Pith. sign in

REVIEW 3 major objections 5 minor 61 references

Electric Field Induced Multi-Space Topological Phase Transitions in Janus Monolayer MnBi2Se2Te2

T0 review · 3 major / 5 minor · reviewed 2026-07-11 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A perpendicular electric field simultaneously switches Chern numbers and skyrmion textures in Janus monolayer MnBi2Se2Te2.

desk verdict Clean computational phase diagram for concurrent electric-field control of Chern number and skyrmions in Janus MBTSe; the hybrid RK-joint counting is an untested extrapolation from collinear bands. read the letter →

arxiv 2607.05631 v1 pith:BV6S6LWM submitted 2026-07-06 cond-mat.mtrl-sci

Kefan Zhang , Jian Wu , Weiyi Pan This is my paper · ORCID
classification cond-mat.mtrl-sci
keywords quantumanomalousHalleffectmagneticskyrmionsJanusmonolayerelectric-fieldcontrolChernnumberDzyaloshinskii-MoriyainteractionMnBi2Se2Te2multi-spacetopology
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 establishes that a single Janus monolayer of MnBi2Se2Te2 can host coexisting quantum anomalous Hall states and magnetic skyrmions that are both reconfigured by one perpendicular electric field. First-principles band calculations show the field closes and reopens gaps, driving the Chern number through 0, −1 and 2, while atomistic spin simulations show the same field tunes the Dzyaloshinskii-Moriya interaction against magnetic anisotropy, converting a uniform ferromagnet into isolated skyrmions and then into a skyrmion-spiral coexistence phase. The joint evolution produces three regimes: pure momentum-space topology, hybrid RK-joint skyrmions whose chiral boundary-state count is set by the local Chern numbers, and pure real-space skyrmions. A sympathetic reader cares because purely electrical, reversible control of both topological orders inside one intrinsic 2-D material would enable multi-functional, low-power spintronic devices without heterostructures or magnetization reversal.

What carries the argument

Electric-field tuning of the competition between the Dzyaloshinskii-Moriya interaction and magnetic anisotropy (which sets skyrmion stability) together with field-induced band anti-crossings at the Q and Γ points (which redistribute Berry curvature and change the Chern number).

What would settle it

A DFT+U recalculation or experiment in which the Chern-number transitions or the FM-to-skyrmion boundary vanish or shift by more than ~0.2 V/Å when U is varied by ±1 eV around 4 eV would falsify the reported critical fields and multi-space phase diagram.

Watch

Extended reading notes

Core claim

Using first-principles calculations and atomistic spin simulations, the authors show that a perpendicular electric field simultaneously drives momentum-space transitions among a topologically trivial state (C = 0), a low-Chern-number quantum anomalous Hall state (C = −1) and a high-Chern-number state (C = 2), and real-space magnetic transitions from a uniform ferromagnetic state through isolated skyrmions to a skyrmion-spiral domain coexistence phase, in Janus monolayer MnBi2Se2Te2.

Load-bearing premise

The Hubbard U of 4 eV chosen for the Mn 3d electrons correctly captures both the band inversions that fix the Chern numbers and the magnetic exchange, DMI and anisotropy parameters that fix the skyrmion phase boundaries.

Editorial extensions

If this is right

  • A single gate voltage can switch both the magnitude and sign of the Chern number, thereby changing the number and chirality of dissipationless edge channels without reversing magnetization.
  • The same voltage can nucleate, enlarge or stabilize isolated skyrmions and later drive the system into a skyrmion-spiral coexistence phase.
  • Hybrid phases host RK-joint skyrmions whose chiral boundary-state count N_CBS equals the difference of inner and outer Chern numbers and is therefore field-tunable (4 → −2 → 0).
  • Collapse barriers of several meV imply that the skyrmion states remain thermally stable at cryogenic temperatures accessible to experiment.
  • The multi-step sequence supplies a materials platform for multi-level topological memory or logic beyond binary skyrmion presence or absence.

