Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Quasi-symmetry-protected topology in a semi-metal


The crystal symmetry of a material dictates the type of topological band structure it may host, and therefore, symmetry is the guiding principle to find topological materials. Here we introduce an alternative guiding principle, which we call ‘quasi-symmetry’. This is the situation where a Hamiltonian has exact symmetry at a lower order that is broken by higher-order perturbation terms. This enforces finite but parametrically small gaps at some low-symmetry points in momentum space. Untethered from the restraints of symmetry, quasi-symmetries eliminate the need for fine tuning as they enforce that sources of large Berry curvature occur at arbitrary chemical potentials. We demonstrate that quasi-symmetry in the semi-metal CoSi stabilizes gaps below 2 meV over a large near-degenerate plane that can be measured in the quantum oscillation spectrum. The application of in-plane strain breaks the crystal symmetry and gaps the degenerate point, observable by new magnetic breakdown orbits. The quasi-symmetry, however, does not depend on spatial symmetries and hence transmission remains fully coherent. These results demonstrate a class of topological materials with increased resilience to perturbations such as strain-induced crystalline symmetry breaking, which may lead to robust topological applications as well as unexpected topology beyond the usual space group classifications.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Prices vary by article type



Prices may be subject to local taxes which are calculated during checkout

Fig. 1: General concept of quasi-symmetry.
Fig. 2: Structural and electronic properties of CoSi.
Fig. 3: Temperature and angle-dependent quantum oscillations of CoSi.
Fig. 4: Resilience of quasi-symmetry to lattice distortion.

Data availability

Source data are available for this paper. Other data that support the findings of this study are available at Zenodo:

Code availability

MATLAB code used for this study is available at Zenodo:


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

    Article  ADS  MathSciNet  Google Scholar 

  2. Lv, B. Q., Qian, T. & Ding, H. Experimental perspective on three-dimensional topological semimetals. Rev. Mod. Phys. 93, 025002 (2021).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  4. Wang, Z., Alexandradinata, A., Cava, R. J. & Bernevig, B. A. Hourglass fermions. Nature 532, 189–194 (2016).

    Article  ADS  Google Scholar 

  5. Bradlyn, B. et al. Beyond Dirac and Weyl fermions: unconventional quasiparticles in conventional crystals. Science 353, aaf5037 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  6. Wang, Z. et al. Dirac semimetal and topological phase transitions in A3Bi (A = Na, K, Rb). Phys. Rev. B 85, 195320 (2012).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  8. Bian, G. et al. Drumhead surface states and topological nodal-line fermions in TlTaSe2. Phys. Rev. B 93, 121113 (2016).

    Article  ADS  Google Scholar 

  9. Rao, Z. et al. Observation of unconventional chiral fermions with long Fermi arcs in CoSi. Nature 567, 496–499 (2019).

    Article  ADS  Google Scholar 

  10. Moll, P. J. W. et al. Transport evidence for Fermi-arc-mediated chirality transfer in the Dirac semimetal Cd3As2. Nature 535, 266–270 (2016).

    Article  ADS  Google Scholar 

  11. Zyuzin, A. A. & Burkov, A. A. Topological response in Weyl semimetals and the chiral anomaly. Phys. Rev. B 86, 115133 (2012).

    Article  ADS  Google Scholar 

  12. Burkov, A. A. Chiral anomaly and transport in Weyl metals. J. Condens. Matter Phys. 27, 113201 (2015).

    Article  ADS  Google Scholar 

  13. Xiong, J. et al. Evidence for the chiral anomaly in the Dirac semimetal Na3Bi. Science 350, 413–416 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Matsushita, T., Fujimoto, S. & Schnyder, A. P. Topological piezoelectric effect and parity anomaly in nodal line semimetals. Phys. Rev. Research 2, 043311 (2020).

    Article  ADS  Google Scholar 

  15. Li, Z. et al. Second harmonic generation in the Weyl semimetal TaAs from a quantum kinetic equation. Phys. Rev. B 97, 085201 (2018).

    Article  ADS  Google Scholar 

  16. Takasan, K., Morimoto, T., Orenstein, J. & Moore, J. E. Current-induced second harmonic generation in inversion-symmetric Dirac and Weyl semimetals. Phys. Rev. B 104, L161202 (2021).

    Article  ADS  Google Scholar 

  17. Burkov, A. A. Giant planar Hall effect in topological metals. Phys. Rev. B 96, 041110 (2017).

    Article  ADS  Google Scholar 

  18. Liu, E. et al. Giant anomalous Hall effect in a ferromagnetic kagome-lattice semimetal. Nat. Phys. 14, 1125–1131 (2018).

    Article  Google Scholar 

  19. Nakatsuji, S., Kiyohara, N. & Higo, T. Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature. Nature 527, 212–215 (2015).

    Article  ADS  Google Scholar 

  20. von Neumann, V. J. & Wigner, E. Uber merkwürdige diskrete Eigenwerte. Phys. Z. 30, 291–293 (1929).

