Higher-Order Topology in Bismuth

Abstract

The mathematical field of topology has become a framework to describe the low-energy electronic structure of crystalline solids. A typical feature of a bulk insulating three-dimensional topological crystal are conducting two-dimensional surface states. This constitutes the topological bulk-boundary correspondence. Here, 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-principle calculations, and the recently introduced framework of topological quantum chemistry. We provide supporting evidence from two complementary experimental techniques. With scanning-tunneling spectroscopy, we probe the unique signatures of the rotational symmetry of the one-dimensional states located at 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.

arXiv comment16 pages, 6 figures, accepted manuscript

References in corpus (54)

• Quantum Spin Hall Effect in Graphene
• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Topological Insulators in Three Dimensions
• Topological Insulators with Inversion Symmetry
• Discovery (theoretical prediction and experimental observation) of a large-gap topological-insulator class with spin-polarized single-Dirac-cone on the surface
• Topological invariants of time-reversal-invariant band structures
• Quantum Spin Hall Effect
• Topological Crystalline Insulators
• Quantized Electric Multipole Insulators
• Higher-Order Topological Insulators
• Topological quantum chemistry
• Topological Crystalline Insulators in the SnTe Material Class
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• First direct observation of Spin-textures in Topological Insulators : Spin-resolved ARPES as a probe of topological quantum spin Hall effect and Berry's phase
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Bismuthene on a SiC Substrate: A Candidate for a New High-Temperature Quantum Spin Hall Paradigm
• Experimental realization of a topological crystalline insulator in SnTe
• Reflection symmetric second-order topological insulators and superconductors
• Observation of a phononic quadrupole topological insulator
• Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
• Nonlocal edge state transport in the quantum spin Hall state
• Higher-order topological insulators and semimetals on the breathing Kagome and pyrochlore lattices
• Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors
• Demonstration of a quantized microwave quadrupole insulator with topologically protected corner states
• Topological crystalline insulator states in Pb(1-x)Sn(x)Se
• The Helical Liquid and the Edge of Quantum Spin Hall Systems
• Surface States of the Topological Insulator Bi_{1-x}Sb_x
• A topological crystalline insulator (TCI) phase via topological phase transition and crystalline mirror symmetry
• Antiferromagnetic topological insulators
• Quantum Spin Hall Effect and Enhanced Magnetic Response by Spin-Orbit Coupling
• Scalable Designs for Quasiparticle-Poisoning-Protected Topological Quantum Computation with Majorana Zero Modes
• Higher-order topological insulators and superconductors protected by inversion symmetry
• Second-order topological insulators and superconductors with an order-two crystalline symmetry
• One-dimensional Topological Edge States of Bismuth Bilayers
• Induced Superconductivity in the Quantum Spin Hall Edge
• Double crystallographic groups and their representations on the Bilbao Crystallographic Server
• Symmetry indicators and anomalous surface states of topological crystalline insulators
• Building Blocks of Topological Quantum Chemistry: Elementary Band Representations
• Topological Quadrupolar Semimetals
• Minimal model for higher-order topological insulators and phosphorene
• Topology of Disconnected Elementary Band Representations
• Measurement of the current-phase relation of superconducting atomic contacts
• Ballistic edge states in Bismuth nanowires revealed by SQUID interferometry
• Robust spin-polarized midgap states at step edges of topological crystalline insulators
• Graph Theory Data for Topological Quantum Chemistry
• Topological crystalline superconductivity and second-order topological superconductivity in nodal-loop materials
• One-Dimensional Edge States with Giant Spin Splitting in a Bismuth Thin Film
• Band Connectivity for Topological Quantum Chemistry: Band Structures As A Graph Theory Problem
• Origin of the large anisotropic g-factor of holes in bismuth
• Topological surface states on Bi(111) based on empirical tight-binding calculations
• Topological Corner States on Kagome Lattice Based Chiral Higher-Order Topological Insulator
• Topological gaps without masses in driven graphene-like systems

