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arxiv: 2605.19184 · v2 · pith:AG5WLGZBnew · submitted 2026-05-18 · ⚛️ physics.comp-ph · cond-mat.mtrl-sci

Floquet-Engineered Odd-Parity Altermagnetic Higher-Order Topology in a Two-Dimensional Antiferromagnet Cr₂CH₂

Xiaorong Zou , Hyeon Suk Shin , Baibiao Huang , Yanmei Zang , Ying Dai , Chengwang Niu , Chang-Jong Kang , Chang Woo Myung This is my paper

Pith reviewed 2026-05-25 05:52 UTC · model grok-4.3

classification ⚛️ physics.comp-ph cond-mat.mtrl-sci
keywords Floquet drivingaltermagnetismhigher-order topological insulatorantiferromagnetcorner statesodd-parity spin splittingCr2CH2periodic driving
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0 comments X

The pith

Periodic driving with circularly polarized light realizes an odd-parity altermagnetic higher-order topological insulator in the Cr₂CH₂ monolayer.

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

This paper demonstrates that periodic driving can turn the equilibrium antiferromagnetic higher-order topological insulator phase of the Cr₂CH₂ monolayer into an odd-parity altermagnetic version of the same phase. The driven system develops f-wave altermagnetism with odd-parity spin splitting while the C₃-protected corner states stay intact across a wide range of light intensities. At stronger driving the material crosses over into an altermagnetic semimetal. A reader would care because the result links dynamic control of magnetism directly to the survival of higher-order topology in two dimensions.

Core claim

In equilibrium, the Cr₂CH₂ monolayer is a two-dimensional antiferromagnetic higher-order topological insulator protected by C₃ rotational symmetry, as indicated by the symmetry indicator χ^(3) = {-2,1} and the presence of robust corner states. Application of circularly polarized light induces a transition to an f-wave altermagnetic state characterized by the symmetry [C₂ || 3-bar_001] and odd-parity spin splitting. The C₃-protected corner states remain intact despite substantial Floquet-induced band renormalization, and increasing the light intensity drives the system into an altermagnetic semimetallic state.

What carries the argument

The C₃ rotational symmetry that continues to protect corner states after Floquet driving induces f-wave altermagnetic order with odd-parity spin splitting.

If this is right

  • The altermagnetic higher-order topology survives Floquet-induced band renormalization over a broad range of driving strengths.
  • Odd-parity spin splitting emerges in the f-wave altermagnetic state under circularly polarized light.
  • Increasing light intensity drives a transition from the altermagnetic HOTI to an altermagnetic semimetal.
  • A direct connection is established between magnetism and topology in a periodically driven antiferromagnetic system.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This Floquet approach could be tested in other C₃-symmetric two-dimensional antiferromagnets to induce similar altermagnetic higher-order topology.
  • Optical control of the transition between the topological and semimetallic regimes may allow tunable spin and topological transport.
  • Scanning tunneling spectroscopy under illumination could directly image whether the corner states remain localized at moderate driving strengths.

Load-bearing premise

The circularly polarized driving field leaves C₃ rotational symmetry unbroken and the equilibrium symmetry indicator continues to protect the corner states after band renormalization.

What would settle it

A calculation or measurement showing that the corner states gap out or move away from zero energy at moderate driving strengths where the symmetry indicator χ^(3) changes.

Figures

Figures reproduced from arXiv: 2605.19184 by Baibiao Huang, Chang-Jong Kang, Chang Woo Myung, Chengwang Niu, Hyeon Suk Shin, Xiaorong Zou, Yanmei Zang, Ying Dai.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Top view and (b) side view of the two-dimensional [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Evolution of the bulk gap of Cr [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Band structure with odd-parity spin split at [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
read the original abstract

