- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Fractional corner charges in threefold-symmetric two-dimensional materials with fragile topology

Olga Arroyo-Gascón1,*, Sergio Bravo2, Leonor Chico3, and Mónica Pacheco2

  • *Contact author: o.arroyo.gascon@usal.es
PDF

Phys. Rev. Research 7, 023282 – Published 20 June, 2025

DOI: https://doi.org/10.1103/qcr8-jxvn

Abstract

We perform a systematic study of the signatures of fragile topology in over 50 nonmagnetic two-dimensional materials with formula AB2, belonging to space group P3¯m1. Using group theory analysis in the framework of topological quantum chemistry, we find fragile bands near the Fermi level for all the materials studied. Since stable topological bands are also present in these systems, the interplay of both phases is discussed, showing that corner charges appear in over 80% of the materials and are linked to fragile topology. Using first-principles calculations, we predict fractionally charged corner charges protected by C3 symmetry. Our work aims to broaden the scope of materials with experimentally accessible fragile bands.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (92)

  1. M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  2. X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  3. T. Wehling, A. Black-Schaffer, and A. Balatsky, Dirac materials, Adv. Phys. 63, 1 (2014).
  4. S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia et al., Discovery of a weyl fermion semimetal and topological fermi arcs, Science 349, 613 (2015).
  5. N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018).
  6. L. Fu and C. L. Kane, Topological insulators with inversion symmetry, Phys. Rev. B 76, 045302 (2007).
  7. H. C. Po, A. Vishwanath, and H. Watanabe, Symmetry-based indicators of band topology in the 230 space groups, Nat. Commun. 8, 50 (2017).
  8. J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and R.-J. Slager, Topological classification of crystalline insulators through band structure combinatorics, Phys. Rev. X 7, 041069 (2017).
  9. B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature (London) 547, 298 (2017).
  10. L. Elcoro, Z. Song, and B. A. Bernevig, Application of induction procedure and Smith decomposition in calculation and topological classification of electronic band structures in the 230 space groups, Phys. Rev. B 102, 035110 (2020).
  11. H. C. Po, Symmetry indicators of band topology, J. Phys.: Condens. Matter 32, 263001 (2020).
  12. J. Cano and B. Bradlyn, Band representations and topological quantum chemistry, Annu. Rev. Condens. Matter Phys. 12, 225 (2021).
  13. M. G. Vergniory, B. J. Wieder, L. Elcoro, S. S. P. Parkin, C. Felser, B. A. Bernevig, and N. Regnault, All topological bands of all nonmagnetic stoichiometric materials, Science 376, eabg9094 (2022).
  14. J. C. Y. Teo, L. Fu, and C. L. Kane, Surface states and topological invariants in three-dimensional topological insulators: Application to Bi1−xSbx, Phys. Rev. B 78, 045426 (2008).
  15. F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, Higher-order topological insulators, Sci. Adv. 4, eaat0346 (2018).
  16. H. C. Po, H. Watanabe, and A. Vishwanath, Fragile topology and Wannier obstructions, Phys. Rev. Lett. 121, 126402 (2018).
  17. L. Zou, H. C. Po, A. Vishwanath, and T. Senthil, Band structure of twisted bilayer graphene: Emergent symmetries, commensurate approximants, and Wannier obstructions, Phys. Rev. B 98, 085435 (2018).
  18. Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, All Magic Angles in Twisted Bilayer Graphene are Topological, Phys. Rev. Lett. 123, 036401 (2019).
  19. H. C. Po, L. Zou, T. Senthil, and A. Vishwanath, Faithful tight-binding models and fragile topology of magic-angle bilayer graphene, Phys. Rev. B 99, 195455 (2019).