Reading between the lines

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

  • If dielectric engineering or mild strain can lower the critical fields from the calculated ~0.3–0.55 V/Å range, the same multi-space control could become practical in gated devices.
  • The Janus substitution strategy demonstrated here can be transferred to other MnBi2Te4-family monolayers in a search for higher-temperature multi-space topology.
  • Simultaneous anomalous-Hall transport and real-space magnetic imaging under a single gate voltage would directly map the predicted four-phase sequence.
  • Field-tunable N_CBS opens a multi-level skyrmion storage scheme in which states are distinguished by the number of circulating boundary modes rather than by skyrmion number alone.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript claims that a perpendicular electric field simultaneously drives momentum-space topological transitions (C = 0, −1, 2) and real-space magnetic transitions (uniform FM → isolated skyrmions → skyrmion–spiral coexistence) in Janus monolayer MnBi2Se2Te2. First-principles DFT+U, Wannier-based Berry curvature/edge-state calculations, and classical LLG/GNEB spin simulations are used to map both sectors versus field. The authors further assert that the coexistence produces hybrid RK-joint skyrmions whose chiral boundary-state count N_CBS = C_inner − C_outer is field-tunable (4 → −2 → 0), yielding a multi-space phase diagram (Fig. 6).

Significance. If the multi-space control is robust, the work supplies a concrete single-material platform in which both Chern number and skyrmion stability are gated by the same perpendicular field—an attractive route for reconfigurable, low-power topological spintronics. The calculations themselves are standard and carefully executed: VASP+U, WannierTools edge states, Spirit LLG and GNEB are applied consistently; the reported Chern numbers match the edge-state counts, and the DMI/MAE trends correctly track the LLG textures. The explicit GNEB collapse barriers and the empirical α guide add quantitative support for the real-space side. These strengths make the paper a useful contribution provided the hybrid topology claim is either demonstrated or carefully caveated.

major comments (3)
  1. §III.D and Fig. 6: The hybrid RK-joint regime (and the sequence N_CBS = 4, −2, 0) rests on assigning C_inner = −C_outer taken from collinear FM band structures (§III.B, Figs. 2–3). All Chern numbers, Berry curvatures and edge states are computed exclusively for uniform out-of-plane FM configurations. Real-space textures are obtained separately from the classical spin Hamiltonian (Eq. 1) via LLG/GNEB. No electronic structure, local Chern marker or Berry curvature is ever evaluated on a non-collinear skyrmion background. Consequently the claim that the FM band inversion and gap survive continuous magnetization rotation—and therefore that field-tunable CBSs exist—is an untested extrapolation. Either a local topological invariant on a skyrmion texture (or a controlled continuum model) must be supplied, or the hybrid claim must be explicitly labeled as a conjecture and the phase diagram revis
  2. Methods §II.A and §III.B–C: The Hubbard U = 4 eV for Mn 3d is taken from bulk MnBi2Te4 literature without a sensitivity scan. Both the critical electric fields for gap closing (≈ −0.3 and +0.55 V Å−1) and the DMI/MAE values that set the skyrmion boundaries depend on the precise position of the Mn d states. A modest U variation (or a hybrid-functional check) is needed to establish that the multi-space phase sequence is not an artifact of this single parameter choice.
  3. §III.B and Fig. 2: The reported critical fields (0.3–0.6 V Å−1) are extremely large for a freestanding monolayer. While dipole corrections are mentioned, no estimate of dielectric screening, substrate effects or dielectric breakdown is given. The practical accessibility of the claimed phase diagram should be addressed, or the fields should be renormalized to a more realistic gated geometry.
minor comments (5)
  1. Abstract and §I: “RK-joint skyrmions” is introduced without a self-contained definition; a one-sentence clarification (or pointer to the earlier definition) would help non-specialist readers.
  2. Fig. 4(b): the empirical parameter α is plotted but its numerical prefactor and literature origin are only briefly mentioned; a short derivation or reference in the caption would improve transparency.
  3. §II.B, Eq. (2): the factor 12S2 in the DMI extraction formula assumes a specific neighbor count; a brief justification (or SM reference) would avoid ambiguity.
  4. Typographical: “dispassionless” → “dissipationless” (p. 1); “symstem” → “system” (p. 4); “Being Consistent” → “Consistent” (p. 7).
  5. Data-availability statement: “not technically feasible” is unusually absolute for DFT/LLG data; a more conventional “available upon reasonable request” would suffice.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Chern numbers, magnetic parameters, and textures are obtained from independent DFT/Wannier and classical spin calculations; multi-space combination is interpretive, not definitional.

