T. Moriyama, W. Zhou, T. Seki, K. Takanashi, and O. Teruo, Phys. Rev. Lett. 121, 167202 (2018)

    2018

  3. [11]

    Wadley, B

    P. Wadley, B. Howells, J. ˇZelezn´ y, C. Andrews, V. Hills, R. P. Campion, V. Nov´ ak, K. Olejn´ ık, F. Maccherozzi, S. S. Dhesi, S. Y. Martin, T. Wagner, J. Wunderlich, F. Freimuth, Y. Mokrousov, J. Kunˇ es, J. S. Chauhan, M. J. Grzybowski, A. W. Rushforth, K. W. Edmonds, B. ...

    2016

  4. [12]

    S. Y. Bodnar, L. ˇSmejkal, I. Turek, T. Jungwirth, O. Gomonay, J. Sinova, A. A. Sapozhnik, H.-J. Elmers, M. Kl¨ aui, and M. Jourdan, Nat. Comm. 9, 348 (2018)

    2018

  5. [13]

    Olejn´ ık, V

    K. Olejn´ ık, V. Schuler, X. Marti, V. Nov´ ak, Z. Kaˇ spar, P. Wadley, R. P. Campion, K. W. Edmonds, B. L. Gal- lagher, J. Garc´ es, M. Baumgartner, P. Gambardella, and T. Jungwirth, Nat. Comm. 8, 15434 (2017)

    2017

  6. [14]

    Nagaosa, J

    N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010)

    2010

  7. [15]

    Kiyohara, T

    N. Kiyohara, T. Tomita, and S. Nakatsuji, Phys. Rev. Appl. 5, 064009 (2016)

    2016

  8. [16]

    Liu and L

    J. Liu and L. Balents, Phys. Rev. Lett. 119, 087202 (2017)

    2017

  9. [17]

    A. K. Nayak, J. E. Fischer, Y. Sun, B. Yan, J. Karel, A. C. Komarek, C. Shekhar, N. Kumar, W. Schnelle, J. K¨ ubler, C. Felser, and S. S. P. Parkin, Sci. Adv. 2, e1501870 (2016)

    2016

  10. [18]

    Nakatsuji, N

    S. Nakatsuji, N. Kiyohara, and T. Higo, Nature 527, 212 (2015)

    2015

  11. [19]

    H. Chen, Q. Niu, and A. H. MacDonald, Phys. Rev. Lett. 112, 017205 (2014)

    2014

  12. [20]

    Z. Liu, H. Chen, J. Wang, J. Liu, K. Wang, Z. Feng, H. Yan, X. Wang, C. Jiang, J. Coey, and A. H. Mac- Donald, Nat. Electro. 1, 172 (2018)

    2018

  13. [21]

    Q. L. He, X. Kou, A. J. Grutter, G. Yin, L. Pan, X. Che, Y. Liu, T. Nie, B. Zhang, S. M. Disseler, B. J. Kirby, W. Ratcliff II, Q. Shao, K. Murata, X. Zhu, G. Yu, Y. Fan, M. Montazeri, X. Han, J. A. Borchers, and K. L. Wang, Nat. Mater. 16, 94 (2017)

    2017

  14. [22]

    P. Tang, Q. Zhou, G. Xu, and S.-C. Zhang, Nat. Phys. 12, 1100 (2016)

    2016

  15. [23]

    Y. Gong, J. Guo, J. Li, K. Zhu, M. Liao, X. Liu, Q. Zhang, L. Gu, L. Tang, X. Feng, D. Zhang, W. Li, C. Song, L. Wang, P. Yu, X. Chen, Y. Wang, H. Yao, W. Duan, Y. Xu, S.-C. Zhang, X. Ma, Q.-K. Xue, and K. He, Chin. Phys. Lett. 36, 076801 (2019)

    2019

  16. [24]

    M. M. Otrokov, I. I. Klimovskikh, H. Bentmann, A. Zeugner, Z. S. Aliev, S. Gass, A. U. Wolter, A. V. Koroleva, D. Estyunin, A. M. Shikin, M. Blanco-Rey, M. Hoffmann, A. Y. Vyazovskaya, S. V. Eremeev, Y. M. Koroteev, I. R. Amiraslanov, M. B. Babanly, N. T. Mamedov, N. A. Abdulla...