Periodic driving provides a platform to dynamically tailor quantum states of matter, yet its impact on symmetry-protected topological phases remains incompletely understood. Here, we demonstrate that periodic driving enables the realization of an odd-parity altermagnetic (AM) higher-order topological insulator (HOTI) phase in the Cr$_2$CH$_2$ monolayer. In equilibrium, Cr$_2$CH$_2$ is a 2D antiferromagnetic (AFM) HOTI protected by $\mathcal C_3$ rotational symmetry, characterized by a symmetry indicator $\chi^{(3)}$ = $\{-2,1\}$ and robust corner states. Under circularly polarized light (CPL), the system develops a f-wave altermagnetic state governed by the symmetry $[C_{2}||\overline{3}_{001}]$ with odd-parity spin splitting. Despite substantial Floquet-induced band renormalization, the $\mathcal C_3$-protected corner states remain intact over a broad range of driving strengths, highlighting the altermagnetic higher-order topology under Floquet driving. As the light intensity increases, the system gradually evolves into an altermagnetic semimetallic state. These results establish a direct connection between magnetism and topology in a periodically driven AFM system, offering a route toward the control of coupled spin and topological transport.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that circularly polarized light driving induces an odd-parity altermagnetic higher-order topological insulator phase in the Cr₂CH₂ monolayer. In equilibrium the system is a C₃-protected AFM HOTI with symmetry indicator χ^(3) = {-2,1} and corner states; under Floquet driving it acquires f-wave altermagnetism with symmetry [C₂ || 3-bar_001] while the C₃-protected corner states remain intact over a broad range of driving amplitudes before the system evolves into an altermagnetic semimetal at high intensity.

Significance. If the central claim is substantiated, the work establishes a concrete route to Floquet-engineer coupled altermagnetic and higher-order topological order in a 2D antiferromagnet, providing a dynamical handle on spin-split topology and corner-state transport. The use of symmetry indicators together with explicit Floquet renormalization constitutes a strength that could be extended to other magnetic 2D materials.

major comments (2)
  1. [Abstract / Floquet-driven phase section] Abstract and the section on the Floquet-driven phase: the assertion that C₃-protected corner states remain intact requires explicit verification that the symmetry indicator χ^(3) = {-2,1} is preserved (or recomputed) on the effective Floquet Hamiltonian; the text states that the driven system develops symmetry [C₂ || 3-bar_001] and reports band renormalization, but does not confirm that the C₃ eigenvalues of the occupied bands or the indicator itself remain unchanged across the reported driving strengths.
  2. [Floquet effective Hamiltonian section] Section describing the effective Hamiltonian under CPL: it is not shown whether the time-periodic drive introduces additional terms that reduce the little-group symmetry at the high-symmetry points relevant to χ^(3), which would undermine the claim that the equilibrium topological protection automatically carries over.
minor comments (2)
  1. [Methods / Figure captions] Figure captions and methods: numerical details on the Floquet-mode truncation, the strength of the vector potential, and the k-mesh used for corner-state calculations should be stated explicitly to allow reproduction of the reported robustness range.
  2. [Symmetry analysis section] Notation: the relation between the equilibrium χ^(3) indicator and the odd-parity spin splitting in the driven f-wave state could be clarified with a short table comparing the relevant irreps before and after driving.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the detailed comments that help improve the manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract / Floquet-driven phase section] Abstract and the section on the Floquet-driven phase: the assertion that C₃-protected corner states remain intact requires explicit verification that the symmetry indicator χ^(3) = {-2,1} is preserved (or recomputed) on the effective Floquet Hamiltonian; the text states that the driven system develops symmetry [C₂ || 3-bar_001] and reports band renormalization, but does not confirm that the C₃ eigenvalues of the occupied bands or the indicator itself remain unchanged across the reported driving strengths.

    Authors: We agree with the referee that explicitly recomputing the symmetry indicator for the Floquet Hamiltonian would provide stronger confirmation. In the revised manuscript, we have added calculations of the C₃ eigenvalues and the symmetry indicator χ^(3) for the effective Hamiltonian at representative driving amplitudes. The indicator remains {-2,1} in the regime where corner states are present, confirming preservation of the topological protection. We have included these results in a new figure and accompanying text in the Floquet-driven phase section. revision: yes

  2. Referee: [Floquet effective Hamiltonian section] Section describing the effective Hamiltonian under CPL: it is not shown whether the time-periodic drive introduces additional terms that reduce the little-group symmetry at the high-symmetry points relevant to χ^(3), which would undermine the claim that the equilibrium topological protection automatically carries over.

