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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.

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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:


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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.

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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.

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

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

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

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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.

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

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