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• Spin-Selective Second-Order Topological Insulators Enabling Cornertronics in 2D Altermagnets
• Anomalous $h/2e$ periodicity and Majorana zero modes in chiral Josephson junctions
• Simulating Higher-Order Topological Insulators in Density Wave Insulators
• Bulk-boundary-transport correspondence of the second-order topological insulators
• Nodal lines and mapping to mirror Chern numbers in Ca${}_2$As family
• Disorder-Induced Phase Transitions in Three-Dimensional Chiral Second-Order Topological Insulator
• Second-order topological insulator under strong magnetic field: Landau levels, Zeeman effect, and magnetotransport
• Localized Wannier function based tight-binding models for two-dimensional allotropes of bismuth
• Remote control of spin polarization of topological corner states
• Theory of Anomalous Floquet Higher-Order Topology: Classification, Characterization, and Bulk-Boundary Correspondence
• Fractional second-order topological insulator from a three-dimensional coupled-wires construction
• Direct demonstration of bulk-boundary correspondence in higher-order topological superconductors with chiral symmetry
• Controlling Majorana modes by $p$-wave pairing in two-dimensional $p+id$ topological superconductors
• Topological Insulators with Hybrid-order Boundary States
• Thermodynamics of a Higher-Order Topological Insulator
• Emergent metallicity at the grain boundaries of higher-order topological insulators
• Hierarchical Topological States in Thermal Diffusive Networks
• Structural, electronic and topological properties of 3D TmBi compound
• Higher-order topology in twisted multilayer systems: a review
• Generic Approach to Intrinsic Magnetic Second-order Topological Insulators via Inverted $p-d$ Orbitals
• Fractional corner charges in threefold-symmetric two-dimensional materials with fragile topology
• STM observation of the hinge-states of bismuth nanocrystals
• Breakdown of bulk-projected isotropy in surface electronic states of topological Kondo insulator SmB${}_6$(001)
• Second-order topological insulator in a coinless discrete-time quantum walk
• Engineering rich two-dimensional higher-order topological phases by flux and periodic driving
• Fermionic tensor networks for higher order topological insulators from charge pumping
• Fragile topological phase on the triangular kagome lattice and its bulk-boundary correspondence
• Zero-Energy State Localized near an Arbitrary Edge in Quadrupole Topological Insulators
• Effects of first- and second-order topological phases on equilibrium crystal shapes
• Corner states of two-dimensional second-order topological insulators with a chiral symmetry and broken time reversal and charge conjugation
• Magnetic impurities along the edge of a quantum spin Hall insulator: Realizing a one-dimensional AIII insulator
• Spin Hall conductivity in Bi${}_{1-x}$Sb${}_x$ as an experimental test of bulk-boundary correspondence
• Kitaev Building-block Construction for Inversion-Protected Higher-order Topological Superconductors
• Pentagonal nanowires from topological crystalline insulators: a platform for intrinsic core-shell nanowires and higher-order topology
• Mixed higher-order topology and nodal and nodeless flat band topological phases in a superconducting multiorbital model
• Observation of Temperature Independent Anomalous Hall Effect in Thin Bismuth from Near Absolute Zero to 300 K Temperature
• Family of third-order topological insulators from Su-Schrieffer-Heeger stacking
• Two-dimensional orbital-obstructed insulators with higher-order band topology
• Unconventional hybrid-order topological insulators
• Strain Solitons in an Epitaxially Strained van der Waals-like Material
• A field-induced reentrant insulator state of a gap-closed topological insulator (Bi_{1-x}Sb_x) in quantum-limit states
• Quantized Thermal Hall Conductance and the Topological Phase Diagram of a Superconducting Bismuth Bilayer
• Higher-order topology in honeycomb lattice with Y-Kekulé distortions
• Second-order topological insulator in periodically driven lattice
• General corner charge formulas in various tetrahedral and cubic space groups
• Theoretical analysis of glide-Z_2 magnetic topological photonic crystals
• Anomalous Hall Effect in Bismuth
• Hinge States of Second-Order Topological Insulators as a Mach-Zehnder Interferometer
• Second-order topological insulator in Bilayer borophene
• Majorana corner states in square and Kagome quantum spin liquids
• Topological phases of amorphous matter
• Optical Berry Curvature of Hinge State and Its Detection
• Higher-order topological superconductors characterized by Fermi level crossings
• Strain-induced two-dimensional topological crystalline insulator
• Coupled pair of one and two dimensional magneto-plasmons on electrons on helium
• Inexorable Edge Kondo Breakdown in Topological Kondo Insulators
• Disorder-induced phase transitions in double HgTe quantum wells
• Adiabatic charge transport in extended SSH models
• In the Pursuit of Majorana Modes in Iron-based High-$T_c$ Superconductors
• Spin Weyl Topological Insulators
• Topological characteristics of gap closing points in nonlinear Weyl semimetals
• Recipe for higher-order topology on the triangular lattice
• Revealing Higher-Order Topological Bulk-boundary Correspondence in Bismuth Crystal with Spin-helical Hinge State Loop and Proximity Superconductivity
• Theory of fractional corner charges in cylindrical crystal shapes
• Fermionic Quantum Simulation on Andreev Bound State Superlattices
• Topological blocking at the Bi(111) surface due to surface relaxation
• Photonic quadrupole topological phases in zero-dimensional cavity with synthetic dimensions
• Tunable corner states in topological insulators with long-range hoppings and diverse shapes
• Closing of the Mott gap near step edges in NiS2
• Interaction- and phonon-induced topological phase transitions in double helical liquids
• Method of Mechanical Exfoliation of Bismuth with Micro-Trench Structures
• Topologically trivial gap-filling in superconducting Fe(Se,Te) by one dimensional defects
• Mapping rules from nodal line semimetal to topological crystalline insulator in face centered cubic lattice