  20. J. Ahn, S. Park, and B.-J. Yang, Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: Application to twisted bilayer graphene at magic angle, Phys. Rev. X 9, 021013 (2019).
  21. B. J. Wieder, B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, L. Elcoro, A. A. Soluyanov, C. Felser, T. Neupert, N. Regnault, and B. A. Bernevig, Topological materials discovery from crystal symmetry, Nat. Rev. Mater. 7, 196 (2022).
  22. W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators, Phys. Rev. B 96, 245115 (2017).
  23. J. Cano, B. Bradlyn, Z. Wang, L. Elcoro, M. G. Vergniory, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topology of disconnected elementary band representations, Phys. Rev. Lett. 120, 266401 (2018).
  24. B. Bradlyn, Z. Wang, J. Cano, and B. A. Bernevig, Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices: An example on the triangular lattice, Phys. Rev. B 99, 045140 (2019).
  25. A. Bouhon, A. M. Black-Schaffer, and R.-J. Slager, Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry, Phys. Rev. B 100, 195135 (2019).
  26. Z.-D. Song, L. Elcoro, and B. A. Bernevig, Twisted bulk-boundary correspondence of fragile topology, Science 367, 794 (2020).
  27. A. Bouhon, T. Bzdušek, and R.-J. Slager, Geometric approach to fragile topology beyond symmetry indicators, Phys. Rev. B 102, 115135 (2020).
  28. A. Altland and M. R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55, 1142 (1997).
  29. A. Kitaev, V. Lebedev, and M. Feigel'man, Periodic table for topological insulators and superconductors, AIP Conf. Proc. 1134, 22 (2009).
  30. Z. Wang, B. J. Wieder, J. Li, B. Yan, and B. A. Bernevig, Higher-order topology, monopole nodal lines, and the origin of large Fermi arcs in transition metal dichalcogenides XTe2 (X = Mo, W), Phys. Rev. Lett. 123, 186401 (2019).
  31. Z.-D. Song, L. Elcoro, Y.-F. Xu, N. Regnault, and B. A. Bernevig, Fragile phases as affine monoids: Classification and material examples, Phys. Rev. X 10, 031001 (2020).
  32. U. Petralanda, Y. Jiang, B. A. Bernevig, N. Regnault, and L. Elcoro, Two-dimensional topological quantum chemistry and catalog of topological materials, arXiv:2411.08950.
  33. M. B. de Paz, M. G. Vergniory, D. Bercioux, A. García-Etxarri, and B. Bradlyn, Engineering fragile topology in photonic crystals: Topological quantum chemistry of light, Phys. Rev. Res. 1, 032005(R) (2019).
  34. J. L. Mañes, Fragile phonon topology on the honeycomb lattice with time-reversal symmetry, Phys. Rev. B 102, 024307 (2020).
  35. Y. Wei, B. Yan, Y. Peng, A. Shi, D. Zhao, R. Peng, Y. Xiang, and J. Liu, Fragile topology in double-site honeycomb lattice photonic crystal, Opt. Lett. 46, 3941 (2021).
  36. Y. Wang, S. Xu, L. Feng, and R. Agarwal, Design of fragile topological flat bands in an optical microcavity array, Phys. Rev. B 110, 045401 (2024).
  37. B. J. Wieder and B. A. Bernevig, The axion insulator as a pump of fragile topology, arXiv:1810.02373.
  38. W. A. Benalcazar, T. Li, and T. L. Hughes, Quantization of fractional corner charge in C n -symmetric higher-order topological crystalline insulators, Phys. Rev. B 99, 245151 (2019).
  39. F. Schindler, M. Brzezińska, W. A. Benalcazar, M. Iraola, A. Bouhon, S. S. Tsirkin, M. G. Vergniory, and T. Neupert, Fractional corner charges in spin-orbit coupled crystals, Phys. Rev. Res. 1, 033074 (2019).
  40. S. Kobayashi and A. Furusaki, Fragile topological insulators protected by rotation symmetry without spin-orbit coupling, Phys. Rev. B 104, 195114 (2021).
  41. J. Zeng, H. Liu, H. Jiang, Q.-F. Sun, and X. C. Xie, Multiorbital model reveals a second-order topological insulator in 1 H transition metal dichalcogenides, Phys. Rev. B 104, L161108 (2021).
