Reflection symmetric second-order topological insulators and superconductors

Abstract

Second-order topological insulators are crystalline insulators with a gapped bulk and gapped crystalline boundaries, but topologically protected gapless states at the intersection of two boundaries. Without further spatial symmetries, five of the ten Altland-Zirnbauer symmetry classes allow for the existence of such second-order topological insulators in two and three dimensions. We show that reflection symmetry can be employed to systematically generate examples of second-order topological insulators and superconductors, although the topologically protected states at corners (in two dimensions) or at crystal edges (in three dimensions) continue to exist if reflection symmetry is broken. A three-dimensional second-order topological insulator with broken time-reversal symmetry shows a Hall conductance quantized in units of $e^2/h$.

13 pages with supplemental material

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• Anomalous Quadrupole Topological Insulators in 2D Nonsymmorphic Sonic Crystals
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• Boundaries of boundaries: a systematic approach to lattice models with solvable boundary states of arbitrary codimension
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• Bulk-boundary correspondence in disordered higher-order topological insulators
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• Appearance of hinge states in second-order topological insulators via the cutting procedure
• Interaction Enabled Fractonic Higher-Order Topological Phases
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• Topological excitonic corner states and nodal phase in bilayer quantum spin Hall insulators
• Hinged Quantum Spin-Hall Effect in Antiferromagnetic Topological Insulators
• Theory of inversion-${\mathbb{Z}}_4$ protected topological chiral hinge states and its applications to layered antiferromagnets
• Higher-order topological spin Hall effect of sound
• Generation of higher-order topological insulators using periodic driving
• Boundary-obstructed topological phases of a massive Dirac fermion in a magnetic field
• Time-reversal-invariant $C_2$-symmetric higher-order topological superconductors
• Quadrupole spin polarization as signature of second-order topological superconductors
• Topological classification of Higher-order topological phases with nested band inversion surfaces
• Magnon corner states in twisted bilayer honeycomb magnets
• Second-order topological corner states in zigzag graphene nanoflake with different types of edge magnetic configurations
• Embedded Topological Insulators
• Corner- and sublattice-sensitive Majorana zero modes on the kagome lattice
• Fractional Chiral Hinge Insulator
• Higher-Order Topological Crystalline Insulating Phase and Quantized Hinge Charge in Topological Electride Apatite
• Klein-bottle quadrupole insulators and Dirac semimetals
• Second-order and real Chern topological insulator in twisted bilayer $\alpha$-graphyne
• Observation of backscattering induced by magnetism in a topological edge state
• Glide-symmetric magnetic topological crystalline insulators with inversion symmetry
• Engineering second-order topological insulators via coupling two first-order topological insulators
• Quasicrystalline second-order topological semimetals
• Average Symmetry Protected Higher-order Topological Amorphous Insulators
• Strain engineered higher order topological phases for spin-3/2 Luttinger fermions
• Three-dimensional topological semimetal phase in layered TaNiTe5 probed by de Haas-van Alphen effect
• Braiding higher-order Majorana corner states through their spin degree of freedom
• Unified characterization for higher-order topological phase transitions
• Sublattice-enriched tunability of bound states in second-order topological insulators and superconductors
• Nodal higher-order topological superconductivity from a ${\mathcal{C}}_4$-symmetric Dirac semimetal
• Mirror-symmetry-protected dynamical quantum phase transitions in topological crystalline insulators
• Higher-order Klein bottle topological insulator in three-dimensional acoustic crystals
• Boundary Topological Superconductors
• Out-of-Time-Order Correlation as a Witness for Topological Phase Transitions
• Topological Crystalline Insulators with $C_2$ Rotation Anomaly
• Dynamical bulk boundary correspondence and dynamical quantum phase transitions in higher order topological insulators
• Universal higher-order bulk-boundary correspondence of triple nodal points
• Constructing higher-order topological states in higher dimension
• Chiral Majorana Hinge Modes in Superconducting Dirac Materials
• Interacting second-order topological insulators in one-dimensional fermions with correlated hopping
• Multiorbital edge and corner states in black phosphorene
• Density-driven higher-order topological phase transitions in amorphous solids
• Surface Bogoliubov-Dirac cones and helical Majorana hinge modes in superconducting Dirac semimetals
• Identifying Higher Order Topology and Fractional Corner Charge Using Entanglement Spectra
• Fractional corner charges in a 2D super-lattice Bose-Hubbard model
• The Bulk-Hinge Correspondence and Three-Dimensional Quantum Anomalous Hall Effect in Second Order Topological Insulators
• Robustness of Helical Hinge States of Weak Second-Order Topological Insulators
• A Dirac fermion model associated with second order topological insulator
• Lattice distortion induced first and second order topological phase transition in rectangular high-T${}_{\mathrm{c}}$ superconducting monolayer
• Topological hinge modes in Dirac semimetals
• Crystalline Responses for Rotation-Invariant Higher-Order Topological Insulators
• Higher-order Weyl superconductors with anisotropic Weyl-point connectivity
• Higher-order Topological Phases of Magnons in van der Waals Honeycomb Ferromagnets
• Higher-order topological superconductors from Weyl semimetals
• Higher-Order Topological Mott Insulator on the Pyrochlore Lattice
• Topological States in Two-Dimensional Su-Schrieffer-Heeger Models
• Phononic Stiefel-Whitney topology with corner vibrational modes in two-dimensional Xenes and ligand-functionalized derivatives
• Boundary edge networks induced by bulk topology
• Generalized Lieb-Schultz-Mattis theorem on bosonic symmetry protected topological phases
• Quantized surface magnetism and higher-order topology: Application to the Hopf insulator
• Hinge mode dynamics of periodically driven higher-order Weyl semimetals
• Interplay between boundary conditions and Wilson's mass in Dirac-like Hamiltonians
• Two-dimensional weak-type topological insulators in inversion symmetric crystals
• Second-Order Topological Insulator in van der Waals Heterostructures of CoBr${}_2$/Pt${}_2$HgSe${}_3$/CoBr${}_2$
• Engineering non-Hermitian Second Order Topological Insulator in Quasicrystals
• Topological states in a dimerized system with staggered magnetic fluxes
• Second-Order Topological Insulator in Two-Dimensional C2N and Its Derivatives
• Hybrid-order topological odd-parity superconductors via Floquet engineering
• Hinge states in a system of coupled Rashba layers
• Topological corner modes in a brick lattice with nonsymmorphic symmetry
• Majorana corner states on the dice lattice
• Role of the edges in a quasicrystalline Haldane model
• Signatures of the Majorana spin in electrical transport through a Majorana nanowire
• Sublattice-sensitive Majorana Modes
• Engineering topologically protected zero-dimensional interface end states in antiferromagnetic heterojunction graphene nanoflakes
• Light bullets in Su-Schrieffer-Heeger photonic topological insulators
• Generalization of Benalcazar-Bernevig-Hughes model to arbitrary dimensions
• Topological phases of the dimerized Hofstadter butterfly
• High-temperature Majorana corner modes in a $d+i{d}^{\prime }$ superconductor heterostructure: Application to twisted bilayer cuprate superconductors
• Odd-Parity Quasiparticle Interference in the Superconductive Surface State of UTe2
• Mirror real Chern insulator in two and three dimensions
• The General Principle behind Magnetization-induced Second-Order Topological Corner States in the Kane-Mele Model
• Floquet second-order topological Anderson insulator hosting corner localized modes
• Pressure-induced superconductivity in weak topological insulator BiSe
• Dephasing enhanced strong Majorana zero modes in 2D and 3D higher-order topological superconductors
• Dynamic bulk-boundary correspondence for anomalous Floquet topology
• Competition of First-Order and Second-Order Topology on the Honeycomb Lattice
• Correlation Effects in Quadrupole Insulators: a Quantum Monte Carlo Study
• Higher-order topological semimetals and nodal superconductors with an order-two crystalline symmetry
• Higher-order Topological Insulators and Semimetals in Three Dimensions without Crystalline Counterparts
• Higher order topological matter and fractional chiral states
• Dirac cones and higher-order topology in quasi-continuous media
• Topological nature of higher-order hinge states revealed by spin transport
• Effects of Disorder On Thouless Pumping In Higher-Order Topological Insulators
• Chemical Principles of Intrinsic Topological Superconductors
• Magnetic Flux Periodicity in Second Order Topological Superconductors
• Long-lived Andreev states as evidence for protected hinge modes in a bismuth nanoring Josephson junction
• Transport Induced Dimer State from Topological Corner States
• Persistence of Monoclinic Crystal Structure in Three-Dimensional Second-Order Topological Insulator Candidate 1T'-MoTe2 Thin Flake without Structural Phase transition
• Tailoring Quadrupole Topological Insulators with Periodic Driving and Disorder
• Dynamical Detection of Topological Spectral Density
• Topological phonon analysis of the 2D buckled honeycomb lattice: an application to real materials
• Interlayer Coupling Induced Topological Phase Transition to Higher Order
• A General Method to Design Acoustic Higher-Order Topological Insulators
• Topological Dipole Conserving Insulators and Multipolar Responses
• Non-Hermitian Origin of Detachable Boundary States in Topological Insulators
• Topological classification of intrinsic 3D superconductors using anomalous surface construction
• Observation of emergent Dirac physics at the surfaces of acoustic higher-order topological insulators
• Topological Edge and Corner States in Biphenylene Network
• Real-space Obstruction in Quantum Spin Hall Insulators
• Topological quantum transition driven by charge-phonon coupling in higher-order topological insulators
• Intertwined Weyl phases emergent from higher-order topology and unconventional Weyl fermions via crystalline symmetry
• Multiple higher-order topological phases with even and odd pairs of zero-energy corner modes in a $C_3$ symmetry broken model
• Meissner effect induced Majorana zero modes at small magnetic field
• Topological insulators and higher-order topological insulators from gauge-invariant 1D lines
• Aharonov-Bohm Effect in Three-dimensional Higher-order Topological Insulator
• Controlling the Deformation of Metamaterials: Corner Modes via Topology
• Topological and dynamical features of periodically driven spin ladders
• Abundance of second order topology in $C_3$ symmetric two-dimensional insulators
• Topological invariants beyond symmetry indicators: Boundary diagnostics for twofold rotationally symmetric superconductors
• Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology
• Network model for higher-order topological phases
• Rotational symmetry breaking and partial Majorana corner states in a high-Tc superconductor based heterostructure
• Hinge Modes and Surface States in Second-Order Topological Three-Dimensional Quantum Hall Systems induced by Charge Density Modulation
• Time evolution of Majorana corner modes in Floquet second-order topological superconductor
• Generating many Majorana corner modes and multiple phase transitions in Floquet second-order topological superconductors
• Magnetic field filtering of the hinge supercurrent in unconventional metal NiTe${}_2$-based Josephson junctions
• Third-order topological insulator in three-dimensional lattice of magnetic vortices
• Spontaneous magnetization in unitary superconductors with time reversal symmetry breaking
• Out of equilibrium chiral higher order topological insulator on a $\pi$-flux square lattice
• Protected Gapless Edge States In Trivial Topology
• Anderson Critical Metal Phase in Trivial States Protected by Average Magnetic Crystalline Symmetry
• Floquet higher-order topological phases in momentum space
• Topology and Magnetism in the Kondo Insulator Phase Diagram
• Rotoinversion-symmetric bulk-hinge correspondence and its applications to higher-order Weyl semimetals
• A semiclassical treatment of spinor topological effects in driven, inhomogeneous insulators under external electromagnetic fields
• Entanglement Signature of Hinge Arcs, Fermi Arcs, and Crystalline Symmetry Protection in Higher-Order Weyl Semimetals
• Green's Function Approach to Interacting Higher-order Topological Insulators
• Magnetic wallpaper Dirac fermions and topological magnetic Dirac insulators
• Multi-Dimensional Quantum Walks: a Playground of Dirac and Schrödinger Particles
• Quantum Monte Carlo study of topological phases on a spin analogue of Benalcazar-Bernevig-Hughes model
• Higher-order topological superconductivity in a topological metal 1T${}^{\prime }$-MoTe${}_2$
• Effect of local Coulomb interaction on Majorana corner modes: weak and strong correlation limits
• Topological Wannier cycles for the bulk and edges
• Realizing Corner States in Artificial Crystals Based on Topological Spin Textures
• Gapless higher-order topology and corner states in Floquet systems
• Transport features of topological corner states in honeycomb lattice with multihollow structure
• Classification of time-reversal symmetric topological superconducting phases for conventional pairing symmetries
• Higher-order topological insulators in a crisscross antiferromagnetic model
• Crystalline-electromagnetic responses of higher order topological semimetals
• Majorana corner modes and flat-band Majorana edge modes in superconductor/topological-insulator/superconductor junctions
• Majorana corner modes in unconventional monolayers of the 1T-PtSe2 family
• Higher-order topological superconductivity in monolayer WTe${}_2$ from repulsive interactions
• Mode-Shell correspondence, a unifying phase space theory in topological physics -- Part I: Chiral number of zero-modes
• Probing Chiral-Symmetric Higher-Order Topological Insulators with Multipole Winding Number
• Higher-order topological corner states induced solely by onsite potentials with mirror symmetry
• Multiorder topological superfluid phase transitions in a two-dimensional optical superlattice
• Topologically localized insulators
• Extrinsic topology of Floquet anomalous boundary states in quantum walks
• Berezinskii-Kosterlitz-Thouless localization-localization transitions in disordered two-dimensional quantized quadrupole insulators
• Corner modes in non-Hermitian next-nearest-neighbor hopping model
• Remote control of spin polarization of topological corner states
• Theory of Anomalous Floquet Higher-Order Topology: Classification, Characterization, and Bulk-Boundary Correspondence
• Direct demonstration of bulk-boundary correspondence in higher-order topological superconductors with chiral symmetry
• Many-body multipole index and bulk-boundary correspondence
• Topological Classical Systems with Generalized Chiral Symmetry
• Two-Dimensional Altermagnetic Iron Oxyhalides: Real Chern Topology and Valley-Spin-Lattice Coupling
• Unveiling Higher-Order Topology via Polarized Topological Charges
• Gapless Fermionic Systems as Phase-space Topological Insulators: Non-perturbative Results from Anomalies
• Second-order topological insulator under strong magnetic field: Landau levels, Zeeman effect, and magnetotransport
• Topological Insulators with Hybrid-order Boundary States
• Fractional second-order topological insulator from a three-dimensional coupled-wires construction
• Competing topological phases in a non-Hermitian time-reversal symmetry-broken Bernevig-Hughes-Zhang model
• Controlling Majorana modes by $p$-wave pairing in two-dimensional $p+id$ topological superconductors
• Second-order topological insulator in a coinless discrete-time quantum walk
• Takagi topological insulator with odd $\mathcal{P}\mathcal{T}$ pairs of corner states
• Hierarchical Topological States in Thermal Diffusive Networks
• Emergent metallicity at the grain boundaries of higher-order topological insulators
• Boundary obstructed topological superconductor in buckled honeycomb lattice under perpendicular electric field
• Fermionic tensor networks for higher order topological insulators from charge pumping
• XFe4Ge2 (X = Y, Lu) and Mn3Pt: Filling-enforced magnetic topological metals
• Engineering rich two-dimensional higher-order topological phases by flux and periodic driving
• Higher-order topology in twisted multilayer systems: a review
• Generic Approach to Intrinsic Magnetic Second-order Topological Insulators via Inverted $p-d$ Orbitals
• Interaction-Driven Topological Switch in a $P$-Band Honeycomb Lattice
• Majorana corner pairs in a two-dimensional $s$-wave cold atomic superfluid
• Magnetic impurities along the edge of a quantum spin Hall insulator: Realizing a one-dimensional AIII insulator
• A topological current divider
• Kitaev Building-block Construction for Inversion-Protected Higher-order Topological Superconductors
• Emergence of Floquet edge states in the coupled Su-Schrieffer-Heeger model
• Additive rule of real and reciprocal space topologies at disclinations
• Chiral Hinge Transport in Disordered Non-Hermitian Second-Order Topological Insulators
• Tunable Majorana corner modes by orbital-dependent exchange interaction in a two-dimensional topological superconductor
• Second-order topology and supersymmetry in two-dimensional 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
• Two-channel Kondo problem in coupled interacting helical liquids
• Higher-order topological superconductor phases in a multilayer system
• Family of third-order topological insulators from Su-Schrieffer-Heeger stacking
• Fluctuating Ensemble Averages and the BTZ Threshold
• Unconventional hybrid-order topological insulators
• Zero-Energy State Localized near an Arbitrary Edge in Quadrupole Topological Insulators
• Topological characteristics of gap closing points in nonlinear Weyl semimetals
• Scattering theory of higher order topological phases
• Second-order topological insulator in Bilayer borophene
• Model non-Hermitian topological operators without skin effect: A general principle of construction
• Exceptional second-order topological insulators
• Strain-induced two-dimensional topological crystalline insulator
• Optical Berry Curvature of Hinge State and Its Detection
• Atiyah-Hirzebruch spectral sequence for topological insulators and superconductors: $E_2$ pages for 1651 magnetic space groups
• Hinge States of Second-Order Topological Insulators as a Mach-Zehnder Interferometer
• Higher-order topological superconductors characterized by Fermi level crossings
• Recipe for higher-order topology on the triangular lattice
• Stability against contact interactions of a topological superconductor in two-dimensional space protected by time-reversal and reflection symmetries
• Corner Modes and Ground-State Degeneracy in Models with Gauge-Like Subsystem Symmetries
• Majorana corner states in square and Kagome quantum spin liquids
• Theoretical analysis of glide-Z_2 magnetic topological photonic crystals
• Majorana corner states in an attractive quantum spin Hall insulator with opposite in-plane Zeeman energy at two sublattice sites
• Second-order topological insulator in periodically driven lattice
• Coexisting surface states in the weak and crystalline topological insulator Bi_2TeI
• Mirror-symmetry protected higher-order topological zero-frequency boundary and corner modes in Maxwell lattices
• Cavity-induced topological edge and corner states
• Photonic quadrupole topological phases in zero-dimensional cavity with synthetic dimensions
• Realization of a three-dimensional quantum Hall effect in a Zeeman-induced second order topological insulator on a torus
• Tunable symmetry-protected higher-order topological states with fermionic atoms in bilayer optical lattices
• Finite-temperature topological invariant for higher-order topological insulators
• Topological States in Chevrel Phase Materials from First-principle Calculations
• Topological semimetal with coexisting nodal points and nodal lines
• Revealing Higher-Order Topological Bulk-boundary Correspondence in Bismuth Crystal with Spin-helical Hinge State Loop and Proximity Superconductivity
• Tunable corner states in topological insulators with long-range hoppings and diverse shapes
• Topologically trivial gap-filling in superconducting Fe(Se,Te) by one dimensional defects
• Persistent Corner Spin Mode at the Quantum Critical Point of a Plaquette Heisenberg Model
• Mixed-parity octupolar pairing and corner Majorana modes in three dimensions
• Higher order topological state induced by $d$-wave competing orders in high-T${}_c$ superconductor based heterostructure
• Observation of dislocation-induced topological modes in a three-dimensional acoustic topological insulator
• Topological parafermion corner states in clock-symmetric non-Hermitian second-order topological insulator
• Anisotropic Penetration Depths of Corner States in a Higher-Order Topological Insulator
• 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
• Hinge Majorana Flat Band in Type-II Dirac Semimetals
• Engineering Majorana corner modes from two-dimensional hexagonal crystals
• Surface theory of a second-order topological insulator beyond the Dirac approximation
• Arboreal Obstructed Atomic Insulating and Metallic Phases of Fermions
• Coexistence of gapless and gapped vortex modes with Majorana corner states in a 2D second-order topological superconductor
• Floquet-Weyl semimetals generated by an optically resonant interband-transition
• Effects of the Vertices on the Topological Bound States in a Quasicrystalline Topological Insulator
• Third-order Orbital Corner State and its Realization in Acoustic Crystals
• Fusion Rules of Majorana-Kramer-Pairs in Time-Reversal-Invariant Topological Superconductors
• Topological defect-mediated corner states and higher-order bulk topology in a two-dimensional crystalline insulator
• Boundary topological insulators and superconductors of Altland-Zirnbauer tenfold classes
• Multi-locational Majorana Zero Modes
• Higher-order topological insulators in two-dimensional antiferromagnetic and altermagnetic chromium-based group-IV chalcogenides
• Extended Haldane model -- a modern gateway to topological insulators
• Intrinsic higher-order topological states in 2D honeycomb Z_2 quantum spin Hall insulators
• Fractional chiral second-order topological insulator from a three-dimensional array of coupled wires
• Euler band topology in superfluids and superconductors
• Anomalous gapped boundaries between surface topological orders in higher-order topological insulators and superconductors with inversion symmetry
• Current switching behavior mediated via hinge modes in higher-order topological phases using altermagnets
• Tunable cornerlike states in topological type-II hyperbolic lattices
• Engineering Two-Dimensional Hybrid-Order Topological Insulators via Trilayer Coupling
• Interaction-Induced Higher-Order Topological Insulator via Floquet Engineering
• Magnetic-flux tunable electronic transport through domain walls in a three-dimensional second-order topological insulator
• Wilson Loop and Topological Properties in 3D Woodpile Photonic Crystal
• Photonic Higher-Order Topological States Induced by Long Range Interactions
• Symmetry-protected gates of Majorana qubits in a high-$T_c$ superconductor platform
• Theory of Topological Superconductivity in Doped IV-VI Semiconductors
• Obstructed Atomic Insulators and Superfluids of Fermions Coupled to ${\mathbb{Z}}_2$ Gauge Fields
• Higher-Order Topology in Monolayer FeSe
• Higher-order Topological Knots and the classification of non-Hermitian lattices under $C_n$ symmetry
• Network model for magnetic higher-order topological phases
• Gapless States Localized along a Staircase Edge in Second-Order Topological Insulators
• Higher-order topological phases protected by noninvertible and subsystem symmetries
• Weak ${\mathbb{Z}}_2$ Supertopology
• Topological squashed entanglement: nonlocal order parameter for one-dimensional topological superconductors
• Coupled-wire construction of non-Abelian higher-order topological phases
• Wilson loop invariants and bulk-boundary correspondence in higher-order topological insulators with two anticommuting mirror symmetries
• Nanoimprinted topological laser in the visible
• Boundary Condition Analysis of First and Second Order Topological Insulators
• Multi-dimensional wave steering with higher-order topological phononic crystal
• Emergent One-Dimensional Helical Channel in Higher-Order Topological Insulators with Step Edges
• Pseudospin-valley-coupled phononic topological insulator with edge and corner states
• Topological energy pumping in a quasi-periodically driven four-level system
• Higher-order Quantum Spin Hall Effect of Light
• Tunable effective length of fractional Josephson junctions
• Dynamical Symmetry Indicators for Floquet Crystals
• Non-Hermitian Higher-Order Dirac Semimetals
• Non-Hermitian Higher-Order Weyl Semimetals
• Classification of topological insulators and superconductors with multiple order-two point group symmetries
• Graphyne as a second-order and real Chern topological insulator in two dimensions
• Projected branes as platforms for crystalline, superconducting, and higher-order topological phases
• Bulk-boundary correspondence in topological two-dimensional non-Hermitian systems: Toeplitz operators and singular values
• Hinge modes of three-dimensional Euler insulators
• Floquet second-order topological insulator in strained graphene
• Second-order topological insulator induced by compensated altermagnetism without bulk spin splitting
• Higher-order topological insulator in cubic semiconductor quantum wells
• Electronic structure and hinge states of strained half-Heusler compounds LiSbZn and LiBiZn
• Corner Flat Bands Induced by $d$-Density Wave and Partial Corner State Modification Due to Competition with $d$-Wave Superconductivity
• Magnetic-Field Tunable Möbius and Higher-Order Topological Insulators in Three-Dimensional Layered Octagonal Quasicrystals
• Semimetals without correspondence between the topological charge of nodal line/surface and Fermi arc
• Majorana vortex phases in time-reversal invariant higher-order topological insulators and topologically trivial insulators
• Hofstadter Butterflies in Topological Insulators
• Proposal for realizing and probing topological crystalline insulators in optical lattices