• Extrinsic higher-order topological corner states in AB-stacked transition-metal dichalcogenides
• Selective branching, quenching, and converting of topological modes
• IrRep: symmetry eigenvalues and irreducible representations of ab initio band structures
• Helical Topological Superconducting Pairing at Finite Excitation Energies
• Anisotropic Penetration Depths of Corner States in a Higher-Order Topological Insulator
• Tunable symmetry-protected higher-order topological states with fermionic atoms in bilayer optical lattices
• Experimental realization of Majorana hinge and corner modes in intrinsic organic topological superconductor without magnetic field at room temperature
• Unconventional Topological Insulators from Extended Topological Band Degeneracies
• Mixed-parity octupolar pairing and corner Majorana modes in three dimensions
• Fragile topology in nodal-line semimetal superconductors
• Finite-temperature topological invariant for higher-order topological insulators
• RKKY interaction in helical higher-order topological insulators
• Boundary topological insulators and superconductors of Altland-Zirnbauer tenfold classes
• Effects of the Vertices on the Topological Bound States in a Quasicrystalline Topological Insulator
• Quantum Coherent Transport of 1D ballistic states in second order topological insulator Bi${}_4$Br${}_4$
• Visualizing near-coexistence of massless Dirac electrons and ultra-massive saddle point electrons
• Magnetic-field driven evolution of zero-energy mode on Bi islands deposited on Fe(Te,Se)
• Electronic band structure from quasiparticle interference and Landau quantization in WTe${}_2$
• Spin separation and filtering assisted by topological corner states in the Kekulé lattice
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Visualizing dissipative charge carrier dynamics at the nanoscale with superconducting charge qubit microscopy
• Quasiperiodic Quadrupole Insulators
• High-field immiscibility of electrons belonging to adjacent twinned bismuth crystals
• Graphene-driven correlated electronic states in one dimensional defects within WS${}_2$
• Enhanced lifetimes of spin chains coupled to chiral edge states
• Anomalous Hall Effect in Thin Bismuth
• Second-order topological insulator and fragile topology in topological circuitry simulation
• Surface theory of a second-order topological insulator beyond the Dirac approximation
• Euler band topology in superfluids and superconductors
• Zeeman field induced corner states in the Kane-Mele-Hubbard model
• Quantum matter and gravitation: photons in a waveguide
• Intrinsic higher-order topological states in 2D honeycomb Z_2 quantum spin Hall insulators
• Extended Haldane model -- a modern gateway to topological insulators
• Interaction-Induced Higher-Order Topological Insulator via Floquet Engineering
• Network model for magnetic higher-order topological phases
• Obstructed Atomic Insulators and Superfluids of Fermions Coupled to ${\mathbb{Z}}_2$ Gauge Fields
• Unveiling Topological Hinge States in the Higher-Order Topological Insulator WTe${}_2$ Based on the Fractional Josephson Effect
• Development of a versatile micro-focused angle-resolved photoemission spectroscopy system with Kirkpatrick-Baez mirror optics
• Theory of Topological Superconductivity in Doped IV-VI Semiconductors
• Hidden wallpaper fermion and third-order topological insulator
• Gapless States Localized along a Staircase Edge in Second-Order Topological Insulators
• Hybrid Higher-Order Skin Topological Modes in the Two-Dimensional Su-Schrieffer-Heeger Model with Nonreciprocal Hoppings
• Local expression of fractional corner charges in obstructed atomic insulators and relationship with the fractional disclination charges
• Current switching behavior mediated via hinge modes in higher-order topological phases using altermagnets
• Tunable cornerlike states in topological type-II hyperbolic lattices
• Higher-order topological insulators in two-dimensional antiferromagnetic and altermagnetic chromium-based group-IV chalcogenides
• Magnetic-flux tunable electronic transport through domain walls in a three-dimensional second-order topological insulator
• Artificial Gauge Field Engineered Excited-State Topology: Control of Dynamical Evolution of Localized Spinons
• Wilson Loop and Topological Properties in 3D Woodpile Photonic Crystal
• Proposal for realizing and probing topological crystalline insulators in optical lattices
• Bulk-boundary correspondence in topological two-dimensional non-Hermitian systems: Toeplitz operators and singular values
• Detecting and Distinguishing Majorana Zero Modes with the Scanning Tunneling Microscope
• Realization of second-order photonic square-root topological insulators
• Classification of topological insulators and superconductors with multiple order-two point group symmetries
• Hinge modes of three-dimensional Euler insulators
• Emergent One-Dimensional Helical Channel in Higher-Order Topological Insulators with Step Edges
• Nanoimprinted topological laser in the visible
• Second-order topological insulator induced by compensated altermagnetism without bulk spin splitting
• Electronic structure and hinge states of strained half-Heusler compounds LiSbZn and LiBiZn
• Topologically protected strong coupling and entanglement between distant quantum emitters
• Topologically protected long-range coherent energy transfer
• Graphyne as a second-order and real Chern topological insulator in two dimensions
• Dynamics of 2D topological quadrupole insulator and Chern insulator induced by real-space topological changes
• Higher-order topological insulator in cubic semiconductor quantum wells
• Corner Flat Bands Induced by $d$-Density Wave and Partial Corner State Modification Due to Competition with $d$-Wave Superconductivity
• Majorana vortex phases in time-reversal invariant higher-order topological insulators and topologically trivial insulators
• Non-Hermitian Higher-Order Dirac Semimetals
• Protected helical transport in magnetically doped quantum wires: Beyond the one-dimensional paradigm
• Magnetically tunable Dirac and Weyl fermions in the Zintl materials family