    Authors: The circularly polarized light drive is chosen to respect the C₃ rotational symmetry of the lattice, and the effective Hamiltonian is derived via the Magnus expansion up to second order, which preserves the relevant symmetries. However, to directly address this concern, we have added an analysis of the little-group representations at the high-symmetry points (Γ, K, K') for the Floquet Hamiltonian, showing that the C₃ eigenvalues are unchanged. This is now detailed in the revised section on the effective Hamiltonian. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via symmetry analysis and Floquet application.

full rationale

The paper applies equilibrium symmetry indicator χ^(3) = {-2,1} under C3 protection to the driven system, then reports that C3-protected corner states remain intact after Floquet renormalization by circularly polarized light. No quoted step reduces a prediction to a fitted input by construction, invokes a self-citation as the sole load-bearing justification, or renames a known result. The central claim rests on explicit symmetry analysis of the effective Hamiltonian and numerical preservation of corner states across driving strengths, which are independent of the target result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract provides limited detail; main assumptions concern symmetry protection and applicability of Floquet theory. No free parameters or invented entities are explicitly introduced.

axioms (2)
  • domain assumption Cr₂CH₂ monolayer is a 2D AFM HOTI protected by C₃ rotational symmetry characterized by symmetry indicator χ^(3) = {-2,1}
    Stated directly in the abstract as the equilibrium characterization.
  • domain assumption Floquet driving with CPL induces an f-wave altermagnetic state governed by symmetry [C₂||3̄₀₀₁] with odd-parity spin splitting
    Claimed as the driven state in the abstract.

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Works this paper leans on

62 extracted references · 62 canonical work pages

  1. [1]

    Jungwirth, X

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

    work page 2016

  2. [2]

    Jungwirth, J

    T. Jungwirth, J. Sinova, A. Manchon, X. Marti, J. Wunderl ich, and C. Felser, Nat. Phys. 14, 200 (2018)

    work page 2018

  3. [3]

    Baltz, A

    V . Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and 5 Y . Tserkovnyak, Rev. Mod. Phys.90, 015005 (2018)

    work page 2018

  4. [4]

    Olejn´ ık, T

    K. Olejn´ ık, T. Seifert, Z. Kaˇ spar, V . Nov´ ak, P . Wadley, R. P . Campion, M. Baumgartner, P . Gambardella, P . Nˇ emec, J. Wun- derlich, J. Sinova, P . Kuˇ zel, M. M¨ uller, T. Kampfrath, and T. Jungwirth, Sci. Adv. 4, eaar3566 (2018)

    work page 2018

  5. [5]

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

    work page 2016

  6. [6]

    Li, S.-Y

    H. Li, S.-Y . Gao, S.-F. Duan, Y .-F. Xu, K.-J. Zhu, S.-J. Ti an, J.-C. Gao, W.-H. Fan, Z.-C. Rao, J.-R. Huang, et al., Phys. Rev. X 9, 041039 (2019)

    work page 2019

  7. [7]

    D.-F. Shao, G. Gurung, S.-H. Zhang, and E. Y . Tsymbal, Phy s. Rev. Lett. 122, 077203 (2019)

    work page 2019

  8. [8]

    C. Niu, H. Wang, N. Mao, B. Huang, Y . Mokrousov, and Y . Dai, Phys. Rev. Lett. 124, 066401 (2020)

    work page 2020

  9. [9]

    Honma, D

    A. Honma, D. Takane, S. Souma, K. Yamauchi, Y . Wang, K. Nakayama, K. Sugawara, M. Kitamura, K. Horiba, H. Ku- migashira, K. Tanaka, T. K. Kim, C. Cacho, T. Oguchi, T. Taka- hashi, Y . Ando, and T. Sato, Nat. Commun. 14, 7396 (2023)

    work page 2023

  10. [10]

    Isshiki, N

    H. Isshiki, N. Budai, A. Kobayashi, R. Uesugi, T. Higo, S. Nakatsuji, and Y . Otani, Phys. Rev. Lett. 132, 216702 (2024)

    work page 2024

  11. [11]

    Nakatsuji, N

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

    work page 2015

  12. [12]