full rationale

The derivation chain is self-contained and non-circular. Momentum-space topology (C = 0, −1, 2) is computed directly from DFT+U band structures of collinear out-of-plane FM configurations via Wannier interpolation, Berry curvature (Eq. 3), AHC quantization, and edge-state Green’s functions (Figs. 2–3, §III.B); no parameters are fitted to force the Chern numbers. Magnetic interactions (J1, d//, Azz) are extracted by energy mapping from the same DFT total energies under electric field (Eqs. 1–2, Fig. 4, §II.B/§III.C) and fed into independent atomistic LLG/GNEB simulations that produce the real-space textures (FM → ISK → SK-SD). The empirical α guide is taken from external literature and is not used to set phase boundaries. The RK-joint skyrmion concept and N_CBS = C_inner − C_outer formula are imported from non-overlapping external citations [18, 60] and applied interpretively to the separately computed phases (Fig. 6, §III.D); they do not feed back into any numerical result. No self-definitional loop, no fitted-input-as-prediction, and no load-bearing self-citation chain exists. The untested extrapolation that FM Chern numbers survive non-collinear textures is an assumption about physical validity, not a circular reduction of the paper’s own equations.

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

The central claim rests on standard DFT+U and classical spin-Hamiltonian machinery plus a handful of numerical choices (U, damping, supercell size) taken from prior literature or set by hand. No new physical entities are postulated; the RK-joint concept is imported from earlier papers. The largest modeling assumptions are the transferability of U = 4 eV and the neglect of higher-order magnetic interactions and thermal fluctuations.

free parameters (4)
  • Hubbard U for Mn 3d = 4 eV
    Fixed at 4 eV following earlier MnBi2Te4 studies; controls both band inversion and magnetic parameters that set the critical electric fields.
  • Gilbert damping = 0.2
    Set to 0.2 for LLG relaxation; affects only dynamics, not the final energy-minimized textures, but is still an arbitrary numerical choice.
  • LLG supercell size = 120×120
    120 × 120 cells chosen to accommodate isolated skyrmions; finite-size effects on skyrmion radius and barrier are not quantified.
  • Empirical skyrmion parameter α prefactor
    α = 4√(J1 Azz)/(π d∥) taken from literature; used only as a qualitative indicator, not fitted to data.
assumptions (4)
  • domain assumption DFT+U with a single static U adequately describes both the topological band inversion and the magnetic exchange/DMI/MAE of the Janus monolayer.
    Invoked throughout §II.A and §III; no hybrid-functional or GW cross-check is provided.
  • domain assumption The classical Heisenberg + DMI + uniaxial MAE spin Hamiltonian (Eq. 1) captures all relevant magnetic energetics; higher-order exchanges and dipolar terms can be neglected.
    Stated in §II.B; only three nearest-neighbor J’s and the in-plane DMI component are retained.
  • domain assumption Out-of-plane DMI component dz has negligible influence on skyrmion stability in C3v triangular lattices.
    Cited from prior literature [45,46] and used to justify focusing solely on d∥ (Eq. 2).
  • standard math Maximally-localized Wannier functions constructed from Bi-p, Mn-d, Se-p, Te-p orbitals faithfully reproduce the near-Fermi topology.
    Standard Wannier90 workflow (§II.C); interpolation quality is not quantified beyond visual band agreement.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Electric Field Induced Multi-Space Topological Phase Transitions in Janus Monolayer MnBi2Se2Te2." pith.science (2026). https://pith.science/paper/BV6S6LWM

@misc{pith2026260705631,
  author       = {Pith},
  title        = {Pith review of: Electric Field Induced Multi-Space Topological Phase Transitions in Janus Monolayer MnBi2Se2Te2},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BV6S6LWM}},
  note         = {Machine review of arXiv:2607.05631}
}
read the original abstract