    MATH  Google Scholar 

  21. Murakami, S., Iso, S., Avishai, Y., Onoda, M. & Nagaosa, N. Tuning phase transition between quantum spin Hall and ordinary insulating phases. Phys. Rev. B 76, 205304 (2007).

    Article  ADS  Google Scholar 

  22. Wilde, M. A. et al. Symmetry-enforced topological nodal planes at the Fermi surface of a chiral magnet. Nature 594, 374–379 (2021).

    Article  ADS  Google Scholar 

  23. Konschuh, S., Gmitra, M. & Fabian, J. Tight-binding theory of the spin-orbit coupling in graphene. Phys. Rev. B 82, 245412 (2010).

    Article  ADS  Google Scholar 

  24. Sichau, J. et al. Resonance microwave measurements of an intrinsic spin-orbit coupling gap in graphene: a possible indication of a topological state. Phys. Rev. Lett. 122, 046403 (2019).

    Article  ADS  Google Scholar 

  25. Yuan, Q. Q. et al. Quasiparticle interference evidence of the topological Fermi arc states in chiral fermionic semimetal CoSi. Sci. Adv. 5, eaaw9485 (2019).

    Article  ADS  Google Scholar 

  26. Sanchez, D. S. et al. Topological chiral crystals with helicoid-arc quantum states. Nature 567, 500–505 (2019).

    Article  ADS  Google Scholar 

  27. Fang, C., Gilbert, M. J., Dai, X. & Bernevig, B. A. Multi-Weyl topological semimetals stabilized by point group symmetry. Phys. Rev. Lett. 108, 266802 (2012).

    Article  ADS  Google Scholar 

  28. Moll, P. J. W. Focused ion beam microstructuring of quantum matter. Annu. Rev. Condens. Matter Phys. 9, 147–162 (2018).

    Article  ADS  Google Scholar 

  29. Xu, X. et al. Crystal growth and quantum oscillations in the topological chiral semimetal CoSi. Phys. Rev. B 100, 045104 (2019).

    Article  ADS  Google Scholar 

  30. Wu, D. S. et al. Single crystal growth and magnetoresistivity of topological semimetal CoSi. Chinese Phys. Lett. 36, 077102 (2019).

    Article  ADS  Google Scholar 

  31. Wang, H. et al. de Haas–van Alphen quantum oscillations and electronic structure in the large-Chern-number topological chiral semimetal CoSi. Phys. Rev. B 102, 115129 (2020).

    Article  ADS  Google Scholar 

  32. Huber, N. et al. Network of topological nodal planes, multifold degeneracies, and Weyl points in CoSi. Preprint at (2021).

  33. Blount, E. I. Bloch electrons in a magnetic field. Phys. Rev. 126, 1636–1653 (1962).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Helm, T. et al. Correlation between Fermi surface transformations and superconductivity in the electron-doped high-Tc superconductor Nd2−xCexCuO4. Phys. Rev. B 92, 094501 (2015).

    Article  ADS  Google Scholar 

  35. Honold, M. M. et al. Magnetic breakdown in the high-field phase of the organic conductor α-(BEDT-TTF)2KHg(SCN)4. Synth. Met. 103, 2093–2094 (1999).

    Article  Google Scholar 

  36. Pezzini, S. et al. Unconventional mass enhancement around the Dirac nodal loop in ZrSiS. Nat. Phys. 14, 178–183 (2017).

    Article  Google Scholar 

  37. Bachmann, M. D. et al. Spatial control of heavy-fermion superconductivity in CeIrIn5. Science 366, 221–226 (2019).

    Article  ADS  Google Scholar 

  38. Sun, Y., Zhang, Y., Liu, C. X., Felser, C. & Yan, B. Dirac nodal lines and induced spin Hall effect in metallic rutile oxides. Phys. Rev. B 95, 235104 (2017).

    Article  ADS  Google Scholar 

  39. Sun, Y., Zhang, Y., Felser, C. & Yan, B. Strong intrinsic spin Hall effect in the TaAs family of Weyl semimetals. Phys. Rev. Lett. 117, 146403 (2016).

    Article  ADS  Google Scholar 

  40. Sodemann, I. & Fu, L. Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials. Phys. Rev. Lett. 115, 216806 (2015).

    Article  ADS  Google Scholar 

  41. Yu, Z. M. et al. Encyclopedia of emergent particles in three-dimensional crystals. Sci. Bull. 67, 375–380 (2022).

    Article  Google Scholar 

  42. Winkler, R. Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems (Springer, 2021).

  43. Yao, M. et al. Observation of giant spin-split Fermi-arc with maximal Chern number in the chiral topological semimetal PtGa. Nat. Commun. 11, 2033 (2020).

    Article  ADS  Google Scholar 

  44. Ma, J. Z. et al. Observation of a singular Weyl point surrounded by charged nodal walls in PtGa. Nat. Commun. 12, 3994 (2021).