    ˇSmejkal, Y

    L. ˇSmejkal, Y . Mokrousov, B. Yan, and A. H. MacDonald, Nat. Phys. 14, 242 (2018)

    work page 2018

  13. [13]

    H. Jani, J. Harrison, S. Hooda, S. Prakash, P . Nandi, J. H u, Z. Zeng, J.-C. Lin, C. Godfrey, G. j. Omar, T. A. Butcher, J. Raabe, S. Finizio, A. V .-Y . Thean, A. Ariando, and P . G. Radaelli, Nat. Mater. 23, 619 (2024)

    work page 2024

  14. [14]

    Mazin and P

    I. Mazin and P . R. X. E. The, Phys. Rev. X 12, 040002 (2022)

    work page 2022

  15. [15]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12, 040501 (2022)

    work page 2022

  16. [16]

    L. Bai, W. Feng, S. Liu, L. ˇSmejkal, Y . Mokrousov, and Y . Yao, Adv. Funct. Mater. 34, 2409327 (2024)

    work page 2024

  17. [17]

    C. Song, H. Bai, Z. Zhou, L. Han, H. Reichlova, J. H. Dil, J . Liu, X. Chen, and F. Pan, Nat. Rev. Mater. 10, 473 (2025)

    work page 2025

  18. [18]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12, 031042 (2022)

    work page 2022

  19. [19]

    Krempask´ y, L

    J. Krempask´ y, L. ˇSmejkal, S. W. D’Souza, M. Hajlaoui, G. Springholz, K. Uhl´ ıˇ rov´ a, F. Alarab, P . C. Constantino u, V . Strocov, D. Usanov, W. R. Pudelko, R. Gonz´ alez-Hern´ andez, A. Birk Hellenes, Z. Jansa, H. Reichlov´ a, Z.ˇSob´ aˇ n, R. D. Gon- zalez Betancourt, P . Wadley, J. Sinova, D. Kriegner, J. Min´ ar, J. H. Dil, and T. Jungwirth, Nat...

    work page 2024

  20. [20]

    Fedchenko, J

    O. Fedchenko, J. Min´ ar, A. Akashdeep, S. W. D’Souza, D.V asi- lyev, O. Tkach, L. Odenbreit, Q. Nguyen, D. Kutnyakhov, N. Wind, L. Wenthaus, M. Scholz, K. Rossnagel, M. Hoesch, M. Aeschlimann, B. Stadtm¨ uller, M. Kl¨ aui, G. Sch¨ onhense, T. Jungwirth, A. B. Hellenes, G. Jakob, L. ˇSmejkal, J. Sinova, and H.-J. Elmers, Sci. Adv. 10, eadj4883 (2024)

    work page 2024

  21. [21]

    Huang, Z

    S. Huang, Z. Qin, F. Zhan, D.-H. Xu, D.-S. Ma, and R. Wang, Phys. Rev. Lett. 136, 126703 (2026)

    work page 2026

  22. [22]

    T. Zhu, D. Zhou, H. Wang, S.-H. Wei, and J. Ruan, Phys. Rev . Lett. 136, 126704 (2026)

    work page 2026

  23. [23]

    Tian, C.-H

    Y . Tian, C.-H. Zhao, C.-B. Wang, B. Zhang, X. Kong, and W.-J. Gong, arXiv:2603.11483 (2026)

    work page arXiv 2026

  24. [24]

    ˇSmejkal, A

    L. ˇSmejkal, A. Marmodoro, K.-H. Ahn, R. Gonz´ alez- Hern´ andez, I. Turek, S. Mankovsky, H. Ebert, S. W. D’Souza, O. ˇSipr, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 131, 256703 (2023)

    work page 2023

  25. [25]

    S. Lee, S. Lee, S. Jung, J. Jung, D. Kim, Y . Lee, B. Seok, J.Kim, B. G. Park, L. ˇSmejkal, C.-J. Kang, and C. Kim, Phys. Rev. Lett. 132, 036702 (2024)

    work page 2024

  26. [26]

