Higher-order topological phases in crystalline and non-crystalline systems: a review

Abstract

In recent years, higher-order topological phases have attracted great interest in various fields of physics. These phases have protected boundary states at lower-dimensional boundaries than the conventional first-order topological phases due to the higher-order bulk-boundary correspondence. In this review, we summarize current research progress on higher-order topological phases in both crystalline and non-crystalline systems. We firstly introduce prototypical models of higher-order topological phases in crystals and their topological characterizations. We then discuss effects of quenched disorder on higher-order topology and demonstrate disorder-induced higher-order topological insulators. We also review the theoretical studies on higher-order topological insulators in amorphous systems without any crystalline symmetry and higher-order topological phases in nonperiodic lattices including quasicrystals, hyperbolic lattices, and fractals, which have no crystalline counterparts. We conclude the review by a summary of experimental realizations of higher-order topological phases and discussions on potential directions for future study.

arXiv comment52 pages, 27 figures. An invited topical review to Journal of Physics: Condensed Matter. Comments are welcome

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• Topological classification under nonmagnetic and magnetic point group symmetry: Application of real-space Atiyah-Hirzebruch spectral sequence to higher-order topology
• Bound states at partial dislocation defects in multipole higher-order topological insulators
• Higher order topological superconductors as generators of quantum codes
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• Topological Anderson insulator phase in a quasicrystal lattice
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• Reducing electronic transport dimension to topological hinge states by increasing geometry size of Dirac semimetal Josephson junctions
• Higher-order topological insulators in hyperbolic lattices
• Non-Commutative Chern Numbers for Generic Aperiodic Discrete Systems
• Realization of a Z classified chiral-symmetric higher-order topological insulator in a coupling-inversion acoustic crystal
• Quasicrystalline Chern Insulators
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• Simulation of Higher-Order Topological Phases and Related Topological Phase Transitions in a Superconducting Qubit
• Photonic Chern insulator through homogenization of an array of particles
• Amorphous topological matter: theory and experiment
• Symmetry-protected Topological Phases at Finite Temperature
• Anyons and Fractional Quantum Hall Effect in Fractal Dimensions
• Three-dimensional higher-order topological acoustic system with multidimensional topological states
• Type-II quadrupole topological insulators
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• Many-Body Invariants for Chern and Chiral Hinge Insulators
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• Geometry and Topology Tango in Ordered and Amorphous Chiral Matter
• Braiding higher-order Majorana corner states through their spin degree of freedom
• Unified characterization for higher-order topological phase transitions
• Application of Convolutional Neural Network to Quantum Percolation in Topological Insulators
• Quasicrystalline second-order topological semimetals
• Average Symmetry Protected Higher-order Topological Amorphous Insulators
• Higher-order Klein bottle topological insulator in three-dimensional acoustic crystals
• Mirror-symmetry-protected dynamical quantum phase transitions in topological crystalline insulators
• Dynamical bulk boundary correspondence and dynamical quantum phase transitions in higher order topological insulators
• Multiorbital edge and corner states in black phosphorene
• Density-driven higher-order topological phase transitions in amorphous solids
• Robustness of Helical Hinge States of Weak Second-Order Topological Insulators
• Topological hinge modes in Dirac semimetals
• Floquet topological superconductors with many Majorana edge modes: topological invariants, entanglement spectrum and bulk-edge correspondence
• Topological Insulator in an Atomic Liquid
• Crystalline Responses for Rotation-Invariant Higher-Order Topological Insulators
• Structural disorder-driven topological phase transition in noncentrosymmetric BiTeI
• Boundary edge networks induced by bulk topology
• Intrinsically Interacting Higher-Order Topological Superconductors
• Topological transitions and Anderson localization of light in disordered atomic arrays
• Second-Order Topological Insulator in van der Waals Heterostructures of CoBr${}_2$/Pt${}_2$HgSe${}_3$/CoBr${}_2$
• Second-Order Topological Insulator in Two-Dimensional C2N and Its Derivatives
• Delocalization of boundary states in disordered topological insulators
• Role of the edges in a quasicrystalline Haldane model
• Detecting Bulk Topology of Quadrupolar Phase from Quench Dynamics
• Competition of First-Order and Second-Order Topology on the Honeycomb Lattice
• Effects of Disorder On Thouless Pumping In Higher-Order Topological Insulators
• Topological nature of higher-order hinge states revealed by spin transport
• Higher-order topological semimetals and nodal superconductors with an order-two crystalline symmetry
• Classification and construction of interacting fractonic higher-order topological phases
• Structural spillage: an efficient method to identify non-crystalline topological materials
• Third-order topological insulator induced by disorder
• Effect of coordination on topological phases on self-similar structures
• Strongly correlated crystalline higher-order topological phases in two-dimensional systems: A coupled-wire study
• Topological diffusive metal in amorphous transition metal monosilicides
• Temperature-driven gapless topological insulator
• Replica Higher-Order Topology of Hofstadter Butterflies in Twisted Bilayer Graphene
• Engineering of Chern insulators and circuits of topological edge states
• Multiple higher-order topological phases with even and odd pairs of zero-energy corner modes in a $C_3$ symmetry broken model
• 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
• Wannier Topology and Quadrupole Moments for a generalized Benalcazar-Bernevig-Hughes Model
• Disordered topological graphs enhancing nonlinear phenomena
• Floquet higher-order topological phases in momentum space
• Higher-order topological insulator in a modified Haldane-Hubbard model
• Green's Function Approach to Interacting Higher-order Topological Insulators
• Quantum Hall effect and Landau levels without spatial long-range correlations
• Disorder-Induced Phase Transitions in Three-Dimensional Chiral Second-Order Topological Insulator
• Many-body multipole index and bulk-boundary correspondence
• Theory of Anomalous Floquet Higher-Order Topology: Classification, Characterization, and Bulk-Boundary Correspondence
• Entanglement Signatures of Multipolar Higher Order Topological Phases
• Bulk-Vortex Correspondence of Higher-Order Topological Superconductors
• Kitaev Building-block Construction for Inversion-Protected Higher-order Topological Superconductors
• Quadrupole Insulator without Corner States in the Energy Spectrum
• Protecting Quantum Superposition and Entanglement with Photonic Higher-Order Topological Crystalline Insulator
• Effects of the Vertices on the Topological Bound States in a Quasicrystalline Topological Insulator
• Elastic fractal higher-order topological states

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