  42. S. Kooi, G. Van Miert, and C. Ortix, The bulk-corner correspondence of time-reversal symmetric insulators, npj Quantum Mater. 6, 1 (2021).
  43. A. Skurativska, S. S. Tsirkin, F. D. Natterer, T. Neupert, and M. H. Fischer, Flat bands with fragile topology through superlattice engineering on single-layer graphene, Phys. Rev. Res. 3, L032003 (2021).
  44. A. Luo, Z. Song, and G. Xu, Fragile topological band in the checkerboard antiferromagnetic monolayer FeSe, npj Comput. Mater. 8, 26 (2022).
  45. S. Qian, G.-B. Liu, C.-C. Liu, and Y. Yao, Cn -symmetric higher-order topological crystalline insulators in atomically thin transition metal dichalcogenides, Phys. Rev. B 105, 045417 (2022).
  46. Y.-F. Chen and D.-X. Yao, Fragile topological phase on the triangular kagome lattice and its bulk-boundary correspondence, Phys. Rev. B 107, 155129 (2023).
  47. M. Costa, B. Focassio, L. M. Canonico, T. P. Cysne, G. R. Schleder, R. B. Muniz, A. Fazzio, and T. G. Rappoport, Connecting higher-order topology with the orbital Hall effect in monolayers of transition metal dichalcogenides, Phys. Rev. Lett. 130, 116204 (2023).
  48. G. Liu, H. Jiang, Z. Guo, X. Zhang, L. Jin, C. Liu, and Y. Liu, Magnetic second-order topological insulators in 2H-transition metal dichalcogenides, Adv. Sci. 10, 2301952 (2023).
  49. V. Nuñez, S. Bravo, J. D. Correa, L. Chico, and M. Pacheco, Higher-order obstructed atomic insulator phase in pentagonal monolayer PdSe2, 2D Mater. 11, 015015 (2023).
  50. B. J. Wieder, Z. Wang, J. Cano, X. Dai, L. M. Schoop, B. Bradlyn, and B. A. Bernevig, Strong and fragile topological Dirac semimetals with higher-order Fermi arcs, Nat. Commun. 11, 627 (2020).
  51. V. Peri, Z.-D. Song, M. Serra-Garcia, P. Engeler, R. Queiroz, X. Huang, W. Deng, Z. Liu, B. A. Bernevig, and S. D. Huber, Experimental characterization of fragile topology in an acoustic metamaterial, Science 367, 797 (2020).
  52. F. N. Ünal, A. Bouhon, and R.-J. Slager, Topological Euler class as a dynamical observable in optical lattices, Phys. Rev. Lett. 125, 053601 (2020).
  53. B. Jiang, A. Bouhon, Z.-K. Lin, X. Zhou, B. Hou, F. Li, R.-J. Slager, and J.-H. Jiang, Experimental observation of non-abelian topological acoustic semimetals and their phase transitions, Nat. Phys. 17, 1239 (2021).
  54. Y. Wu, Z.-K. Lin, Y. Yang, Z. Song, F. Li, and J.-H. Jiang, Probing fragile topology with dislocations, Sci. Bull. 69, 3657 (2024).
  55. B. Jiang, A. Bouhon, S.-Q. Wu, Z.-L. Kong, Z.-K. Lin, R.-J. Slager, and J.-H. Jiang, Observation of an acoustic topological euler insulator with meronic waves, Sci. Bull. 69, 1653 (2024).
  56. W. Zhao, Y.-B. Yang, Y. Jiang, Z. Mao, W. Guo, L. Qiu, G. Wang, L. Yao, L. He, Z. Zhou, Y. Xu, and L. Duan, Quantum simulation for topological Euler insulators, Commun. Phys. 5, 223 (2022).
  57. D. Rhodes, R. Schönemann, N. Aryal, Q. Zhou, Q. R. Zhang, E. Kampert, Y.-C. Chiu, Y. Lai, Y. Shimura, G. T. McCandless, J. Y. Chan, D. W. Paley, J. Lee, A. D. Finke, J. P. C. Ruff, S. Das, E. Manousakis, and L. Balicas, Bulk Fermi surface of the Weyl type-II semimetallic candidate γ−MoTe2, Phys. Rev. B 96, 165134 (2017).
  58. D. Di Sante, P. K. Das, C. Bigi, Z. Ergönenc, N. Gürtler, J. A. Krieger, T. Schmitt, M. N. Ali, G. Rossi, R. Thomale, C. Franchini, S. Picozzi, J. Fujii, V. N. Strocov, G. Sangiovanni, I. Vobornik, R. J. Cava, and G. Panaccione, Three-dimensional electronic structure of the Type-II Weyl semimetal WTe2, Phys. Rev. Lett. 119, 026403 (2017).
  59. Y. Wu, N. H. Jo, D. Mou, L. Huang, S. L. Bud'ko, P. C. Canfield, and A. Kaminski, Three-dimensionality of the bulk electronic structure in WTe2, Phys. Rev. B 95, 195138 (2017).