Manipulating coexisting multi-space topological states within a single material is a critical frontier for low-power spintronic applications, for which perpendicular electric-field gating offers a highly precise tuning mechanism. Using first-principles calculations and atomistic spin simulations, it is demonstrated that Janus monolayer MnBi2Se2Te2 can exhibit simultaneous momentum-space and real-space topological phase transitions under an external electric field. Specifically, the electric field drives momentum-space transitions across a topologically trivial state (C = 0), a low-Chern-number quantum anomalous Hall state (C = -1), and a high-Chern-number state (C = 2). Concurrently, by actively tuning the competition between the Dzyaloshinskii-Moriya interaction and magnetic anisotropy, the electric field induces real-space magnetic transitions from a uniform ferromagnetic state to an isolated skyrmion state, and ultimately to a skyrmion-spiral domain coexistence phase. Electric-field-induced variations in both the QAHE and magnetic textures may give rise to multiple topological phase transitions, involving distinct topological regimes: a purely k-space topology, RK-joint skyrmions, and pure real-space skyrmions. These findings establish a powerful material platform and efficient route for the synergistic control of coexisting topological orders, making it highly promising for next generation multi-functional spintronic devices.

Figures

Figures reproduced from arXiv: 2607.05631 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Side view of the MBTSe crystal structure and constituent atomic species. (b) Top view of the MBTSe lattice [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Evolution of the local bandgap as a function of the perpendicular external electric field for the out-of-plane spin [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Calculated chiral edge states for semi-infinite MBTSe at perpendicular electric fields of (a) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Evolution of the nearest-neighbor exchange interaction [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Collapse energy barrier of an isolated skyrmion [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Comprehensive multi-space topological phase diagram of Janus monolayer MBTSe under a perpendicular electric field. [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

61 extracted references · 2 canonical work pages

  1. [1]

    through bandgap closing and reopening under electric field. In real space, the electric field induces magnetic transitions from a uniform ferromagnetic (FM) state to a skyrmion phase, which is also revealed by the variation of the skyrmion collapse barrier. This coupled evolu- tion leads to the formation of multi-space joint topolog- ical structures. Thes...

  2. [2]

    J. M. Kosterlitz and D. J. Thouless, J. Phys. C: Solid State Phys.6, 1181 (1973)

    1973

  3. [3]

    D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett.49, 405 (1982)

    1982

  4. [4]

    M. Z. Hasan and C. L. Kane, Rev. Mod. Phys.82, 3045 (2010)

    2010

  5. [5]

    Qi and S.-C

    X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys.83, 1057 (2011)

    2011

  6. [6]

    Nagaosa and Y

    N. Nagaosa and Y. Tokura, Nat. Nanotechnol.8, 899 (2013)

    2013

  7. [7]

    Tokura and N

    Y. Tokura and N. Kanazawa, Chem. Rev.121, 2857 (2020)

    2020

  8. [8]

    A. Fert, N. Reyren, and V. Cros, Nat. Rev. Mater.2, 17031 (2017)

    2017

Show all 61 references
  1. [9]

    Q. L. He, T. L. Hughes, N. P. Armitage, Y. Tokura, and K. L. Wang, Nat. Mater.21, 15 (2022)

    2022

  2. [10]

    M.-G. Han, J. D. Thomsen, J. P. Philbin, J. Mun, E. Park, F. Camino, L. Dekanovsky, C. Liu, Z. Sofer, P. Narang,et al., Nano Lett.25, 5174 (2025)

    2025

  3. [11]

    F. D. M. Haldane, Phys. Rev. Lett.61, 2015 (1988)

    2015

  4. [12]

    H. Weng, R. Yu, X. Hu, X. Dai, and Z. Fang, Adv. Phys. 64, 227 (2015)

    2015

  5. [13]

    Liu, S.-C

    C.-X. Liu, S.-C. Zhang, and X.-L. Qi, Annu. Rev. Con- dens. Matter Phys.7, 301 (2016)

    2016

  6. [14]

    Chang, J

    C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang,et al., Science 340, 167 (2013)

    2013

  7. [15]

    Z. Xu, W. Duan, and Y. Xu, Nano Lett.23, 305 (2022)

    2022

  8. [16]

