$(d-2)$-dimensional edge states of rotation symmetry protected topological states

Abstract

We study fourfold rotation invariant gapped topological systems with time-reversal symmetry in two and three dimensions ($d=2,3$). We show that in both cases nontrivial topology is manifested by the presence of the $(d-2)$-dimensional edge states, existing at a point in 2D or along a line in 3D. For fermion systems without interaction, the bulk topological invariants are given in terms of the Wannier centers of filled bands, and can be readily calculated using a Fu-Kane-like formula when inversion symmetry is also present. The theory is extended to strongly interacting systems through explicit construction of microscopic models having robust $(d-2)$-dimensional edge states.

v5, accepted in prl, the explicit Z8 Fu-Kane-like formulas provided

References in corpus (7)

• Topological Crystalline Insulators
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• Reflection symmetric second-order topological insulators and superconductors
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• Wannier representation of Z_2 topological insulators
• Quantum Spin Hall Effect in Strip of Stripes Model
• Integer and Fractional Quantum Anomalous Hall Effect in a Strip of Stripes Model

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• Theoretical analysis of glide-Z_2 magnetic topological photonic crystals
• Second-order topological insulator in periodically driven lattice
• Atiyah-Hirzebruch spectral sequence for topological insulators and superconductors: $E_2$ pages for 1651 magnetic space groups
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• Hinge States of Second-Order Topological Insulators as a Mach-Zehnder Interferometer
• Higher-order topological superconductors characterized by Fermi level crossings
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• Second-order topological insulator in Bilayer borophene
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• Topological characteristics of gap closing points in nonlinear Weyl semimetals
• Recipe for higher-order topology on the triangular lattice
• Higher-order topology in honeycomb lattice with Y-Kekulé distortions
• Theory of fractional corner charges in cylindrical crystal shapes
• Homotopy invariant in time-reversal and twofold rotation symmetric systems
• Topological parafermion corner states in clock-symmetric non-Hermitian second-order topological insulator
• Observation of dislocation-induced topological modes in a three-dimensional acoustic topological insulator
• 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
• Cavity-induced topological edge and corner states
• Mirror-symmetry protected higher-order topological zero-frequency boundary and corner modes in Maxwell lattices
• Intertwined topological phases in TaAs2 nanowires with giant magnetoresistance and quantum coherent surface transport
• Tunable corner states in topological insulators with long-range hoppings and diverse shapes
• Unconventional Topological Insulators from Extended Topological Band Degeneracies
• Hinge Majorana Flat Band in Type-II Dirac Semimetals
• Floquet topological phases with time-reversal and space inversion symmetries and dynamical detection of topological charges
• Realization of a three-dimensional quantum Hall effect in a Zeeman-induced second order topological insulator on a torus
• Finite-temperature topological invariant for higher-order topological insulators
• Selective branching, quenching, and converting of topological modes
• Persistent Corner Spin Mode at the Quantum Critical Point of a Plaquette Heisenberg Model
• Revealing Higher-Order Topological Bulk-boundary Correspondence in Bismuth Crystal with Spin-helical Hinge State Loop and Proximity Superconductivity
• Tunable symmetry-protected higher-order topological states with fermionic atoms in bilayer optical lattices
• Perfectly localized Majorana corner modes in fermionic lattices
• Multi-locational Majorana Zero Modes
• Topological defect-mediated corner states and higher-order bulk topology in a two-dimensional crystalline insulator
• Floquet-Anderson localization in the Thouless pump and how to avoid it
• Spin separation and filtering assisted by topological corner states in the Kekulé lattice
• Chiral Majorana Edge Modes and Vortex Majorana Zero Modes in Superconducting Antiferromagnetic Topological Insulator
• Topological Aspects of Dirac Fermions in a Kagomé Lattice
• Second-order topological insulator and fragile topology in topological circuitry simulation
• Arboreal Obstructed Atomic Insulating and Metallic Phases of Fermions
• Third-order Orbital Corner State and its Realization in Acoustic Crystals
• Global and Local Topological Crystalline Markers for Rotation-Symmetric Insulators
• Surface theory of a second-order topological insulator beyond the Dirac approximation
• Observation of Square-Root Higher-Order Topological States in Photonic Waveguide Arrays
• Boundary topological insulators and superconductors of Altland-Zirnbauer tenfold classes
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Anomalous gapped boundaries between surface topological orders in higher-order topological insulators and superconductors with inversion symmetry
• Photonic Higher-Order Topological States Induced by Long Range Interactions
• 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
• The Quadrupole Moment of Higher-Order Topological Insulator at Finite temperature
• 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
• Higher order topology in a Creutz ladder
• Tunable cornerlike states in topological type-II hyperbolic lattices
• Engineering Two-Dimensional Hybrid-Order Topological Insulators via Trilayer Coupling
• 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
• Zeeman field induced corner states in the Kane-Mele-Hubbard model
• Wilson Loop and Topological Properties in 3D Woodpile Photonic Crystal
• Network model for magnetic higher-order topological phases
• Symmetry-protected gates of Majorana qubits in a high-$T_c$ superconductor platform
• Hybrid Higher-Order Skin Topological Modes in the Two-Dimensional Su-Schrieffer-Heeger Model with Nonreciprocal Hoppings
• Theory of Topological Superconductivity in Doped IV-VI Semiconductors
• Higher-Order Topology in Monolayer FeSe
• Hidden wallpaper fermion and third-order topological insulator
• Gapless States Localized along a Staircase Edge in Second-Order Topological Insulators
• Interaction-Induced Higher-Order Topological Insulator via Floquet Engineering
• Obstructed Atomic Insulators and Superfluids of Fermions Coupled to ${\mathbb{Z}}_2$ Gauge Fields
• Higher-order Topological Knots and the classification of non-Hermitian lattices under $C_n$ symmetry
• Realization of second-order photonic square-root topological insulators
• Higher-order topological insulator in cubic semiconductor quantum wells
• Dynamical Symmetry Indicators for Floquet Crystals
• Proposal for realizing and probing topological crystalline insulators in optical lattices
• Graphyne as a second-order and real Chern topological insulator in two dimensions
• Hofstadter Butterflies in Topological Insulators
• Deconfined Quantum Phase Transition of a Higher-Order Symmetry-Protected Topological State
• Robust zero-energy states in two-dimensional Su-Schrieffer-Heeger topological insulators
• Magnetic-Field Tunable Möbius and Higher-Order Topological Insulators in Three-Dimensional Layered Octagonal Quasicrystals
• Floquet second-order topological insulator in strained graphene
• Hinge modes of three-dimensional Euler insulators
• Bulk-boundary correspondence in topological two-dimensional non-Hermitian systems: Toeplitz operators and singular values
• Coupled-wire construction of non-Abelian higher-order topological phases
• Majorana vortex phases in time-reversal invariant higher-order topological insulators and topologically trivial insulators
• Nanoimprinted topological laser in the visible
• Topological energy pumping in a quasi-periodically driven four-level system
• Classification of topological insulators and superconductors with multiple order-two point group symmetries
• Higher-order topological phases protected by noninvertible and subsystem symmetries
• Boundary Condition Analysis of First and Second Order Topological Insulators
• Projected branes as platforms for crystalline, superconducting, and higher-order topological phases
• Electronic structure and hinge states of strained half-Heusler compounds LiSbZn and LiBiZn
• Emergent One-Dimensional Helical Channel in Higher-Order Topological Insulators with Step Edges
• Topological Photonics on a Small Scale
• Non-Hermitian Higher-Order Weyl Semimetals
• Non-Hermitian Higher-Order Dirac Semimetals
• Corner Flat Bands Induced by $d$-Density Wave and Partial Corner State Modification Due to Competition with $d$-Wave Superconductivity