    Reimers, L

    S. Reimers, L. Odenbreit, L. ˇSmejkal, V . N. Strocov, P . Con- stantinou, A. B. Hellenes, R. Jaeschke Ubiergo, W. H. Campos , V . K. Bharadwaj, A. Chakraborty, T. Denneulin, W. Shi, R. E. Dunin-Borkowski, S. Das, M. Kl¨ aui, J. Sinova, and M. Jour- dan, Nat. Commun. 15, 2116 (2024)

    work page 2024

  27. [27]

    O. J. Amin, A. Dal Din, E. Golias, Y . Niu, A. Zakharov, S. C. Fromage, C. J. B. Fields, S. L. Heywood, R. B. Cousins, F. Maccherozzi, J. Krempask´ y, J. H. Dil, D. Kriegner, B. Ki- raly, R. P . Campion, A. W. Rushforth, K. W. Edmonds, S. S. Dhesi, L. ˇSmejkal, T. Jungwirth, and P . Wadley, Nature 636, 348 (2024)

    work page 2024

  28. [28]

    Y . Zhu, T. Chen, Y . Li, L. Qiao, X. Ma, C. Liu, T. Hu, H. Gao, and W. Ren, Nano Lett. 24, 472 (2024)

    work page 2024

  29. [29]

    Gonz´ alez-Hern´ andez, L.ˇSmejkal, K

    R. Gonz´ alez-Hern´ andez, L.ˇSmejkal, K. V´ yborn´ y, Y . Yahagi, J. Sinova, T. Jungwirth, and J. ˇZelezn´ y, Phys. Rev. Lett.126, 127701 (2021)

    work page 2021

  30. [30]

    Z. Feng, X. Zhou, L. ˇSmejkal, L. Wu, Z. Zhu, H. Guo, R. Gonz´ alez-Hern´ andez, X. Wang, H. Yan, P . Qin, X. Zhang, H. Wu, H. Chen, Z. Meng, L. Liu, Z. Xia, J. Sinova, T. Jung- wirth, and Z. Liu, Nat. Electron. 5, 735 (2022)

    work page 2022

  31. [31]

    Liao, Y .-C

    C.-T. Liao, Y .-C. Wang, Y .-C. Tien, S.-Y . Huang, and D. Q u, Phys. Rev. Lett. 133, 056701 (2024)

    work page 2024

  32. [32]

    Ma and J.-F

    H.-Y . Ma and J.-F. Jia, Phys. Rev. B 110, 064426 (2024)

    work page 2024

  33. [33]

    Y .-X. Li, Y . Liu, and C.-C. Liu, Phys. Rev. B 109, L201109 (2024)

    work page 2024

  34. [34]

    J. Wang, X. Yang, Z. Yang, J. Lu, P . Ho, W. Wang, Y . S. Ang, Z. Cheng, and S. Fang, Adv. Funct. Mater. 36, 2505145 (2026)

    work page 2026

  35. [35]

    G´ omez-Le´ on and G

    A. G´ omez-Le´ on and G. Platero, Phys. Rev. Lett. 110, 200403 (2013)

    work page 2013

  36. [36]

    Bukov, L

    M. Bukov, L. D’Alessio, and A. Polkovnikov, Adv. Phys. 64, 139 (2015)

    work page 2015

  37. [37]

    Oka and S

    T. Oka and S. Kitamura, Annu. Rev. Condens. Matter Phys. 10, 387 (2019)

    work page 2019

  38. [38]

    M. S. Rudner and N. H. Lindner, Nat. Rev. Phys. 2, 229 (2020)

    work page 2020

  39. [39]

    H. Liu, H. Cao, and S. Meng, Prog. Surf. Sci 98, 100705 (2023)

    work page 2023

  40. [40]

    Yan and Z

    Z. Yan and Z. Wang, Phys. Rev. Lett. 117, 087402 (2016)

    work page 2016

  41. [41]

    Zhou and J

    C. Zhou and J. Zhou, Nano Lett. 24, 7311 (2024)

    work page 2024

  42. [42]

    X. Wan, Z. Ning, D.-H. Xu, R. Wang, and B. Zheng, Phys. Rev . B 109, 085148 (2024)

    work page 2024

  43. [43]

    Pena and C

    A. Pena and C. Radu, Phys. Rev. B 109, 075121 (2024)

    work page 2024

  44. [44]

