- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

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

Classification of mass terms in kagome semimetals

Simone Ciceri1,2,3,*, Matteo Massaro4,1, and Lars Fritz1,†

  • *Contact author: simone.ciceri@tu-berlin.de
  • †Contact author: l.fritz@uu.nl
PDF

Phys. Rev. B 112, 035155 – Published 21 July, 2025

DOI: https://doi.org/10.1103/2gzg-vnf9

Abstract

In the last years, kagome materials have received massive attention by virtue of being candidate hosts for a large variety of quantum phases: spin liquids, unconventional superconductivity, and topological phases of matter, to name the more exotic. One of the most interesting features is tunability: changing the filling, the noninteracting band structure can be tuned from flat bands to conventional metallic phases as well as to semimetals. In this paper, we concentrate on the latter. At specific lattice filling, the electronic bands have a semimetallic structure, hosting Dirac, massless quasiparticles, such as in graphene or other layered two dimensional materials. Specifically, we determine what terms can be added to the nearest-neighbor hopping that opens at the gap at the said Dirac point. These terms can, in principle, arise through external perturbations, interactions, or collective instabilities. We classify the 16 possible gap-opening terms according to the broken symmetries. Furthermore, we identify concrete microscopic realizations, allowing for an interpretation of these phases.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (42)

