Abstract
We show that the surface dispersions of topological semimetals map to helicoidal structures, where the bulk nodal points project to the branch points of the helicoids whose equal-energy contours are Fermi arcs. This mapping is demonstrated in the recently discovered Weyl semimetals and leads us to predict new types of topological semimetals, whose surface states are represented by double- and quad-helicoid surfaces. Each helicoid or multi-helicoid is shown to be the non-compact Riemann surface representing a multi-valued holomorphic function (generating function). The intersection of multiple helicoids, or the branch cut of the generating function, appears on high-symmetry lines in the surface Brillouin zone, where surface states are guaranteed to be doubly degenerate by a glide reflection symmetry. We predict the heterostructure superlattice [(SrIrO3)2(CaIrO3)2] to be a topological semimetal with double-helicoid surface states.
This is a preview of subscription content, access via your institution
Access options
Subscribe to this journal
Receive 12 print issues and online access
$209.00 per year
only $17.42 per issue
Buy this article
- Purchase on Springer Link
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
References
Murakami, S. Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase. New J. Phys. 9, 356 (2007).
Chiu, C.-K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Preprint at http://arXiv.org/abs/1505.03535 (2015).
Wan, X., 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).
Hosur, P., Parameswaran, S. A. & Vishwanath, A. Charge transport in Weyl semimetals. Phys. Rev. Lett. 108, 046602 (2012).
Son, D. T. & Spivak, B. Z. Chiral anomaly and classical negative magnetoresistance of Weyl metals. Phys. Rev. B 88, 104412 (2013).
Liu, C.-X., Ye, P. & Qi, X.-L. Chiral gauge field and axial anomaly in a Weyl semimetal. Phys. Rev. B 87, 235306 (2013).
Burkov, A. A., Hook, M. D. & Balents, L. Topological nodal semimetals. Phys. Rev. B 84, 235126 (2011).
Burkov, A. A. & Balents, L. Weyl semimetal in a topological insulator multilayer. Phys. Rev. Lett. 107, 127205 (2011).
Xu, G., Weng, H., Wang, Z., Dai, X. & Fang., Z. Chern semimetal and the quantized anomalous Hall effect in HgCr2Se4 . Phys. Rev. Lett. 107, 186806 (2011).
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).
Lu, L., Fu, L., Joannopoulos, J. D. & Soljacic, M. Weyl points and line nodes in gyroid photonic crystals. Nature Photon. 7, 294–299 (2013).
Liu, J. & Vanderbilt, D. Weyl semimetals from noncentrosymmetric topological insulators. Phys. Rev. B 90, 155316 (2014).
Weng, H., Fang, C., Fang, Z., Bernevig, B. A. & Dai, X. Weyl semimetal phase in noncentrosymmetric transition-metal monophosphides. Phys. Rev. X 5, 011029 (2015).
Huang, S.-M. et al. An inversion breaking Weyl semimetal state in the taas material class. Nature Commun. 6, 7373 (2015).
Soluyanov, A. A. et al. Type-ii Weyl semimetals. Nature 527, 495–498 (2015).
Young, S. M. et al. Dirac semimetal in three dimensions. Phys. Rev. Lett. 108, 140405 (2012).
Wang, Z. et al. Dirac semimetal and topological phase transitions in A3Bi (A = Na, K, Rb). Phys. Rev. B 85, 195320 (2012).
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).
Yang, B.-J. & Nagaosa, N. Classification of stable three-dimensional Dirac semimetals with nontrivial topology. Nature Commun. 5, 4898 (2014).
Zeng, M. et al. Topological semimetals and topological insulators in rare earth monopnictides. Preprint at http://arXiv.org/abs/1504.03492 (2015).
Chiu, C.-K. & Schnyder, A. P. Classification of reflection-symmetry-protected topological semimetals and nodal superconductors. Phys. Rev. B 90, 205136 (2014).
Phillips, M. & Aji, V. Tunable line node semimetals. Phys. Rev. B 90, 115111 (2014).
Mullen, K., Uchoa, B. & Glatzhofer, D. T. Line of Dirac nodes in hyperhoneycomb lattices. Phys. Rev. Lett. 115, 026403 (2015).
Weng, H. et al. Topological node-line semimetal in three-dimensional graphene networks. Phys. Rev. B 92, 045108 (2015).
Xie, L. S. et al. A new form of Ca3P2 with a ring of Dirac nodes. APL Mater. 3, 083602 (2015).
