Skip to main content
SpringerLink
Log in

Topological hinge modes in Dirac semimetals

  • Research Article
  • Published:
Frontiers of Physics Aims and scope Submit manuscript

Abstract

Dirac semimetals (DSMs) are an important class of topological states of matter. Here, focusing on DSMs of band inversion type, we investigate their boundary modes from the effective model perspective. We show that in order to properly capture the boundary modes, k-cubic terms must be included in the effective model, which would drive an evolution of surface degeneracy manifold from a nodal line to a nodal point. Sizable k-cubic terms are also needed for better exposing the topological hinge modes in the spectrum. Using first-principles calculations, we demonstrate that this feature and the topological hinge modes can be clearly exhibited in β-CuI. We extend the discussion to magnetic DSMs and show that the time-reversal symmetry breaking can gap out the surface bands and hence is beneficial for the experimental detection of hinge modes. Furthermore, we show that magnetic DSMs serve as a parent state for realizing multiple other higher-order topological phases, including higher-order Weyl-point/nodal-line semimetals and higher-order topological insulators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Z. Hasan and C. L. Kane, Topological insulators, Rev. Mod. Phys. 82(4), 3045 (2010)

    Article  ADS  Google Scholar 

  2. X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83(4), 1057 (2011)

    Article  ADS  Google Scholar 

  3. S. Q. Shen, Topological Insulators, Vol. 174, Springer Berlin Heidelberg, Berlin, Heidelberg, 2012

    Book  Google Scholar 

  4. B. A. Bernevig and T. L. Hughes, Topological Insulators and Topological Superconductors, Princeton University Press, 2013

  5. A. Bansil, H. Lin, and T. Das, Topological band theory, Rev. Mod. Phys. 88(2), 021004 (2016)

    Article  ADS  Google Scholar 

  6. C. K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88(3), 035005 (2016)

    Article  ADS  Google Scholar 

  7. S. A. Yang, Dirac and Weyl materials: Fundamental aspects and some spintronics applications, Spin 6(2), 1640003 (2016)

    Article  ADS  Google Scholar 

  8. X. Dai, Weyl fermions go into orbit, Nat. Phys. 12(8), 727 (2016)

    Article  Google Scholar 

  9. A. A. Burkov, Topological semimetals, Nat. Mater. 15(11), 1145 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  10. N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90(1), 015001 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  11. J. Qi, H. Liu, H. Jiang, and X. C. Xie, Dephasing effects in topological insulators, Front. Phys. 14(4), 43403 (2019)

    Article  ADS  Google Scholar 

  12. F. D. M. Haldane, Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”, Phys. Rev. Lett. 61(18), 2015 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  13. X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates, Phys. Rev. B 83(20), 205101 (2011)

    Article  ADS  Google Scholar 

  14. S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, and A. M. Rappe, Dirac semimetal in three dimensions, Phys. Rev. Lett. 108(14), 140405 (2012)

    Article  ADS  Google Scholar 

  15. Z. Wang, Y. Sun, X. Q. Chen, C. Franchini, G. Xu, H. Weng, X. Dai, and Z. Fang, Dirac semimetal and topological phase transitions in A3Bi (A = Na, K, Rb), Phys. Rev. B 85(19), 195320 (2012)

    Article  ADS  Google Scholar 

  16. Z. Wang, H. Weng, Q. Wu, X. Dai, and Z. Fang, Three-dimensional Dirac semimetal and quantum transport in Cd3As2, Phys. Rev. B 88(12), 125427 (2013)

    Article  ADS  Google Scholar 

  17. S. Li, Z. M. Yu, Y. Yao, and S. A. Yang, Type-II topological metals, Front. Phys. 15(4), 43201 (2020)

    Article  ADS  Google Scholar 

  18. J. A. Steinberg, S. M. Young, S. Zaheer, C. L. Kane, E. J. Mele, and A. M. Rappe, Bulk Dirac points in distorted spinels, Phys. Rev. Lett. 112(3), 036403 (2014)

    Article  ADS  Google Scholar 

  19. Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng, D. Prabhakaran, S. K. Mo, Z. X. Shen, Z. Fang, X. Dai, Z. Hussain, and Y. L. Chen, Discovery of a three-dimensional topological Dirac semimetal, Na3Bi, Science 343(6173), 864 (2014)

    Article  ADS  Google Scholar 

  20. Z. K. Liu, J. Jiang, B. Zhou, Z. J. Wang, Y. Zhang, H. M. Weng, D. Prabhakaran, S. K. Mo, H. Peng, P. Dudin, T. Kim, M. Hoesch, Z. Fang, X. Dai, Z. X. Shen, D. L. Feng, Z. Hussain, and Y. L. Chen, A stable three-dimensional topological Dirac semimetal Cd3As2, Nat. Mater. 13(7), 677 (2014)

    Article  ADS  Google Scholar 

  21. M. Neupane, S. Y. Xu, R. Sankar, N. Alidoust, G. Bian, C. Liu, I. Belopolski, T. R. Chang, H. T. Jeng, H. Lin, A. Bansil, F. Chou, and M. Z. Hasan, Observation of a three-dimensional topological Dirac semimetal phase in high-mobility Cd3As2, Nat. Commun. 5(1), 3786 (2014)