    2018 arXiv

  17. [25]

    Zhang, M

    D. Zhang, M. Shi, T. Zhu, D. Xing, H. Zhang, and J. Wang, Phys. Rev. Lett. 122, 206401 (2019)

    2019

  18. [26]

    J. Li, Y. Li, S. Du, Z. Wang, B.-L. Gu, S.-C. Zhang, K. He, W. Duan, and Y. Xu, Sci. Adv. 5, eaaw5685 (2019)

    2019

  19. [27]

    M. M. Otrokov, I. P. Rusinov, M. Blanco-Rey, M. Hoff- mann, A. Y. Vyazovskaya, S. V. Eremeev, A. Ernst, P. M. Echenique, A. Arnau, and E. V. Chulkov, Phys. Rev. Lett. 122, 107202 (2019)

    2019

  20. [28]

    Zeugner, F

    A. Zeugner, F. Nietschke, A. U. B. Wolter, S. Gaß, R. C. Vidal, T. R. F. Peixoto, D. Pohl, C. Damm, A. Lubk, R. Hentrich, S. K. Moser, C. Fornari, C. H. Min, S. Schatz, K. Kißner, M. ¨Unzelmann, M. Kaiser, F. Scar- avaggi, B. Rellinghaus, K. Nielsch, C. Hess, B. B¨ uchner, F. ...

    2019

  21. [29]

    J. Cui, M. Shi, H. Wang, F. Yu, T. Wu, X. Luo, J. Ying, and X. Chen, Phys. Rev. B 99, 155125 (2019)

    2019

  22. [30]

    X.-Y. Dong, S. Kanungo, B. Yan, and C.-X. Liu, Phys. Rev. B 94, 245135 (2016)

    2016

  23. [31]

    Y. Deng, Y. Yu, M. Z. Shi, J. Wang, X. H. Chen, and Y. Zhang, arXiv:1904.11468 (2019)

    2019 arXiv

  24. [32]

    C. Liu, Y. Wang, H. Li, Y. Wu, Y. Li, J. Li, K. He, Y. Xu, J. Zhang, and Y. Wang, arXiv:1905.00715 (2019)

    2019 arXiv

  25. [33]

    J. Ge, Y. Liu, J. Li, H. Li, T. Luo, Y. Wu, Y. Xu, and J. Wang, arXiv:1907.09947 (2019)

    2019 arXiv

  26. [34]

    Kresse and J

    G. Kresse and J. Furthmller, Comput. Mater. Sci. 6, 15 7 (1996)

    1996

  27. [35]

    Grimme, J

    S. Grimme, J. Comput. Chem. 27, 1787 (2006),

    2006

  28. [36]

    J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 118, 8207 (2003),

    2003

  29. [37]

    Pizzi, V

    G. Pizzi, V. Vitale, R. Arita, S. Bl¨ ugel, F. Freimuth, G. G´ eranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune, J. Iba˜ nez−Azpiroz, H. Lee, J.-M. Lihm, D. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa, Y. Nohara, Y. Nomura, L. Paulatto, S. Ponc´ e, T. Pon- wei...

    2019 arXiv

  30. [38]

    Q. S. Wu, S. N. Zhang, H.-F. Song, M. Troyer, and A. A. Soluyanov, Comput. Phys. Commu. 224, 405 (2018)

    2018

  31. [39]

    S. Wu, J. S. Ross, G.-B. Liu, G. Aivazian, A. Jones, Z. Fei, W. Zhu, D. Xiao, W. Yao, D. Cobden, and X. Xu, Nat. Phys. 9, 149 (2013)

    2013

  32. [40]

    Murakami, New J

    S. Murakami, New J. Phys. 9, 356 (2007)

    2007

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