  60. F. Schindler, Z. Wang, M. G. Vergniory, A. M. Cook, A. Murani, S. Sengupta, A. Y. Kasumov, R. Deblock, S. Jeon, I. Drozdov, H. Bouchiat, S. Guéron, A. Yazdani, B. A. Bernevig, and T. Neupert, Higher-order topology in bismuth, Nat. Phys. 14, 918 (2018).
  61. Y. Han, C. Cui, X.-P. Li, T.-T. Zhang, Z. Zhang, Z.-M. Yu, and Y. Yao, Cornertronics in two-dimensional second-order topological insulators, Phys. Rev. Lett. 133, 176602 (2024).
  62. G. Long, M. Pan, H. Zeng, and H. Huang, Second-order topological insulators in two-dimensional monolayers of the 1T-phase PtSe2 material class, Phys. Rev. Mater. 8, 044203 (2024).
  63. P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos et al., QUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materials, J. Phys.: Condens. Matter 21, 395502 (2009).
  64. P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. D. Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio, Jr., A. Ferretti, A. Floris, G. Fratesi, G. Fugallo et al., Advanced capabilities for materials modelling with Quantum ESPRESSO, J. Phys.: Condens. Matter 29, 465901 (2017).
  65. S. Haastrup, M. Strange, M. Pandey, T. Deilmann, P. S. Schmidt, N. F. Hinsche, M. N. Gjerding, D. Torelli, P. M. Larsen, A. C. Riis-Jensen, J. Gath, K. W. Jacobsen, J. J. Mortensen, T. Olsen, and K. S. Thygesen, The Computational 2D materials database: High-throughput modeling and discovery of atomically thin crystals, 2D Mater. 5, 042002 (2018).
  66. M. N. Gjerding, A. Taghizadeh, A. Rasmussen, S. Ali, F. Bertoldo, T. Deilmann, N. R. Knøsgaard, M. Kruse, A. H. Larsen, S. Manti, T. G. Pedersen, U. Petralanda, T. Skovhus, M. K. Svendsen, J. J. Mortensen, T. Olsen, and K. S. Thygesen, Recent progress of the computational 2D materials database (C2DB), 2D Mater. 8, 044002 (2021).
  67. M. G. Vergniory, L. Elcoro, C. Felser, N. Regnault, B. A. Bernevig, and Z. Wang, A complete catalogue of high-quality topological materials, Nature (London) 566, 480 (2019).
  68. M. Iraola, J. L. Mañes, B. Bradlyn, M. K. Horton, T. Neupert, M. G. Vergniory, and S. S. Tsirkin, IrRep: Symmetry eigenvalues and irreducible representations of ab initio band structures, Comput. Phys. Commun. 272, 108226 (2022).
  69. J. M. Soler, E. Artacho, J. D. Gale, A. García, J. Junquera, P. Ordejón, and D. Sánchez-Portal, The SIESTA method for ab initio order-N materials simulation, J. Phys.: Condens. Matter 14, 2745 (2002).
  70. A. García, N. Papior, A. Akhtar, E. Artacho, V. Blum, E. Bosoni, P. Brandimarte, M. Brandbyge, J. I. Cerdá, F. Corsetti, R. Cuadrado, V. Dikan, J. Ferrer, J. Gale, P. García-Fernández, V. M. García-Suárez, S. García, G. Huhs, S. Illera, R. Korytár et al., siesta: Recent developments and applications, J. Chem. Phys. 152, 204108 (2020).
  71. M. I. Aroyo, A. Kirov, C. Capillas, J. M. Perez-Mato, and H. Wondratschek, Bilbao crystallographic server. II. Representations of crystallographic point groups and space groups, Acta Cryst. A 62, 115 (2006).
  72. M. I. Aroyo, J. M. Perez-Mato, C. Capillas, E. Kroumova, S. Ivantchev, G. Madariaga, A. Kirov, and H. Wondratschek, Bilbao Crystallographic Server: I. Databases and crystallographic computing programs, Z. Kristallogr. - Cryst. Mater. 221, 15 (2006).
  73. M. Aroyo, J. Perez-Mato, D. Orobengoa, E. Tasci, G. De La Flor, and A. Kirov, Crystallography online: Bilbao Crystallographic Server, Bulg. Chem. Commun. 43, 183 (2011).
  74. M. Zhang, Y. Zhu, X. Wang, Q. Feng, S. Qiao, W. Wen, Y. Chen, M. Cui, J. Zhang, C. Cai, and L. Xie, Controlled synthesis of ZrS2 monolayer and few layers on hexagonal boron nitride, J. Am. Chem. Soc. 137, 7051 (2015).
  75. S. Mañas-Valero, V. García-López, A. Cantarero, and M. Galbiati, Raman spectra of ZrS2 and ZrSe2 from bulk to atomically thin layers, Appl. Sci. 6, 264 (2016).