    Article  ADS  Google Scholar 

  22. S. Jeon, B. B. Zhou, A. Gyenis, B. E. Feldman, I. Kimchi, A. C. Potter, Q. D. Gibson, R. J. Cava, A. Vishwanath, and A. Yazdani, Landau quantization and quasiparticle interference in the three-dimensional Dirac semimetal Cd3As2, Nat. Mater. 13(9), 851 (2014)

    Article  ADS  Google Scholar 

  23. S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolotnyy, B. Büchner, and R. J. Cava, Experimental realization of a three-dimensional Dirac semimetal, Phys. Rev. Lett. 113(2), 027603 (2014)

    Article  ADS  Google Scholar 

  24. T. Liang, Q. Gibson, M. N. Ali, M. Liu, R. J. Cava, and N. P. Ong, Ultrahigh mobility and giant magnetoresistance in the Dirac semimetal Cd3As2, Nat. Mater. 14(3), 280 (2015)

    Article  ADS  Google Scholar 

  25. S. Y. Xu, C. Liu, S. K. Kushwaha, R. Sankar, J. W. Krizan, I. Belopolski, M. Neupane, G. Bian, N. Alidoust, T. R. Chang, H. T. Jeng, C. Y. Huang, W. F. Tsai, H. Lin, P. P. Shibayev, F. C. Chou, R. J. Cava, and M. Z. Hasan, Observation of Fermi arc surface states in a topological metal, Science 347(6219), 294 (2015)

    Article  ADS  Google Scholar 

  26. J. Xiong, S. K. Kushwaha, T. Liang, J. W. Krizan, M. Hirschberger, W. Wang, R. J. Cava, and N. P. Ong, Evidence for the chiral anomaly in the Dirac semimetal Na3Bi, Science 350(6259), 413 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. M. Kargarian, M. Randeria, and Y. M. Lu, Are the surface Fermi arcs in Dirac semimetals topologically protected, Proc. Natl. Acad. Sci. USA 113(31), 8648 (2016)

    Article  ADS  Google Scholar 

  28. F. Zhang, C. L. Kane, and E. J. Mele, Surface state magnetization and chiral edge states on topological insulators, Phys. Rev. Lett. 110(4), 046404 (2013)

    Article  ADS  Google Scholar 

  29. W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Quantized electric multipole insulators, Science 357(6346), 61 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen, and P. W. Brouwer, Reflection-symmetric second-order topological insulators and superconductors, Phys. Rev. Lett. 119(24), 246401 (2017)

    Article  ADS  Google Scholar 

  31. Z. Song, Z. Fang, and C. Fang, (d−2)-dimensional edge states of rotation symmetry protected topological states, Phys. Rev. Lett. 119(24), 246402 (2017)

    Article  ADS  Google Scholar 

  32. F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, T. Neupert, B. Andrei Bernevig, and T. Neupert, Higher-order topological insulators, Sci. Adv. 4(6), eaat0346 (2018)

    Article  ADS  Google Scholar 

  33. 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(9), 918 (2018)

    Article  Google Scholar 

  34. X. L. Sheng, C. Chen, H. Liu, Z. Chen, Z. M. Yu, Y. X. Zhao, and S. A. Yang, Two-dimensional second-order topological insulator in graphdiyne, Phys. Rev. Lett. 123(25), 256402 (2019)

    Article  ADS  Google Scholar 

  35. H. X. Wang, Z. K. Lin, B. Jiang, G. Y. Guo, and J. H. Jiang, Higher-order Weyl semimetals, Phys. Rev. Lett. 125(14), 146401 (2020)

    Article  ADS  Google Scholar 

  36. S. A. A. Ghorashi, T. Li, and T. L. Hughes, Higher-order Weyl semimetals, Phys. Rev. Lett. 125(26), 266804 (2020)

    Article  ADS  Google Scholar 

  37. H. Qiu, M. Xiao, F. Zhang, and C. Qiu, Higher-order Dirac sonic crystals, Phys. Rev. Lett. 127(14), 146601 (2021)

    Article  ADS  Google Scholar 

  38. C. Chen, X. T. Zeng, Z. Chen, Y. X. Zhao, X. L. Sheng, and S. A. Yang, Second-order real nodal-line semimetal in three-dimensional graphdiyne, Phys. Rev. Lett. 128(2), 026405 (2022)

    Article  ADS  Google Scholar 

  39. H. D. Scammell, J. Ingham, M. Geier, and T. Li, Intrinsic first- and higher-order topological superconductivity in a doped topological insulator, Phys. Rev. B 105(19), 195149 (2022)

    Article  ADS  Google Scholar 

  40. 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(1), 627 (2020)

    Article  ADS  Google Scholar 

  41. Y. Fang and J. Cano, Classification of Dirac points with higher-order Fermi arcs, Phys. Rev. B 104(24), 245101 (2021)