    Y.-F. Zhao, R. Zhang, R. Mei, L.-J. Zhou, H. Yi, Y.- Q. Zhang, J. Yu, R. Xiao, K. Wang, N. Samarth,et al., Nature588, 419 (2020)

    2020

  9. [17]

    Y. Deng, Y. Yu, M. Z. Shi, Z. Guo, Z. Xu, J. Wang, X. H. Chen, and Y. Zhang, Science367, 895 (2020)

    2020

  10. [18]

    H. Chen, J. Lu, N. Wang, X. Zhao, G. Hu, X. Yuan, and J. Ren, Appl. Phys. Lett.124, 10.1063/5.0174081 (2024)

    2024 doi

  11. [19]

    Y. Li, S. Xu, J. Wang, C. Wang, B. Yang, H. Lin, W. Duan, and B. Huang, Proc. Natl. Acad. Sci. U.S.A. 119, e2122952119 (2022)

    2022

  12. [20]

    X. Yang, Y. Shen, J. Liu, L. Lv, M. Zhou, Y. Zhang, X. Meng, Z. Zhou, and Y. Zheng, Appl. Phys. Lett.122, 10.1063/5.0147538 (2023)

    2023 doi

  13. [21]

    Serlin, C

    M. Serlin, C. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. Young, Science367, 900 (2020)

    2020

  14. [22]

    S. Du, P. Tang, J. Li, Z. Lin, Y. Xu, W. Duan, and A. Rubio, Phys. Rev. Res.2, 022025 (2020)

    2020

  15. [23]

    You, X.-J

    J.-Y. You, X.-J. Dong, B. Gu, and G. Su, Phys. Rev. B 103, 104403 (2021)

    2021

  16. [24]

    Zhang, Y

    X. Zhang, Y. Zhou, K.-M. Song, T.-E. Park, J. Xia, M. Ezawa, X. Liu, W. Zhao, G. Zhao, and S. Woo, J. Phys.: Condens. Matter32, 143001 (2020)

    2020

  17. [25]

    Tokura, M

    Y. Tokura, M. Kawasaki, and N. Nagaosa, Nat. Phys. 13, 1056 (2017)

    2017

  18. [26]

    W. Du, K. Dou, Z. He, Y. Dai, B. Huang, and Y. Ma, Nano Lett.22, 3440 (2022)

    2022

  19. [27]

    J. Pang, X. Niu, M. Zhang, Y. Tang, Y. Zhang, and L. Bellaiche, Phys. Rev. B108, 134430 (2023)

    2023

  20. [28]

    F. Wei, W. Zhang, G. Chen, Z. Ren, S. Yan, and S. Kang, Phys. Rev. B113, 045411 (2026)

    2026

  21. [29]

    A. Fert, V. Cros, and J. Sampaio, Nat. Nanotechnol.8, 152 (2013)

    2013

  22. [30]

    Sampaio, V

    J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert, Nat. Nanotechnol.8, 839 (2013)

    2013

  23. [31]

    Romming, C

    N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, Science341, 636 (2013)

    2013

  24. [32]

    Paul and S

    S. Paul and S. Heinze, npj Comput. Mater.8, 105 (2022)

    2022

  25. [33]

    Desplat, S

    L. Desplat, S. Meyer, J. Bouaziz, P. Buhl, S. Lounis, B. Dup´ e, and P.-A. Hervieux, Phys. Rev. B104, L060409 (2021)

    2021

  26. [34]

    D. Li, S. Haldar, T. Drevelow, and S. Heinze, Phys. Rev. B107, 104428 (2023)

    2023

  27. [35]

    I. L. Fernandes, J. Chico, and S. Lounis, J. Phys.: Con- dens. Matter32, 425802 (2020)

    2020

  28. [36]

    I. L. Fernandes, J. Bouaziz, S. Bl¨ ugel, and S. Lounis, Nat. Commun.9, 4395 (2018)

    2018

  29. [37]

    Jiang, D

    J. Jiang, D. Xiao, F. Wang, J.-H. Shin, D. Andreoli, J. Zhang, R. Xiao, Y.-F. Zhao, M. Kayyalha, L. Zhang, et al., Nat. Mater.19, 732 (2020)

