Fu, L., Kane, C. L. & Mele, E. J. Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007).
Xia, Y. et al. Observation of a large-gap topological-insulator class with a single Dirac cone on the surface. Nat. Phys. 5, 398 (2009).
Liu, Z. et al. Discovery of a three-dimensional topological Dirac semimetal, Na3Bi. Science 343, 864–867 (2014).
Liu, Z. et al. A stable three-dimensional topological Dirac semimetal Cd3As2. Nat. Mater. 13, 677–681 (2014).
Lv, B. et al. Experimental discovery of Weyl semimetal TaAs. Phys. Rev. X 5, 031013 (2015).
Lv, B. Q. et al. Observation of Weyl nodes in TaAs. Nat. Phys. 11, 724–727 (2015).
Xu, S.-Y. et al. Discovery of a weyl fermion state with fermi arcs in niobium arsenide. Nat. Phys. 11, 748–754 (2015).
Xu, S.-Y. et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015).
Bradlyn, B. et al. Beyond Dirac and Weyl fermions: unconventional quasiparticles in conventional crystals. Science 353, aaf5037 (2016).
Chang, G. et al. Unconventional chiral fermions and large topological fermi arcs in RhSi. Phys. Rev. Lett. 119, 206401 (2017).
Sanchez, D. S. et al. Topological chiral crystals with helicoid-arc quantum states. Nature 567, 500–505 (2019).
Rao, Z. et al. Observation of unconventional chiral fermions with long Fermi arcs in CoSi. Nature 567, 496–499 (2019).
Schröter, N. B. et al. Chiral topological semimetal with multifold band crossings and long Fermi arcs. Nat. Phys. 15, 759–765 (2019).
Schröter, N. B. et al. Observation and control of maximal Chern numbers in a chiral topological semimetal. Science 369, 179–183 (2020).
Vergniory, M., Elcoro, L., Felser, C., Bernevig, B. & Wang, Z. The (high quality) topological materials in the world. Nature 566, 480–485 (2019).
Tang, F., Po, H. C., Vishwanath, A. & Wan, X. Efficient topological materials discovery using symmetry indicators. Nat. Phys. 15, 470–476 (2019).
Zhang, T. et al. Catalogue of topological electronic materials. Nature 566, 475–479 (2019).
Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90, 015001 (2018).
Wieder, B. J. et al. Topological materials discovery from crystal symmetry. Nat. Rev. Mater. 7, 196–216 (2021).
Chang, G. et al. Topological quantum properties of chiral crystals. Nat. Mater. 17, 978–985 (2018).
Flicker, F. et al. Chiral optical response of multifold fermions. Phys. Rev. B 98, 155145 (2018).
Rees, D. et al. Helicity-dependent photocurrents in the chiral Weyl semimetal RhSi. Sci. Adv. 6, eaba0509 (2020).
De Juan, F., Grushin, A. G., Morimoto, T. & Moore, J. E. Quantized circular photogalvanic effect in Weyl semimetals. Nat. Comm. 8, 15995 (2017).
Ni, Z. et al. Linear and nonlinear optical responses in the chiral multifold semimetal RhSi. npj Quantum Mater. 5, 96 (2020).
Zhang, C.-L. et al. Ultraquantum magnetoresistance in the Kramers-Weyl semimetal candidate β-Ag2Se. Phys. Rev. B 96, 165148 (2017).
Wan, B. et al. Theory for the negative longitudinal magnetoresistance in the quantum limit of Kramers Weyl semimetals. J. Condens. Matter Phys. 30, 505501 (2018).
Ni, Z. et al. Giant topological longitudinal circular photo-galvanic effect in the chiral multifold semimetal CoSi. Nat. Comm. 12, 154 (2021).
Wang, Z. Z. et al. Charge density wave transport in (TaSe4)2I. Solid State Commun. 46, 325–328 (1983).
Maki, M., Kaiser, M., Zettl, A. & Grüner, G. Charge density wave transport in a novel inorganic chain compound (TaSe4)2I. Solid State Commun. 46, 497–500 (1983).
Shi, W. et al. A charge-density-wave topological semimetal. Nat. Phys. 17, 381–387 (2021).
Gooth, J. et al. Axionic charge-density wave in the Weyl semimetal (TaSe4)2I. Nature 575, 315–319 (2019).
Wang, Z. & Zhang, S.-C. Chiral anomaly, charge density waves, and axion strings from Weyl semimetals. Phys. Rev. B 87, 161107 (2013).
Sinchenko, A. A., Ballou, R., Lorenzo, J. E., Grenet, T. & Monceau, P. Does (TaSe4)2I really harbor an axionic charge density wave? Appl. Phys. Lett. 120, 063102 (2022).
Crepaldi, A. et al. Optically induced changes in the band structure of the weyl charge-density-wave compound (TaSe4)2I. J. Phys. Mater. 5, 044006 (2022).
Nguyen, Q. L. et al. Ultrafast x-ray scattering reveals composite amplitude collective mode in the weyl charge density wave material (TaSe4)2I. Phys. Rev. Lett. 131, 076901 (2023).
Kim, S. et al. Observation of a massive phason in a charge-density-wave insulator. Nat. Mater. 22, 429–433 (2023).
Lin, M.-K. et al. Unconventional spectral gaps induced by charge density waves in the weyl semimetal (TaSe4)2I. Nano Lett. 24, 8778 (2024).