    Article  ADS  Google Scholar 

  42. B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 314(5806), 1757 (2006)

    Article  ADS  Google Scholar 

  43. C. Le, X. Wu, S. Qin, Y. Li, R. Thomale, F. C. Zhang, and J. Hu, Dirac semimetal in β-CuI without surface Fermi arcs, Proc. Natl. Acad. Sci. USA 115(33), 8311 (2018)

    Article  ADS  Google Scholar 

  44. Y. Shan, G. Li, G. Tian, J. Han, C. Wang, S. Liu, H. Du, and Y. Yang, Description of the phase transitions of cuprous iodide, J. Alloys Compd. 477(1–2), 403 (2009)

    Article  Google Scholar 

  45. P. Tang, Q. Zhou, G. Xu, and S. C. Zhang, Dirac fermions in an antiferromagnetic semimetal, Nat. Phys. 12(12), 1100 (2016)

    Article  Google Scholar 

  46. G. Hua, S. Nie, Z. Song, R. Yu, G. Xu, and K. Yao, Dirac semimetal in type-IV magnetic space groups, Phys. Rev. B 98, 201116(R) (2018)

    Article  ADS  Google Scholar 

  47. K. Wang, J. X. Dai, L. B. Shao, S. A. Yang, and Y. X. Zhao, Boundary Criticality of PT-invariant topology and second-order nodal-line semimetals, Phys. Rev. Lett. 125(12), 126403 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  48. S. Nie, J. Chen, C. Yue, C. Le, D. Yuan, W. Zhang, and H. Weng, Tunable Dirac semimetals with higher-order Fermi arcs in Kagome lattices Pd3Pb2X2 (X = S, Se), arXiv: 2203.03162 (2022)

  49. G. Kresse and J. Hafner, Ab initio molecular-dynamics simulation of the liquid-metal-amorphous-semiconductor transition in germanium, Phys. Rev. B 49(20), 14251 (1994)

    Article  ADS  Google Scholar 

  50. G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54(16), 11169 (1996)

    Article  ADS  Google Scholar 

  51. P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50(24), 17953 (1994)

    Article  ADS  Google Scholar 

  52. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77(18), 3865 (1996)

    Article  ADS  Google Scholar 

  53. H. J. Monkhorst and J. D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13(12), 5188 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  54. N. Marzari and D. Vanderbilt, Maximally localized generalized Wannier functions for composite energy bands, Phys. Rev. B 56(20), 12847 (1997)

    Article  ADS  Google Scholar 

  55. I. Souza, N. Marzari, and D. Vanderbilt, Maximally localized Wannier functions for entangled energy bands, Phys. Rev. B 65(3), 035109 (2001)

    Article  ADS  Google Scholar 

  56. M. P. L. Sancho, J. M. L. Sancho, and J. Rubio, Quick iterative scheme for the calculation of transfer matrices: application to Mo(100), J. Phys. F Met. Phys. 14(5), 1205 (1984)

    Article  ADS  Google Scholar 

  57. M. P. L. Sancho, J. M. L. Sancho, J. M. L. Sancho, and J. Rubio, Highly convergent schemes for the calculation of bulk and surface Green functions, J. Phys. F Met. Phys. 15(4), 851 (1985)

    Article  ADS  Google Scholar 

  58. Q. Wu, S. Zhang, H. F. Song, M. Troyer, and A. A. Soluyanov, WannierTools: An open-source software package for novel topological materials, Comput. Phys. Commun. 224, 405 (2018)

    Article  ADS  MATH  Google Scholar 

Download references

Acknowledgements

We thank Zhijun Wang and D. L. Deng for helpful discussions. We acknowledge computational support from HPC of Beihang University. This work is supported by the NSFC (Grants No. 12174018, No. 12074024, No. 11774018), the National Key R&D Program of China (No. 2022YFA1402600), the Fundamental Research Funds for the Central Universities and the Singapore Ministry of Education AcRF Tier 2 (MOE-T2EP50220-0011).

Author information

Authors and Affiliations

  1. School of Physics, Beihang University, Beijing, 100191, China

    Xu-Tao Zeng, Ziyu Chen, Cong Chen, Bin-Bin Liu & Xian-Lei Sheng

  2. Department of Physics, The University of Hong Kong, Hong Kong, China

    Cong Chen

  3. Peng Huanwu Collaborative Center for Research and Education, Beihang University, Beijing, 100191, China

    Xian-Lei Sheng

  4. Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore, 487372, Singapore

    Shengyuan A. Yang

  5. Center for Quantum Transport and Thermal Energy Science, School of Physics and Technology, Nanjing Normal University, Nanjing, 210023, China

    Shengyuan A. Yang

Authors
  1. Xu-Tao Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ziyu Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Cong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Bin-Bin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xian-Lei Sheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Shengyuan A. Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xian-Lei Sheng.

Additional information

Note added

Recently, a preprint appeared [48], which reported higher-order Dirac semimetal state in the materials Pd3Pb2X2 (X = S, Se).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zeng, XT., Chen, Z., Chen, C. et al. Topological hinge modes in Dirac semimetals. Front. Phys. 18, 13308 (2023). https://doi.org/10.1007/s11467-022-1221-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11467-022-1221-y

Keywords

Search

Navigation