Kim, Y., Wieder, B. J., Kane, C. L. & Rappe, A. M. Dirac line nodes in inversion-symmetric crystals. Phys. Rev. Lett. 115, 036806 (2015).
Yu, R., Weng, H., Fang, Z., Dai, X. & Hu, X. Topological node-line semimetal and Dirac semimetal state in antiperovskite cu3PdN. Phys. Rev. Lett. 115, 036807 (2015).
Rhim, J.-W. & Kim, Y. B. Landau level quantization and almost flat modes in three-dimensional semimetals with nodal ring spectra. Phys. Rev. B 92, 045126 (2015).
Carter, J.-M., Shankar, V. V., Zeb, M. A. & Kee, H.-Y. Semimetal and topological insulator in perovskite iridates. Phys. Rev. B 85, 115105 (2012).
Chen, Y., Lu, Y.-M. & Kee, H.-Y. Topological crystalline metal in orthorhombic perovskite iridates. Nature Commun. 6, 6593 (2015).
Fang, C., Chen, Y., Kee, H.-Y. & Fu, L. Topological nodal line semimetals with and without spin-orbital coupling. Phys. Rev. B 92, 081201 (2015).
Rau, J. G., Lee, E. K.-H. & Kee, H.-Y. Spin-orbit physics giving rise to novel phases in correlated systems: Iridates and related materials. Annu. Rev. Condens. Matter Phys. 7, 57–82 (2016).
Lu, L. et al. Experimental observation of Weyl points. Science 349, 622–624 (2015).
Xu, S.-Y. et al. Experimental realization of a topological Weyl semimetal phase with Fermi arc surface states in TaAs. Science 349, 613–617 (2015).
Lv, B. Q. Experimental discovery of Weyl semimetal TaAs. Phys. Rev. X 5, 031013 (2015).
Shekhar, C. et al. Extremely large magnetoresistance and ultrahigh mobility in the topological Weyl semimetal candidate NbP. Nature Phys. 11, 645–649 (2015).
Lv, B. Q. et al. Observation of Weyl nodes in TaAs. Nature Phys. 11, 724–727 (2015).
Yang, L. X. et al. Weyl semimetal phase in the non-centrosymmetric compound TaAs. Nature Phys. 11, 728–732 (2015).
Xu, S.-Y. et al. Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide. Nature Phys. 11, 748–754 (2015).
Zhang, C. et al. Tantalum monoarsenide: an exotic compensated semimetal. Preprint at http://arXiv.org/abs/1502.00251 (2015).
Huang, X. et al. Observation of the chiral-anomaly-induced negative magnetoresistance in 3D Weyl semimetal TaAs. Phys. Rev. X 5, 031023 (2015).
Liu, Z. K. et al. A stable three-dimensional topological Dirac semimetal Cd3As2 . Nature Mater. 13, 677–681 (2014).
Liu, Z. K. Discovery of a three-dimensional topological Dirac semimetal, Na3Bi. Science 343, 864–867 (2014).
Neupane, M. et al. Observation of a three-dimensional topological Dirac semimetal phase in high-mobility Cd3As2 . Nature Commun. 5, 3786 (2014).
He, L. P. et al. Quantum transport evidence for the three-dimensional Dirac semimetal phase in Cd3As2 . Phys. Rev. Lett. 113, 246402 (2014).
Jeon, S. et al. Landau quantization and quasiparticle interference in the three-dimensional Dirac semimetal Cd3As2 . Nature Mater. 13, 851–856 (2014).
Xu, S.-Y. et al. Observation of Fermi arc surface states in a topological metal. Science 347, 294–298 (2015).
Xiong, J. et al. Evidence for the chiral anomaly in the Dirac semimetal Na3Bi. Science 350, 413–416 (2015).
Bian, G. et al. Topological nodal-line fermions in the spin–orbit metal PbTaSe2 . Nature Commun. 7, 10556 (2016).
Potter, A. C., Kimchi, I. & Vishwanath, A. Quantum oscillations from surface Fermi arcs in Weyl and Dirac semimetals. Nature Commun. 5, 5161 (2014).
Kargarian, M., Randeria, M. & Lu, Y.-M. Are the double Fermi arcs of Dirac semimetals topologically protected? Preprint at http://arXiv.org/abs/1509.02180v1 (2015).
Matsuno, J. et al. Engineering a spin-orbital magnetic insulator by tailoring superlattices. Phys. Rev. Lett. 114, 247209 (2015).
Weyl, H. The Concept of a Riemann Surface (Dover, 2009).
Li, S. & Andreev, A. V. Spiraling Fermi arcs in Weyl materials. Phys. Rev. B 92, 201107 (2015).
Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part II (Dover, 1996).
Peskin, M. E. An Introduction to Quantum Field Theory (Westview, 1995).
Parameswaran, S. A., Turner, A. M., Arovas, D. P. & Vishwanath, A. Topological order and absence of band insulators at integer filling in non-symmorphic crystals. Nature Phys. 9, 299–303 (2013).
Freed, D. S. & Moore, G. W. Twisted equivariant matter. Ann. Henri Poincare 14, 1927–2023 (2013).
Liu, C.-X., Zhang, R.-X. & VanLeeuwen, B. K. Topological nonsymmorphic crystalline insulators. Phys. Rev. B 90, 085304 (2014).
Fang, C. & Fu, L. New classes of three-dimensional topological crystalline insulators: Nonsymmorphic and magnetic. Phys. Rev. B 91, 161105 (2015).
Shiozaki, K., Sato, M. & Gomi, K. Z2 topology in nonsymmorphic crystalline insulators: Möbius twist in surface states. Phys. Rev. B 91, 155120 (2015).
Varjas, D., de Juan, F. & Lu, Y.-M. Bulk invariants and topological response in insulators and superconductors with nonsymmorphic symmetries. Phys. Rev. B 92, 195116 (2015).
Watanabe, H., Po, H. C., Vishwanath, A. & Zaletel, M. P. Filling constraints for spin-orbit coupled insulators in symmorphic and non-symmorphic crystals. Proc. Natl Acad. Sci. USA 112, 14551–14556 (2015).
Lu, L. et al. Symmetry-protected topological photonic crystal in three dimensions. Nature Phys. 12, 337–340 (2016).
Wang, Z., Alexandradinata, A., Cava, R. J. & Bernevig, B. A. Hourglass fermions. Nature 532, 189–194 (2016).
Kramers, H. Théorie générale de la rotation paramagnétique dans les cristaux. Proc. Amsterdam Akad. 33, 959–972 (1930).
Fu, L. & Kane, C. Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302 (2007).
Kress, G. & Furthmuller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169 (1996).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
Blochl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953 (1994).
Sancho, M. P. L., Sancho, J. M. L. & Rubio, J. Highly convergent schemes for the calculation of bulk and surface Green functions. J. Phys. F: Met. Phys. 15, 851–858 (1985).
Acknowledgements
We thank T. H. Hsieh for discussions. C.F. thanks Y. Chen for helpful discussions on the tight-binding model. C.F. thanks J.L. for fruitful discussions on potential material systems. C.F. and L.F. were supported by S3TEC Solid State Solar Thermal Energy Conversion Center, an Energy Frontier Research Center funded by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under Award no. DE-SC0001299/DE-FG02-09ER46577. C.F. was also supported by the National Thousand-Young-Talents Program of China. L.L. was supported in part by USARO through the ISN under Contract no. W911NF-13-D-0001, in part by the MRSEC Program of the NSF under Award no. DMR-1419807, and in part by the MIT S3TEC EFRC of DOE under Grant no. DE-SC0001299. J.L. was supported by the STC Center for Integrated Quantum Materials, NSF Grant no. DMR-1231319.
Author information
Authors and Affiliations
Contributions
C.F. and L.L. conceived the mapping between surface state dispersions and helicoids; L.F. planned the project; C.F. and L.F. performed the band topology analysis; J.L. performed the first-principles calculation and all authors contributed to the preparation of the manuscript.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
About this article
Cite this article
Fang, C., Lu, L., Liu, J. et al. Topological semimetals with helicoid surface states. Nature Phys 12, 936–941 (2016). https://doi.org/10.1038/nphys3782
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nphys3782
This article is cited by
-
Tunable topologically driven Fermi arc van Hove singularities
Nature Physics (2023)
-
Photonic helicoid-like surface states in chiral metamaterials
Scientific Reports (2023)
-
Parallel and anti-parallel helical surface states for topological semimetals
Scientific Reports (2023)
-
Type-II Weyl points and one-way interface transmission in a three-dimensional gyromagnetic photonic crystal
Science China Physics, Mechanics & Astronomy (2023)
-
Consecutive topological transitions of helical Fermi arcs at saddle points in CoSi
Science China Physics, Mechanics & Astronomy (2022)