  1. M. Kang, S. Fang, L. Ye, H. C. Po, J. Denlinger, C. Jozwiak, A. Bostwick, E. Rotenberg, E. Kaxiras, J. G. Checkelsky et al., Topological flat bands in frustrated kagome lattice CoSn, Nat. Commun. 11, 4004 (2020).
  2. M. Norman, Colloquium: Herbertsmithite and the search for the quantum spin liquid, Rev. Mod. Phys. 88, 041002 (2016).
  3. L. Savary and L. Balents, Quantum spin liquids: A review, Rep. Prog. Phys. 80, 016502 (2017).
  4. C. Mielke III, Y. Qin, J.-X. Yin, H. Nakamura, D. Das, K. Guo, R. Khasanov, J. Chang, Z. Wang, S. Jia et al., Nodeless kagome superconductivity in LaRu3Si2, Phys. Rev. Mater. 5, 034803 (2021).
  5. H.-M. Guo and M. Franz, Topological insulator on the kagome lattice, Phys. Rev. B 80, 113102 (2009).
  6. F. Pollmann, K. Roychowdhury, C. Hotta, and K. Penc, Interplay of charge and spin fluctuations of strongly interacting electrons on the kagome lattice, Phys. Rev. B 90, 035118 (2014).
  7. X. Feng, Y. Zhang, K. Jiang, and J. Hu, Low-energy effective theory and symmetry classification of flux phases on the kagome lattice, Phys. Rev. B 104, 165136 (2021).
  8. S. M. Young and C. L. Kane, Dirac semimetals in two dimensions, Phys. Rev. Lett. 115, 126803 (2015).
  9. S. Rao, Weyl semi-metals: A short review, arXiv:1603.02821.
  10. N. Armitage, E. Mele, and A. Vishwanath, Weyl and dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018).
  11. I. Mazin, H. O. Jeschke, F. Lechermann, H. Lee, M. Fink, R. Thomale, and R. Valentí, Theoretical prediction of a strongly correlated Dirac metal, Nat. Commun. 5, 4261 (2014).
  12. C. Chamon, C.-Y. Hou, C. Mudry, S. Ryu, and L. Santos, Masses and Majorana fermions in graphene, Phys. Scr. T146, 014013 (2012).
  13. G. W. Semenoff, Condensed-matter simulation of a three-dimensional anomaly, Phys. Rev. Lett. 53, 2449 (1984).
  14. C. Chamon, Solitons in carbon nanotubes, Phys. Rev. B 62, 2806 (2000).
  15. F. D. M. Haldane, Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”, Phys. Rev. Lett. 61, 2015 (1988).
  16. G. W. Semenoff, Chiral symmetry breaking in graphene, Phys. Scr. T146, 014016 (2012).
  17. C. L. Kane and E. J. Mele, Quantum spin Hall effect in graphene, Phys. Rev. Lett. 95, 226801 (2005).
  18. J.-S. Park and H. J. Choi, Band-gap opening in graphene: A reverse-engineering approach, Phys. Rev. B 92, 045402 (2015).
  19. R. Ciola, K. Pongsangangan, R. Thomale, and L. Fritz, Chiral symmetry breaking through spontaneous dimerization in kagomé metals, Phys. Rev. B 104, 245138 (2021).
  20. R. J. Flores, Classifying spaces for wallpaper groups, in Geometric Group Theory (Springer, Basel, 2007), pp. 51–64.
  21. S. Ciceri, Mass generation in graphene and kagome metals, Master's thesis, Utrecht University, 2023.
  22. L. Ye, M. Kang, J. Liu, F. von Cube, C. R. Wicker, T. Suzuki et al., Massive Dirac fermions in a ferromagnetic kagome metal, Nat. Commun. 555, 638 (2018).
  23. Y. Okamura, K. Shoriki, Y. Nomura, Y. Fujishiro, A. Kitaori, N. Kanazawa, R. Arita, Y. Tokura, and Y. Takahashi, Observation of anomalous Hall resonance of massive Dirac fermions in topological kagome-lattice magnet, npj Quantum Mater. 8, 57 (2023).
  24. Z. Liu, M. Li, Q. Wang, G. Wang, C. Wen, K. Jiang, X. Lu, S. Yan, Y. Huang, D. Shen et al., Orbital-selective Dirac fermions and extremely flat bands in frustrated kagome-lattice metal cosn, Nat. Commun. 11, 4002 (2020).
  25. D. F. Liu, E. K. Liu, Q. N. Xu, J. L. Shen, Y. W. Li, D. Pei, A. J. Liang, P. Dudin, T. K. Kim, C. Cacho, Y. F. Xu, Y. Sun, L. X. Yang, Z. K. Liu, C. Felser, S. S. P. Parkin, and Y. L. Chen, Direct observation of the spin–orbit coupling effect in magnetic Weyl semimetal Co3Sn2S2, npj Quantum Mater. 7, 11 (2022).
  26. K. Nakayama, Y. Li, T. Kato, Multiple energy scales and anisotropic energy gap in the charge-density-wave phase of the kagome superconductor CsV3Sb5, Phys. Rev. B 104, L161112 (2021).
  27. S. Regmi, T. Fernando, Y. Zhao, A. P. Sakhya, G. Dhakal, I. Bin Elius, H. Vazquez, J. D. Denlinger, J. Yang, J.-H. Chu et al., Spectroscopic evidence of flat bands in breathing kagome semiconductor Nb3I8, Commun. Mater. 3, 100 (2022).
  28. Y. Xie, K. Ji, J. He, X. Shen, D. Wang, and J. Zhang, Manipulation of topology by electric field in breathing kagome lattice, Phys. Rev. Lett. (2025), doi:10.1103/29ht-pwyt.
  29. C. M. Pasco, I. El Baggari, E. Bianco, L. F. Kourkoutis, and T. M. McQueen, Tunable magnetic transition to a singlet ground state in a 2D van der Waals layered trimerized kagomé magnet, ACS Nano 13, 9457 (2019).
  30. H. Zhang, Z. Shi, Z. Jiang, M. Yang, J. Zhang, Z. Meng, T. Hu, F. Liu, L. Cheng, Y. Xie et al., Topological flat bands in 2D breathing-kagome lattice Nb3TeCl7, Adv. Mater. 35, 2301790 (2023).
  31. G. Liu, S.-L. Zhu, S. Jiang, F. Sun, and W. M. Liu, Simulating and detecting the quantum spin Hall effect in the kagome optical lattice, Phys. Rev. A 82, 053605 (2010).
  32. N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous Hall effect, Rev. Mod. Phys. 82, 1539 (2010).
  33. Y. Taguchi, Y. Oohara, H. Yoshizawa, N. Nagaosa, and Y. Tokura, Spin chirality, Berry phase, and anomalous Hall effect in a frustrated ferromagnet, Science 291, 2573 (2001).
  34. J. Ye, Y. B. Kim, A. J. Millis, B. I. Shraiman, P. Majumdar, and Z. Tešanović, Berry phase theory of the anomalous Hall effect: Application to colossal magnetoresistance manganites, Phys. Rev. Lett. 83, 3737 (1999).
  35. F. D. M. Haldane, Berry curvature on the Fermi surface: Anomalous Hall effect as a topological Fermi-liquid property, Phys. Rev. Lett. 93, 206602 (2004).
  36. K. K. Kolincio, M. Hirschberger, J. Masell, T. H. Arima, N. Nagaosa, and Y. Tokura, Kagomé lattice promotes chiral spin fluctuations, Phys. Rev. Lett. 130, 136701 (2023).
  37. Y.-X. Jiang, J.-X. Yin, M. M. Denner, N. Shumiya, B. R. Ortiz, G. Xu, Z. Guguchia, J. He, M. S. Hossain, X. Liu, J. Ruff, L. Kautzsch, S. S. Zhang, G. Chang, I. Belopolski, Q. Zhang, T. A. Cochran, D. Multer, M. Litskevich, Z.-J. Cheng et al., Unconventional chiral charge order in kagome superconductor Kv3Sb5, Nat. Mater. 20, 1353 (2021).
  38. P. Mendels and F. Bert, Gapless ground state in the archetypal quantum kagome antiferromagnet ZnCu3(OH)6Cl2, J. Phys. Soc. Jpn. 79, 011001 (2010).
  39. T.-H. Han, J. S. Helton, S. Chu, D. G. Nocera, J. A. Rodriguez-Rivera, C. Broholm, and Y. S. Lee, Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet, Nature (London) 492, 406 (2012).
  40. A. Bolens and N. Nagaosa, Topological states on the breathing kagome lattice, Phys. Rev. B 99, 165141 (2019).
  41. M. Massaro, Strain engineering in graphene and kagomé materials, Master's thesis, Utrecht University, 2023.
  42. A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81, 109 (2009).

Outline

Information

Phys. Rev. B 112, 035155– Published 21 July, 2025
  • Received 19 November 2024
  • Revised 22 June 2025
  • Accepted 8 July 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