    Y . H. Wang, H. Steinberg, P . Jarillo-Herrero, and N. Ged ik, Science 342, 453 (2013)

    work page 2013

  45. [45]

    J. W. McIver, B. Schulte, F. U. Stein, T. Matsuyama, G. Jo tzu, G. Meier, and A. Cavalleri, Nat. Phys. 16, 38 (2020)

    work page 2020

  46. [46]

    S. Zhou, C. Bao, B. Fan, H. Zhou, Q. Gao, H. Zhong, T. Lin, H. Liu, P . Y u, P . Tang, S. Meng, W. Duan, and S. Zhou, Nature 614, 75 (2023)

    work page 2023

  47. [47]

    Schindler, A

    F. Schindler, A. M. Cook, M. G. V ergniory, Z. Wang, S. S. Parkin, B. A. Bernevig, and T. Neupert, Sci. Adv. 4, eaat0346 (2018)

    work page 2018

  48. [48]

    Y . Ren, Z. Qiao, and Q. Niu, Phys. Rev. Lett. 124, 166804 (2020)

    work page 2020

  49. [49]

    Xie, H.-X

    B. Xie, H.-X. Wang, X. Zhang, P . Zhan, J.-H. Jiang, M. Lu, and Y . Chen, Nat. Rev. Phys.3, 520 (2021)

    work page 2021

  50. [50]

    Xiao and B

    J. Xiao and B. Yan, Nat. Rev. Phys. 3, 283 (2021)

    work page 2021

  51. [51]

    Y . Xu, Z. Song, Z. Wang, H. Weng, and X. Dai, Phys. Rev. Lett. 122, 256402 (2019)

    work page 2019

  52. [52]

    C. Chen, Z. Song, J.-Z. Zhao, Z. Chen, Z.-M. Y u, X.-L. She ng, and S. A. Yang, Phys. Rev. Lett. 125, 056402 (2020)

    work page 2020

  53. [53]

    Wang, X.-P

    X. Wang, X.-P . Li, J. Li, C. Xie, J. Wang, H. Y uan, W. Wang, Z. Cheng, Z.-M. Y u, and G. Zhang, Adv. Funct. Mater. 33, 2304499 (2023)

    work page 2023

  54. [54]

    R. Li, N. Mao, X. Wu, B. Huang, Y . Dai, and C. Niu, Nano Lett. 23, 91 (2023). 6

    work page 2023

  55. [55]

    Kresse and J

    G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993)

    work page 1993

  56. [56]

    Kresse and J

    G. Kresse and J. Furthm¨ uller, Phys. Rev. B 54, 11169 (1996)

    work page 1996

  57. [57]

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

    work page 1996

  58. [58]

    Pizzi, V

    G. Pizzi, V . Vitale, R. Arita, S. Bl¨ ugel, F. Freimuth, G. G´ eranton, M. Gibertini, D. Gresch, C. Johnson, T. Koret- sune, J. Iba˜ nez-Azpiroz, H. Lee, J.-M. Lihm, D. Marchand, A. Marrazzo, Y . Mokrousov, J. I. Mustafa, Y . Nohara, Y . No- mura, L. Paulatto, S. Ponc´ e, T. Ponweiser, J. Qiao, F. Th¨ ol e, S. S. Tsirkin, M. Wierzbowska, N. Marzari, D. V...

    work page 2020

  59. [59]

    M. Wang, S. Li, Y . Wang, X. Dong, F. Li, W. Ji, and C. Zhang, Phys. Rev. B 112, 174414 (2025)

    work page 2025

  60. [60]

    W. A. Benalcazar, T. Li, and T. L. Hughes, Phys. Rev. B 99, 245151 (2019)

    work page 2019

  61. [61]

    Schindler, M

    F. Schindler, M. Brzezi´ nska, W. A. Benalcazar, M. Irao la, A. Bouhon, S. S. Tsirkin, M. G. V ergniory, and T. Neupert, Phys. Rev. Res. 1, 033074 (2019)

    work page 2019

  62. [62]

    T. Li, P . Zhu, W. A. Benalcazar, and T. L. Hughes, Phys. Re v. B 101, 115115 (2020)

    work page 2020