    2020

  30. [38]

    Zhang, F

    S. Zhang, F. Kronast, G. van der Laan, and T. Hesjedal, Nano Lett.18, 1057 (2018)

    2018

  31. [39]

    Kresse and J

    G. Kresse and J. Furthm¨ uller, Phys. Rev. B54, 11169 (1996)

    1996

  32. [40]

    P. E. Bl¨ ochl, Phys. Rev. B50, 17953 (1994)

    1994

  33. [41]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett.77, 3865 (1996). 10

    1996

  34. [42]

    S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B57, 1505 (1998)

    1998

  35. [43]

    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

  36. [44]

    Zhang, W

    H. Zhang, W. Yang, Y. Wang, and X. Xu, Phys. Rev. B 103, 094433 (2021)

    2021

  37. [45]

    See Supplementary Material for additional calculation details and supporting figures, Supplementary Material

  38. [46]

    Moriya, Phys

    T. Moriya, Phys. Rev.120, 91 (1960)

    1960

  39. [47]

    H. Niu, H. Y. Kwon, T. Ma, Z. Cheng, C. Ophus, B. Miao, L. Sun, Y. Wu, K. Liu, S. S. Parkin,et al., Nat. Commun.15, 10199 (2024)

    2024

  40. [48]

    Marzari, A

    N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Rev. Mod. Phys.84, 1419 (2012)

    2012

  41. [49]

    A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, Comput. Phys. Commun. 185, 2309 (2014)

    2014

  42. [50]

    X. Wang, J. R. Yates, I. Souza, and D. Vanderbilt, Phys. Rev. B74, 195118 (2006)

    2006

  43. [51]

    J. R. Yates, X. Wang, D. Vanderbilt, and I. Souza, Phys. Rev. B75, 195121 (2007)

    2007

  44. [52]

    M. P. Lopez Sancho, J. M. Lopez Sancho, J. M. Lopez Sancho, and J. Rubio, J. Phys. F: Met. Phys.15, 851 (1985)

    1985

  45. [53]

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

    2018

  46. [54]

    G. P. M¨ uller, M. Hoffmann, C. Dißelkamp, D. Sch¨ urhoff, S. Mavros, M. Sallermann, N. S. Kiselev, H. J´ onsson, and S. Bl¨ ugel, Phys. Rev. B99, 224414 (2019)

    2019

  47. [55]

    P. F. Bessarab, V. M. Uzdin, and H. J´ onsson, Comput. Phys. Commun.196, 335 (2015)

    2015

  48. [56]

    G. P. M¨ uller, P. F. Bessarab, S. M. Vlasov, F. Lux, N. S. Kiselev, S. Bl¨ ugel, V. M. Uzdin, and H. J´ onsson, Phys. Rev. Lett.121, 197202 (2018)

    2018

  49. [57]

    Liang, W

    J. Liang, W. Wang, H. Du, A. Hallal, K. Garcia, M. Chshiev, A. Fert, and H. Yang, Phys. Rev. B101, 184401 (2020)

    2020

  50. [58]

    Zhang, H

    Y. Zhang, H. Zhang, J. Pang, Y. Ma, M. Zhang, X. Xu, and L. Bellaiche, Phys. Rev. B107, 024402 (2023)

    2023

  51. [59]

    Banerjee, J

    S. Banerjee, J. Rowland, O. Erten, and M. Randeria, Phys. Rev. X4, 031045 (2014)

    2014

  52. [60]

    Soumyanarayanan, M

    A. Soumyanarayanan, M. Raju, A. Gonzalez Oyarce, A. K. Tan, M.-Y. Im, A. P. Petrovi´ c, P. Ho, K. Khoo, M. Tran, C. Gan,et al., Nat. Mater.16, 898 (2017)

    2017

  53. [61]

    Z. He, W. Du, K. Dou, Y. Dai, B. Huang, and Y. Ma, Adv. Funct. Mater.35, 2414922 (2025)

    2025

Pith tools

Reviewed July 11, 2026 · model on record in the stance chip above.