  76. G. Ye, Y. Gong, S. Lei, Y. He, B. Li, X. Zhang, Z. Jin, L. Dong, J. Lou, R. Vajtai, W. Zhou, and P. M. Ajayan, Synthesis of large-scale atomic-layer SnS2 through chemical vapor deposition, Nano Res. 10, 2386 (2017).
  77. Y. Wang, L. Li, W. Yao, S. Song, J. T. Sun, J. Pan, X. Ren, C. Li, E. Okunishi, Y.-Q. Wang, E. Wang, Y. Shao, Y. Y. Zhang, H.-t. Yang, E. F. Schwier, H. Iwasawa, K. Shimada, M. Taniguchi, Z. Cheng, S. Zhou et al., Monolayer PtSe2, a new semiconducting transition-metal-dichalcogenide, epitaxially grown by direct selenization of Pt, Nano Lett. 15, 4013 (2015).
  78. S. Sinha, T. Zhu, A. France-Lanord, Y. Sheng, J. C. Grossman, K. Porfyrakis, and J. H. Warner, Atomic structure and defect dynamics of monolayer lead iodide nanodisks with epitaxial alignment on graphene, Nat. Commun. 11, 823 (2020).
  79. E. Chen, W. Xu, J. Chen, and J. H. Warner, 2D layered noble metal dichalcogenides (Pt, Pd, Se, S) for electronics and energy applications, Mater. Today Adv. 7, 100076 (2020).
  80. Q.-Q. Yuan, F. Zheng, Z.-Q. Shi, Q.-Y. Li, Y.-Y. Lv, Y. Chen, P. Zhang, and S.-C. Li, Direct growth of van der Waals tin diiodide monolayers, Adv. Sci. 8, 2100009 (2021).
  81. L. Liu, D. Zemlyanov, and Y. P. Chen, Epitaxial growth of monolayer PdTe2 and patterned PtTe2 by direct tellurization of Pd and Pt surfaces, 2D Mater. 8, 045033 (2021).
  82. L. Elcoro, B. Bradlyn, Z. Wang, M. G. Vergniory, J. Cano, C. Felser, B. A. Bernevig, D. Orobengoa, G. de la Flor, and M. I. Aroyo, Double crystallographic groups and their representations on the bilbao crystallographic server, J. Appl. Crystallogr. 50, 1457 (2017).
  83. See Supplemental Material at http://link.aps.org/supplemental/10.1103/qcr8-jxvn for topological quantum chemistry analysis of SG 164 monolayers, as well as additional figures for topological semimetals ZrTe2 and NiTe2 and other flake figures. The irreducible representations and symmetry operations at high symmetry points of all the valence bands of the 52 materials studied using Quantum ESPRESSO and the IrRep code are available and can be downloaded as a zip file.
  84. P. Tsipas, D. Tsoutsou, S. Fragkos, R. Sant, C. Alvarez, H. Okuno, G. Renaud, R. Alcotte, T. Baron, and A. Dimoulas, Massless Dirac fermions in ZrTe2 semimetal grown on InAs(111) by van der Waals epitaxy, ACS Nano 12, 1696 (2018).
  85. J. Henke, M. Kurttutan, J. Kruthoff, and J. van Wezel, Topological invariants of rotationally symmetric crystals, Phys. Rev. B 104, L201110 (2021).
  86. E. Lee, R. Kim, J. Ahn, and B.-J. Yang, Two-dimensional higher-order topology in monolayer graphdiyne, npj Quantum Mater. 5, 1 (2020).
  87. E. Khalaf, W. A. Benalcazar, T. L. Hughes, and R. Queiroz, Boundary-obstructed topological phases, Phys. Rev. Res. 3, 013239 (2021).
  88. M. Hitomi, T. Kawakami, and M. Koshino, Multiorbital edge and corner states in black phosphorene, Phys. Rev. B 104, 125302 (2021).
  89. J. Sødequist, U. Petralanda, and T. Olsen, Abundance of second order topology in C3 symmetric two-dimensional insulators, 2D Mater. 10, 015009 (2023).
  90. H. Sheng, Y. Xie, Q. Wu, H. Weng, X. Dai, B. A. Bernevig, Z. Fang, and Z. Wang, Majorana corner modes in unconventional monolayers of the 1T-PtSe2 family, Phys. Rev. B 110, 035151 (2024).
  91. O. Arroyo-Gascón, S. Bravo, M. Pacheco, and L. Chico, Two-dimensional orbital-obstructed insulators with higher-order band topology, arXiv:2412.09561v1.
  92. B. Liu, G. Zhao, Z. Liu, and Z. F. Wang, Two-dimensional quadrupole topological insulator in γ-graphyne, Nano Lett. 19, 6492 (2019).

Outline

Information

Phys. Rev. Research 7, 023282– Published 20 June, 2025
  • Received 11 March 2024
  • Accepted 7 May 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation