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Title: Higher-order topology in bismuth

Abstract

The mathematical field of topology has become a framework in which to describe the low-energy electronic structure of crystalline solids. Typical of a bulk insulating three-dimensional topological crystal are conducting two-dimensional surface states. This constitutes the topological bulk–boundary correspondence. We establish that the electronic structure of bismuth, an element consistently described as bulk topologically trivial, is in fact topological and follows a generalized bulk–boundary correspondence of higher-order: not the surfaces of the crystal, but its hinges host topologically protected conducting modes. These hinge modes are protected against localization by time-reversal symmetry locally, and globally by the three-fold rotational symmetry and inversion symmetry of the bismuth crystal. We support our claim theoretically and experimentally. Our theoretical analysis is based on symmetry arguments, topological indices, first-principles calculations, and the recently introduced framework of topological quantum chemistry. We provide supporting evidence from two complementary experimental techniques. With scanning-tunnelling spectroscopy, we probe the signatures of the rotational symmetry of the one-dimensional states located at the step edges of the crystal surface. With Josephson interferometry, we demonstrate their universal topological contribution to the electronic transport. Our work establishes bismuth as a higher-order topological insulator.

Authors:
 [1]; ORCiD logo [2];  [3];  [1];  [4];  [5];  [6];  [4];  [7]; ORCiD logo [8];  [4];  [4]; ORCiD logo [7];  [9]; ORCiD logo [1]
  1. Univ. of Zurich (Switzerland). Dept. of Physics
  2. Princeton Univ., NJ (United States). Dept. of Physics
  3. Donostia International Physics Center (DIPC), Donostia (Spain); Univ. of the Basque Country, Bilbao (Spain). Dept. of Applied Physics II and Faculty of Science and Technology; Basque Foundation for Science (IKERBASQUE), Bilbao (Spain)
  4. Univ. Paris-Sud, Orsay (France). Lab. of Solid Physics
  5. Univ. Paris-Sud, Orsay (France). Center for Nuclear Sciences and Matter Sciences (CSNSM)
  6. Univ. Paris-Sud, Orsay (France). Lab. of Solid Physics; Russian Academy of Sciences (RAS), Moscow (Russian Federation). Inst. of Microelectronics Technology and High Purity Materials
  7. Princeton Univ., NJ (United States). Joseph Henry Lab. and Dept. of Physics
  8. Brookhaven National Lab. (BNL), Upton, NY (United States). Condensed Matter Physics and Materials Science Dept.
  9. Princeton Univ., NJ (United States). Joseph Henry Lab. and Dept. of Physics; Freie Univ., Berlin (Germany). Dept. of Physics; Max Planck Inst. of Microstructure Physics, Halle (Germany)
Publication Date:
Research Org.:
Brookhaven National Lab. (BNL), Upton, NY (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES); Swiss National Science Foundation (SNF); European Union (EU); Ministry of Economy and Competitiveness (MINECO) (Spain); National Science Foundation (NSF); National Research Agency (ANR) (France); US Army Research Office (ARO); Simons Foundation; Packard Foundation; Schmidt Fund
OSTI Identifier:
1476292
Report Number(s):
BNL-209127-2018-JAAM
Journal ID: ISSN 1745-2473
Grant/Contract Number:  
SC0012704; 200021_169061; ERC-StG-Neupert-757867-PARATOP; IS2016-75862-P; DMR-142054; DMR-1608848; W911NF-12-1-046; sc0016239; DMR-1643312; ONR-N00014-14-1-0330; W911NF-12-1-0461; DMR-1420541
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Nature Physics
Additional Journal Information:
Journal Volume: 14; Journal Issue: 9; Journal ID: ISSN 1745-2473
Publisher:
Nature Publishing Group (NPG)
Country of Publication:
United States
Language:
English
Subject:
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY

Citation Formats

Schindler, Frank, Wang, Zhijun, Vergniory, Maia G., Cook, Ashley M., Murani, Anil, Sengupta, Shamashis, Kasumov, Alik Yu., Deblock, Richard, Jeon, Sangjun, Drozdov, Ilya, Bouchiat, Hélène, Guéron, Sophie, Yazdani, Ali, Bernevig, B. Andrei, and Neupert, Titus. Higher-order topology in bismuth. United States: N. p., 2018. Web. doi:10.1038/s41567-018-0224-7.
Schindler, Frank, Wang, Zhijun, Vergniory, Maia G., Cook, Ashley M., Murani, Anil, Sengupta, Shamashis, Kasumov, Alik Yu., Deblock, Richard, Jeon, Sangjun, Drozdov, Ilya, Bouchiat, Hélène, Guéron, Sophie, Yazdani, Ali, Bernevig, B. Andrei, & Neupert, Titus. Higher-order topology in bismuth. United States. doi:10.1038/s41567-018-0224-7.
Schindler, Frank, Wang, Zhijun, Vergniory, Maia G., Cook, Ashley M., Murani, Anil, Sengupta, Shamashis, Kasumov, Alik Yu., Deblock, Richard, Jeon, Sangjun, Drozdov, Ilya, Bouchiat, Hélène, Guéron, Sophie, Yazdani, Ali, Bernevig, B. Andrei, and Neupert, Titus. Mon . "Higher-order topology in bismuth". United States. doi:10.1038/s41567-018-0224-7. https://www.osti.gov/servlets/purl/1476292.
@article{osti_1476292,
title = {Higher-order topology in bismuth},
author = {Schindler, Frank and Wang, Zhijun and Vergniory, Maia G. and Cook, Ashley M. and Murani, Anil and Sengupta, Shamashis and Kasumov, Alik Yu. and Deblock, Richard and Jeon, Sangjun and Drozdov, Ilya and Bouchiat, Hélène and Guéron, Sophie and Yazdani, Ali and Bernevig, B. Andrei and Neupert, Titus},
abstractNote = {The mathematical field of topology has become a framework in which to describe the low-energy electronic structure of crystalline solids. Typical of a bulk insulating three-dimensional topological crystal are conducting two-dimensional surface states. This constitutes the topological bulk–boundary correspondence. We establish that the electronic structure of bismuth, an element consistently described as bulk topologically trivial, is in fact topological and follows a generalized bulk–boundary correspondence of higher-order: not the surfaces of the crystal, but its hinges host topologically protected conducting modes. These hinge modes are protected against localization by time-reversal symmetry locally, and globally by the three-fold rotational symmetry and inversion symmetry of the bismuth crystal. We support our claim theoretically and experimentally. Our theoretical analysis is based on symmetry arguments, topological indices, first-principles calculations, and the recently introduced framework of topological quantum chemistry. We provide supporting evidence from two complementary experimental techniques. With scanning-tunnelling spectroscopy, we probe the signatures of the rotational symmetry of the one-dimensional states located at the step edges of the crystal surface. With Josephson interferometry, we demonstrate their universal topological contribution to the electronic transport. Our work establishes bismuth as a higher-order topological insulator.},
doi = {10.1038/s41567-018-0224-7},
journal = {Nature Physics},
issn = {1745-2473},
number = 9,
volume = 14,
place = {United States},
year = {2018},
month = {7}
}

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Works referenced in this record:

One-Dimensional Edge States with Giant Spin Splitting in a Bismuth Thin Film
journal, February 2015


Bismuthene on a SiC substrate: A candidate for a high-temperature quantum spin Hall material
journal, June 2017


Generalized Gradient Approximation Made Simple
journal, October 1996

  • Perdew, John P.; Burke, Kieron; Ernzerhof, Matthias
  • Physical Review Letters, Vol. 77, Issue 18, p. 3865-3868
  • DOI: 10.1103/PhysRevLett.77.3865

Observation of a phononic quadrupole topological insulator
journal, January 2018

  • Serra-Garcia, Marc; Peri, Valerio; Süsstrunk, Roman
  • Nature, Vol. 555, Issue 7696
  • DOI: 10.1038/nature25156

New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance
journal, August 1980


Higher-Order Topological Insulators and Semimetals on the Breathing Kagome and Pyrochlore Lattices
journal, January 2018


Quantization of the Hall Conductance for General, Multiparticle Schrödinger Hamiltonians
journal, January 1985


Robust spin-polarized midgap states at step edges of topological crystalline insulators
journal, December 2016


Higher-order topological insulators
journal, June 2018

  • Schindler, Frank; Cook, Ashley M.; Vergniory, Maia G.
  • Science Advances, Vol. 4, Issue 6
  • DOI: 10.1126/sciadv.aat0346

Quantum Spin Hall Insulator State in HgTe Quantum Wells
journal, November 2007


Double crystallographic groups and their representations on the Bilbao Crystallographic Server
journal, September 2017

  • Elcoro, Luis; Bradlyn, Barry; Wang, Zhijun
  • Journal of Applied Crystallography, Vol. 50, Issue 5
  • DOI: 10.1107/S1600576717011712

Experimental Realization of a Three-Dimensional Topological Insulator, Bi2Te3
journal, June 2009


Ballistic edge states in Bismuth nanowires revealed by SQUID interferometry
journal, July 2017

  • Murani, Anil; Kasumov, Alik; Sengupta, Shamashis
  • Nature Communications, Vol. 8, Issue 1
  • DOI: 10.1038/ncomms15941

Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
journal, December 2006

  • Bernevig, B. A.; Hughes, T. L.; Zhang, S.-C.
  • Science, Vol. 314, Issue 5806, p. 1757-1761
  • DOI: 10.1126/science.1133734

From ultrasoft pseudopotentials to the projector augmented-wave method
journal, January 1999


Second-order topological insulators and superconductors with an order-two crystalline symmetry
journal, May 2018


Induced superconductivity in the quantum spin Hall edge
journal, August 2014

  • Hart, Sean; Ren, Hechen; Wagner, Timo
  • Nature Physics, Vol. 10, Issue 9
  • DOI: 10.1038/nphys3036

Measurement of the Current-Phase Relation of Superconducting Atomic Contacts
journal, September 2007


Topological Crystalline Insulators
journal, March 2011


Band connectivity for topological quantum chemistry: Band structures as a graph theory problem
journal, January 2018


Topological crystalline insulator states in Pb1−xSnxSe
journal, September 2012

  • Dziawa, P.; Kowalski, B. J.; Dybko, K.
  • Nature Materials, Vol. 11, Issue 12
  • DOI: 10.1038/nmat3449

Electronic structure of the semimetals Bi and Sb
journal, July 1995


Higher-order topological insulators and superconductors protected by inversion symmetry
journal, May 2018


Magnetic field resistant quantum interferences in Josephson junctions based on bismuth nanowires
journal, December 2014


Quantum Spin Hall Effect and Enhanced Magnetic Response by Spin-Orbit Coupling
journal, December 2006


Active Control of Type-I Edge-Localized Modes with n = 1 Perturbation Fields in the JET Tokamak
journal, June 2007


Mechanical and electronic properties of superhard Re B 2
journal, October 2007


Topological crystalline superconductivity and second-order topological superconductivity in nodal-loop materials
journal, March 2018


Perdew, Burke, and Ernzerhof Reply:
journal, January 1998


Topological phases and the quantum spin Hall effect in three dimensions
journal, May 2009


Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors
journal, August 2006


Building blocks of topological quantum chemistry: Elementary band representations
journal, January 2018


Topological crystalline insulators in the SnTe material class
journal, January 2012

  • Hsieh, Timothy H.; Lin, Hsin; Liu, Junwei
  • Nature Communications, Vol. 3, Article No. 982
  • DOI: 10.1038/ncomms1969

Origin of the Large Anisotropic g Factor of Holes in Bismuth
journal, November 2015


Fully unconstrained noncollinear magnetism within the projector augmented-wave method
journal, November 2000


Observation of Unconventional Quantum Spin Textures in Topological Insulators
journal, February 2009


Quantized Hall Conductance in a Two-Dimensional Periodic Potential
journal, August 1982


Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set
journal, July 1996


Graph theory data for topological quantum chemistry
journal, August 2017


Surface states and topological invariants in three-dimensional topological insulators: Application to Bi1xSbx
journal, July 2008

  • Teo, Jeffrey C. Y.; Fu, Liang; Kane, C. L.
  • Physical Review B, Vol. 78, Issue 4, Article No. 045426
  • DOI: 10.1103/PhysRevB.78.045426

Z2 Topological Order and the Quantum Spin Hall Effect
journal, September 2005


Ab initio molecular dynamics for open-shell transition metals
journal, November 1993


Quantized Hall conductance as a topological invariant
journal, March 1985


Quantum Spin Hall Effect in Graphene
journal, November 2005


Observation of a large-gap topological-insulator class with a single Dirac cone on the surface
journal, May 2009

  • Xia, Y.; Qian, D.; Hsieh, D.
  • Nature Physics, Vol. 5, Issue 6, p. 398-402
  • DOI: 10.1038/nphys1274

Experimental realization of a topological crystalline insulator in SnTe
journal, September 2012

  • Tanaka, Y.; Ren, Zhi; Sato, T.
  • Nature Physics, Vol. 8, Issue 11
  • DOI: 10.1038/nphys2442

Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators
journal, December 2017

  • Benalcazar, Wladimir A.; Bernevig, B. Andrei; Hughes, Taylor L.
  • Physical Review B, Vol. 96, Issue 24
  • DOI: 10.1103/PhysRevB.96.245115

Topological quantum chemistry
journal, July 2017

  • Bradlyn, Barry; Elcoro, L.; Cano, Jennifer
  • Nature, Vol. 547, Issue 7663
  • DOI: 10.1038/nature23268

Antiferromagnetic topological insulators
journal, June 2010


Reflection-Symmetric Second-Order Topological Insulators and Superconductors
journal, December 2017


( d 2 ) -Dimensional Edge States of Rotation Symmetry Protected Topological States
journal, December 2017


Helical Liquid and the Edge of Quantum Spin Hall Systems
journal, March 2006


Nondegenerate Metallic States on Bi(114): A One-Dimensional Topological Metal
journal, March 2009


Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes
journal, June 2017


Quantum Spin Hall Effect
journal, March 2006


Topology of Disconnected Elementary Band Representations
journal, June 2018


Quantized electric multipole insulators
journal, July 2017

  • Benalcazar, Wladimir A.; Bernevig, B. Andrei; Hughes, Taylor L.
  • Science, Vol. 357, Issue 6346
  • DOI: 10.1126/science.aah6442

Topological phase transition of single-crystal Bi based on empirical tight-binding calculations
journal, December 2016


Topological invariants of time-reversal-invariant band structures
journal, March 2007


One-dimensional topological edge states of bismuth bilayers
journal, August 2014

  • Drozdov, Ilya K.; Alexandradinata, A.; Jeon, Sangjun
  • Nature Physics, Vol. 10, Issue 9
  • DOI: 10.1038/nphys3048

A quantized microwave quadrupole insulator with topologically protected corner states
journal, March 2018

  • Peterson, Christopher W.; Benalcazar, Wladimir A.; Hughes, Taylor L.
  • Nature, Vol. 555, Issue 7696
  • DOI: 10.1038/nature25777

Nonlocal Transport in the Quantum Spin Hall State
journal, July 2009


Observation of a topological crystalline insulator phase and topological phase transition in Pb1−xSnxTe
journal, January 2012

  • Xu, Su-Yang; Liu, Chang; Alidoust, N.
  • Nature Communications, Vol. 3, Issue 1
  • DOI: 10.1038/ncomms2191

Minimal models for Wannier-type higher-order topological insulators and phosphorene
journal, July 2018


    Works referencing / citing this record:

    Topology on a new facet of bismuth
    journal, June 2019

    • Hsu, Chuang-Han; Zhou, Xiaoting; Chang, Tay-Rong
    • Proceedings of the National Academy of Sciences, Vol. 116, Issue 27
    • DOI: 10.1073/pnas.1900527116

    Observation of a Majorana zero mode in a topologically protected edge channel
    journal, June 2019


    Photonic crystal nanocavity based on a topological corner state
    journal, January 2019


    Topology on a new facet of bismuth
    journal, June 2019

    • Hsu, Chuang-Han; Zhou, Xiaoting; Chang, Tay-Rong
    • Proceedings of the National Academy of Sciences, Vol. 116, Issue 27
    • DOI: 10.1073/pnas.1900527116

    Observation of a Majorana zero mode in a topologically protected edge channel
    journal, June 2019


    Photonic crystal nanocavity based on a topological corner state
    journal, January 2019


    Three-dimensional superconductors with hybrid higher-order topology
    journal, March 2019


    Topological Phononics: From Fundamental Models to Real Materials
    journal, August 2019

    • Liu, Yizhou; Chen, Xiaobin; Xu, Yong
    • Advanced Functional Materials, Vol. 30, Issue 8
    • DOI: 10.1002/adfm.201904784

    Large‐Scale Synthesis of Strain‐Tunable Semiconducting Antimonene on Copper Oxide
    journal, December 2019

    • Niu, Tianchao; Meng, Qingling; Zhou, Dechun
    • Advanced Materials, Vol. 32, Issue 4
    • DOI: 10.1002/adma.201906873

    Higher‐Order Topological States in Surface‐Wave Photonic Crystals
    journal, March 2020


    Polar and phase domain walls with conducting interfacial states in a Weyl semimetal MoTe2
    journal, September 2019


    Symmetry-protected hierarchy of anomalous multipole topological band gaps in nonsymmorphic metacrystals
    journal, January 2020


    Strong and fragile topological Dirac semimetals with higher-order Fermi arcs
    journal, January 2020


    Two-dimensional higher-order topology in monolayer graphdiyne
    journal, January 2020


    Robust zero-energy modes in an electronic higher-order topological insulator
    journal, September 2019


    Efficient topological materials discovery using symmetry indicators
    journal, February 2019


    Symmetry-enforced chiral hinge states and surface quantum anomalous Hall effect in the magnetic axion insulator Bi2–xSmxSe3
    journal, March 2019


    Origin of linear magnetoresistance in polycrystalline Bi films
    journal, January 2020

    • Wang, Nan; Zhang, Liqiang; Wang, Tianlin
    • Journal of Applied Physics, Vol. 127, Issue 2
    • DOI: 10.1063/1.5127570

    Charge and spin transport on surfaces and atomic layers measured by multi-probe techniques
    journal, March 2019


    Spin- and angle-resolved photoemission on topological materials
    journal, April 2019


    Second-order Dirac superconductors and magnetic field induced Majorana hinge modes
    journal, July 2019


    Phase-tunable second-order topological superconductor
    journal, August 2019


    Higher-order Floquet topological phases with corner and bulk bound states
    journal, August 2019


    Classification of crystalline insulators without symmetry indicators: Atomic and fragile topological phases in twofold rotation symmetric systems
    journal, September 2019


    Statistical analysis of the Chern number in the interacting Haldane-Hubbard model
    journal, September 2019


    Topological states on fractal lattices
    journal, October 2019


    Topoelectrical circuit octupole insulator with topologically protected corner states
    journal, November 2019


    Second-order topological phases protected by chiral symmetry
    journal, December 2019


    Doping-induced topological phase transition in Bi: The role of quantum electronic stress
    journal, January 2020


    Hinged quantum spin Hall effect in antiferromagnetic topological insulators
    journal, January 2020


    Measurement-only quantum computation with Floquet Majorana corner modes
    journal, February 2020


    Majorana corner flat bands in two-dimensional second-order topological superconductors
    journal, March 2020


    Fractional disclination charge in two-dimensional C n -symmetric topological crystalline insulators
    journal, March 2020


    Appearance of hinge states in second-order topological insulators via the cutting procedure
    journal, March 2020


    Higher-order topological insulators protected by inversion and rotoinversion symmetries
    journal, August 2018


    Connecting higher-order topological insulators to lower-dimensional topological insulators
    journal, November 2018


    Inversion-symmetry protected chiral hinge states in stacks of doped quantum Hall layers
    journal, December 2018


    Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices: An example on the triangular lattice
    journal, January 2019


    Boundary edge networks induced by bulk topology
    journal, April 2019


    Quantization of fractional corner charge in C n -symmetric higher-order topological crystalline insulators
    journal, June 2019


    Topological Switch between Second-Order Topological Insulators and Topological Crystalline Insulators
    journal, September 2018


    Majorana Kramers Pairs in Higher-Order Topological Insulators
    journal, November 2018


    Microwave Signature of Topological Andreev level Crossings in a Bismuth-based Josephson Junction
    journal, February 2019


    Second-Order Topological Superconductors with Mixed Pairing
    journal, June 2019


    Higher-Order Topology of the Axion Insulator EuIn 2 As 2
    journal, June 2019


    Floquet Second-Order Topological Insulators from Nonsymmorphic Space-Time Symmetries
    journal, July 2019


    Nonlinear Second-Order Topological Insulators
    journal, August 2019


    Higher-Order Topological Odd-Parity Superconductors
    journal, October 2019


    Higher-Order Topology, Monopole Nodal Lines, and the Origin of Large Fermi Arcs in Transition Metal Dichalcogenides X Te 2 ( X = Mo , W )
    journal, October 2019


    Topological Phases without Crystalline Counterparts
    journal, November 2019


    Higher-Order Topological Mott Insulators
    journal, November 2019


    Two-Dimensional Second-Order Topological Insulator in Graphdiyne
    journal, December 2019


    Higher-Order Topological Insulators in Quasicrystals
    journal, January 2020


    Unhinging the Surfaces of Higher-Order Topological Insulators and Superconductors
    journal, January 2020


    Bulk and Surface Electronic Structure of the Dual-Topology Semimetal Pt 2 HgSe 3
    journal, March 2020


    Discovering two-dimensional topological insulators from high-throughput computations
    journal, February 2019


    Dual topology in jacutingaite Pt 2 HgSe 3
    journal, July 2019


    Topological crystalline insulators with C 2 rotation anomaly
    journal, August 2019


    Out of equilibrium higher-order topological insulator: Floquet engineering and quench dynamics
    journal, December 2019


    Topological mechanics from supersymmetry
    journal, December 2019


    Z Q Berry phase for higher-order symmetry-protected topological phases
    journal, January 2020


    Hinge states in a system of coupled Rashba layers
    journal, January 2020


    Bulk-edge and bulk-hinge correspondence in inversion-symmetric insulators
    journal, March 2020


    Higher-Order Bulk-Boundary Correspondence for Topological Crystalline Phases
    journal, January 2019