Christensen, J. A. et al. Disorder and diffuse scattering in single-chirality (TaSe4)2I crystals. Phys. Rev. Mater. 8, 034202 (2024).
Yi, H. et al. Surface charge induced dirac band splitting in a charge density wave material (TaSe4)2I. Phys. Rev. Res. 3, 013271 (2021).
Grüner, G. The dynamics of charge-density waves. Rev. Mod. Phys. 60, 1129–1181 (1988).
Voit, J. Electronic structure of solids with competing periodic potentials. Science 290, 501–503 (2000).
Tournier-Colletta, C. et al. Electronic instability in a zero-gap semiconductor: the charge-density wave in (TaSe4)2I. Phys. Rev. Lett. 110, 236401 (2013).
van Smaalen, S., Lam, E. J. & Lüdecke, J. Structure of the charge-density wave in (TaSe4)2I. J. Phys.:Condens. Matter 13, 9923 (2001).
Bradlyn, B. et al. Topological quantum chemistry. Nature 547, 298–305 (2017).
Vergniory, M.G. et al. All topological bands of all nonmagnetic stoichiometric materials. Science 376, 816 (2022).
See Supplementary Information.
Favre-Nicolin, V. et al. Structural evidence for ta-tetramerization displacements in the charge-density-wave compound (TaSe4)2I from x-ray anomalous diffraction. Phys. Rev. Lett. 87, 015502 (2001).
Fujishita, H., Shapiro, S. M., Sato, M. & Hoshino, S. A neutron scattering study of the quasi-one-dimensional conductor (TaSe4)2I. J. Phys. C Solid State Phys. 19, 3049–3057 (1986).
Li, X.-P. et al. Type-III Weyl semimetals: (TaSe4)2I. Phys. Rev. B 103, L081402 (2021).
Perfetti, L. et al. Spectroscopic indications of polaronic carriers in the quasi-one-dimensional conductor (TaSe4)2I. Phys. Rev. Lett. 87, 216404 (2001).
Dardel, B. et al. Unusual photoemission spectral function of quasi-one-dimensional metals. Phys. Rev. Lett. 67, 3144 (1991).
Gatti, G. et al. Radial spin texture of the Weyl fermions in chiral tellurium. Phys. Rev. Lett. 125, 216402 (2020).
Sakano, M. et al. Radial spin texture in elemental tellurium with chiral crystal structure. Phys. Rev. Lett. 124, 136404 (2020).
Wang, Y. H. et al. Observation of a warped helical spin texture in Bi2Se3 from circular dichroism angle-resolved photoemission spectroscopy. Phys. Rev. Lett. 107, 207602 (2011).
Fu, L. Hexagonal warping effects in the surface states of the topological insulator Bi2Te3. Phys. Rev. Lett. 103, 266801 (2009).
Jung, W. et al. Warping effects in the band and angular-momentum structures of the topological insulator Bi2Te3. Phys. Rev. B 84, 245435 (2011).
Ryu, H. et al. Photon energy dependent circular dichroism in angle-resolved photoemission from Au(111) surface states. Phys. Rev. B 95, 115144 (2017).
Crepaldi, A. et al. Momentum and photon energy dependence of the circular dichroic photoemission in the bulk rashba semiconductors BiTeX (X=I, Br, Cl). Phys. Rev. B 89, 125408 (2014).
Liu, Y., Bian, G., Miller, T. & Chiang, T.-C. Visualizing electronic chirality and berry phases in graphene systems using photoemission with circularly polarized light. Phys. Rev. Lett. 107, 166803 (2011).
Zhang, Y., Lin, L.-F., Moreo, A., Dong, S. & Dagotto, E. First-principles study of the low-temperature charge density wave phase in the quasi-one-dimensional Weyl chiral compound (TaSe4)2I. Phys. Rev. B 101, 174106 (2020).
Lorenzo, J. E. et al. A neutron scattering study of the quasi-one-dimensional conductor (TaSe4)2I. J. Phys. Condens. Matter 10, 5039 (1998).
Hohenberg, P. & Kohn, W. Inhomogeneous electron gas. Phys. Rev. 136, B864 (1964).
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169 (1996).
Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, 1758–1775 (1999).
Gajdoš, M., Hummer, K., Kresse, G., Furthmüller, J. & Bechstedt, F. Linear optical properties in the projector-augmented wave methodology. Phys. Rev. B 73, 045112 (2006).
Steiner, S., Khmelevskyi, S., Marsmann, M. & Kresse, G. Calculation of the magnetic anisotropy with projected-augmented-wave methodology and the case study of disordered Fe1−xCox alloys. Phys. Rev. B 93, 224425 (2016).
Jain, A. et al. The materials project: a materials genome approach to accelerating materials innovation. APL Mater. 1, 011002 (2013).
Ong, S. P. et al. Python Materials Genomics (pymatgen): a robust, open-source python library for materials analysis. Comput. Mater. Sci. 68, 314–319 (2013).
Setyawan, W. & Curtarolo, S. High-throughput electronic band structure calculations: challenges and tools. Comput. Mater. Sci. 49, 299–312 (2010).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
Monkhorst, H. J. & Pack, J. D. Special points for brillouin-zone integrations. Phys. Rev. B 13, 5188 (1976).
Momma, K. & Izumi, F. VESTA3 for three-dimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr. 44, 1272–1276 (2011).