    Article  ADS  Google Scholar 

  45. Xu, S. et al. Quantum oscillations and electronic structure in the large-Chern-number topological chiral semimetal PtGa. Chinese Phys. Lett. 37, 107504 (2020).

    Article  ADS  Google Scholar 

  46. Schröter, N. B. M. et al. Chiral topological semimetal with multifold band crossings and long Fermi arcs. Nat. Phys. 15, 759–765 (2019).

    Article  Google Scholar 

  47. Schröter, N. B. M. et al. Observation and control of maximal Chern numbers in a chiral topological semimetal. Science 369, 179–183 (2020).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Sessi, P. et al. Handedness-dependent quasiparticle interference in the two enantiomers of the topological chiral semimetal PdGa. Nat. Commun. 11, 3507 (2020).

    Article  ADS  Google Scholar 

  49. Chang, G. et al. Unconventional chiral fermions and large topological Fermi arcs in RhSi. Phys. Rev. Lett. 119, 206401 (2017).

    Article  ADS  Google Scholar 

  50. Rees, D. et al. Helicity-dependent photocurrents in the chiral Weyl semimetal RhSi. Sci. Adv. 6, eaba0509 (2020).

    Article  ADS  Google Scholar 

  51. Mühlbauer, S. et al. Skyrmion lattice in a chiral magnet. Science 323, 915–919 (2009).

    Article  ADS  Google Scholar 

  52. Bradlyn, B. et al. Topological quantum chemistry. Nature 547, 298–305 (2017).

    Article  ADS  Google Scholar 

  53. Zhang, T. et al. Catalogue of topological electronic materials. Nature 566, 475–479 (2019).

    Article  ADS  Google Scholar 

  54. Vergniory, M. G. et al. A complete catalogue of high-quality topological materials. Nature 566, 480–485 (2019).

    Article  ADS  Google Scholar 

  55. Tang, F., Po, H. C., Vishwanath, A. & Wan, X. G. Comprehensive search for topological materials using symmetry indicators. Nature 566, 486–489 (2019).

    Article  ADS  Google Scholar 

Download references


We would like to acknowledge J. Harms for the assistance on graphic design. This work was funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (MiTopMat; grant agreement no. 715730). This project received funding by the Swiss National Science Foundation (grant no. PP00P2_176789). C.L. and L.H. are supported by the Office of Naval Research (grant no. N00014-18-1-2793) and Kaufman New Initiative research grant no. KA2018-98553 of the Pittsburgh Foundation. K.M. and C.F. acknowledge financial support from the ERC advanced grant no. 742068 ‘TOP-MAT’, European Union’s Horizon 2020 research and innovation programme (grant nos. 824123 and 766566) and Deutsche Forschungsgemeinschaft (DFG) through SFB 1143. Additionally, K.M. acknowledges Max Plank Society for funding support under Max Plank-India partner group project. B.A.B. acknowledges funding from the ERC under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 101020833). B.A.B. is also supported by the US Department of Energy (grant no. DE-SC0016239) and partially supported by the National Science Foundation (EAGER grant no. DMR 1643312), a Simons Investigator grant (no. 404513), the Office of Naval Research (ONR grant no. N00014-20-1-2303), the Packard Foundation, the Schmidt Fund for Innovative Research, the BSF Israel US foundation (grant no. 2018226), the Gordon and Betty Moore Foundation through grant no. GBMF8685 towards the Princeton theory program and a Guggenheim Fellowship from the John Simon Guggenheim Memorial Foundation. B.A.B. and C.L. are supported by the NSF-MERSEC (grant no. MERSEC DMR 2011750). B.A.B. gratefully acknowledges financial support from the Schmidt DataX Fund at Princeton University made possible through a major gift from the Schmidt Futures Foundation. B.A.B. received additional support from the Max Planck Society. Further support was provided by the NSF-MRSEC (no. DMR-1420541), BSF Israel US foundation (no. 2018226) and the Princeton Global Network Funds.

Author information

Authors and Affiliations



The crystals were synthesized and characterized by K.M., C.S. and C.F. The experimental design, FIB microstructuring and magnetotransport measurements were performed by C.G., C.P., J.D., X.H. and P.J.W.M. L.H., C.L. and B.A.B. developed and applied the general theoretical framework, and the analysis of experimental results was done by C.G., C.P. and P.J.W.M. The band structures were calculated by Y.S., F.-R.F. and C.F. All the authors were involved in writing the paper.

Corresponding authors

Correspondence to Chaoxing Liu, B. Andrei Bernevig or Philip J. W. Moll.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Christian Pfleiderer and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–28 and Supplementary Sections I–VII.

Source data

Source Data Fig. 2

Source data for Fig. 2b.

Source Data Fig. 3

Source data for Fig. 3.

Source Data Fig. 4

Source data for Fig. 4c,d.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, C., Hu, L., Putzke, C. et al. Quasi-symmetry-protected topology in a semi-metal. Nat. Phys. 18, 813–818 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing