Topological classification of crystalline insulators through band structure combinatorics

Abstract

We present a method for efficiently enumerating all allowed, topologically distinct, electronic band structures within a given crystal structure. The algorithm applies to crystals with broken time-reversal, particle-hole, and chiral symmetries in any dimension. The presented results match the mathematical structure underlying the topological classification of these crystals in terms of K-theory, and therefore elucidate this abstract mathematical framework from a simple combinatorial perspective. Using a straightforward counting procedure, we classify the allowed topological phases in any possible two-dimensional crystal in class A. We also show how the same procedure can be used to classify the allowed phases for any three-dimensional space group. Employing these classifications, we study transitions between topological phases within class A that are driven by band inversions at high symmetry points in the first Brillouin zone. This enables us to list all possible types of phase transitions within a given crystal structure, and identify whether or not they give rise to intermediate Weyl semimetallic phases.

24 pages, 9 figures

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• Characterization of higher-order topological superconductors using Bott indices
• Second Euler number in four dimensional synthetic matter
• Localization anisotropy and complex geometry in two-dimensional insulators
• Kitaev Building-block Construction for Inversion-Protected Higher-order Topological Superconductors
• Unconventional hybrid-order topological insulators
• Effects of first- and second-order topological phases on equilibrium crystal shapes
• Isolated nearly flat higher Chern band in monolayer transition metal trihalides
• Atiyah-Hirzebruch spectral sequence for topological insulators and superconductors: $E_2$ pages for 1651 magnetic space groups
• Topological Signatures in Nodal Semimetals through Neutron Scattering
• Topological Descendants of Multicritical EuTl2
• Discriminant indicator with generalized rotational symmetry
• Evidence of phase stability, topological phonon and temperature-induced topological phase transition in rocksalt SnS and SnSe
• Recipe for higher-order topology on the triangular lattice
• Correlated Zak insulator in organic antiferromagnets
• Higher-order topological superconductors characterized by Fermi level crossings
• Broken time-reversal symmetry in cubic skutterudite-like superconductor Y${}_3$Ru${}_4$Ge${}_{13}$
• Layer Construction of Three-Dimensional Z2 Monopole Charge Nodal Line Semimetals and prediction of the abundant candidate materials
• General corner charge formulas in various tetrahedral and cubic space groups
• Polarization textures in crystal supercells with topological bands
• Topological insulators on fractal lattices: A general principle of construction
• Quasiperiodicity protects quantized transport in disordered systems without gaps
• Hong-Ou-Mandel Interference in a temporal-average-inversion-symmetric chain
• Nonreciprocal Weyl semimetal waveguide
• Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators
• Topological Transitions in a Model for Proximity Induced Superconductivity
• Tuning topological surface states by cleavage angle in topological crystalline insulators
• Quantum geometric bounds in spinful systems with trivial band topology
• Enhancing the Hyperpolarizability of Crystals with Quantum Geometry
• Visualising the connection between edge states and the mobility edge in adiabatic and non-adiabatic topological charge transport
• Projective Representations, Bogomolov Multiplier, and Their Applications in Physics
• Exciton Berryology
• Topological phase in the extended Haldane-Hubbard model with sublattice-dependent repulsion
• Non-Hermitian ${\mathbb{Z}}_4$ skin effect protected by glide symmetry
• The destiny of obstructed atomic insulator under correlation
• Topological Anderson insulators by latent symmetry
• Takagi Topological Insulator on the Honeycomb Lattice
• Design of a room-temperature topological exciton-polariton laser in a ZnO/TiO${}_2$-photonic crystal slab
• Electronic Crystal Phases in the Presence of Non-Uniform Berry Curvature and Tunable Berry Flux: The ${\lambda }_N$-Jellium model
• Higher-order topological phases for time-reversal-symmetry breaking superconductivity in UTe${}_2$
• Representation Theory for Massless Quasiparticles in Bogoliubov-de Gennes Systems
• Mapping rules from nodal line semimetal to topological crystalline insulator in face centered cubic lattice
• Simple realization of a fragile topological lattice with quasi flat-bands in a microcavity array
• Non-trivial topological valence bands of common diamond and zinc-blende semiconductors
• Phase diagram of the interacting Haldane model with spin-dependent sublattice potentials
• Tight-binding model with sublattice-asymmetric spin-orbit coupling for square-net nodal line Dirac semimetals
• Topologically-enhanced exciton transport
• Crystalline invariants of fractional Chern insulators
• Multiband superconductivity in the topological Kramers nodal-line semimetals
• Dynamic melting and condensation of topological dislocation modes
• Hybrid symmetry class topological insulators
• Theory of fractional corner charges in cylindrical crystal shapes
• Controlling topology through targeted composite symmetry manipulation in magnetic systems
• Effective field theories for interacting boundaries of 3D topological crystalline insulators through bosonisation
• One-dimensional ${\mathbb{Z}}_4$ topological superconductor
• Microscopic Green's function approach for generalized Dirac Hamiltonians
• Homotopy invariant in time-reversal and twofold rotation symmetric systems
• From quantum geometry to non-linear optics and gerbes: Recent advances in topological band theory
• Intrinsic (Axion) Statistical Topological Insulator
• Selective branching, quenching, and converting of topological modes
• Hall effect of ferro/antiferromagnetic wallpaper fermions
• Topological Micro-Electro-Mechanical Systems
• Field-driven band asymmetry and non-reciprocal transport in a helimagnet
• Correlated interaction effects in three-dimensional semi-Dirac semimetal
• Hermitian and non-Hermitian topology in active matter
• Band structure picture for topology in strongly correlated systems with the ghost Gutzwiller ansatz
• Symmetry Enforced Dirac Points in Antiferromagnetic Semiconductors
• Realization of Hopf-link structure in phonon spectra: Symmetry guidance and High-throughput investigation
• Gapless Topological Peierls-like instabilities in more than one dimension
• Global and Local Topological Crystalline Markers for Rotation-Symmetric Insulators
• Topological insulators and geometry of vector bundles
• Magnon-mediated superconductivity on ferromagnetic wallpaper fermions
• Wavefunction textures in twisted bilayer graphene from first principles
• High-chirality and multiquaternion Weyl nodes in hexagonal ReO${}_3$
• Interaction-driven phase transition in one dimensional mirror-symmetry protected topological insulator
• Fermiology and transport properties of the proposed topological crystalline insulator SrAg4Sb2
• Hilbert band complexes and their applications
• Topological charge excitations and Green's function zeros in paramagnetic Mott insulators
• Atomically thin obstructed atomic insulators with robust edge modes and quantized spin Hall effect
• Anomalous multi-gap topological phases in periodically driven quantum rotors
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Role of effective mass anisotropy in realizing a hybrid nodal-line fermion state
• Breakdown of boundary criticality and exotic topological semimetals in $\mathcal{P}\mathcal{T}$-invariant systems
• Perfectly localized Majorana corner modes in fermionic lattices
• Spinless topological chirality from Umklapp scattering in twisted 3D structures
• Visualizing topological transport
• Interacting Virtual Topological Phases in Defect-Rich 2D Materials
• Reentrant topological phase in half-Heusler compounds
• Predicting Many Crystal Properties via an Adaptive Transformer-based Framework
• Network model for magnetic higher-order topological phases
• Nodal higher-order topological superconductivity from C6-symmetric Dirac semimetals
• Disorder-induced delocalization and reentrance in a Chern-Hopf insulator
• Universal Broadening of Zero Modes: A General Framework and Identification
• Spacetime symmetry indicators for two-dimensional Floquet topological insulators
• Higher-order Topological Knots and the classification of non-Hermitian lattices under $C_n$ symmetry
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Topological phases protected by projective space-time inversion symmetry in alkaline-earth-metal-like atoms
• Anomalous wave-packet transport on boundaries of Floquet topological systems
• Disclinations, dislocations, and emanant flux at Dirac criticality
• Predicted versatile topological nodal magnons in Tb-based icosahedral quasicrystal 1/1 approximants
• Impurity-induced Mott ring states and Mott zeros ring states in the Hubbard operator formalism
• Anomalous gapped boundaries between surface topological orders in higher-order topological insulators and superconductors with inversion symmetry
• Topological band insulators without translational symmetry
• Preserving Topology while Breaking Chirality: From Chiral Orthogonal to Anti-symmetric Hermitian Ensemble
• Laser-induced topological phases in monolayer amorphous carbon
• Semi-Dirac spin liquids and frustrated quantum magnetism on the trellis lattice
• Topological Phases in Non-Hermitian Nonlinear-Eigenvalue Systems
• Fragile topological insulators protected by rotation symmetry without spin-orbit coupling
• Local expression of fractional corner charges in obstructed atomic insulators and relationship with the fractional disclination charges
• Majorana Zero Modes and Topological Nature in Bi2Ta3S6-family Superconductors
• Fragile dislocation modes in obstructed atomic topological phases
• Hybrid Higher-Order Skin Topological Modes in the Two-Dimensional Su-Schrieffer-Heeger Model with Nonreciprocal Hoppings
• Crystalline finite-size topology
• Topological Invariants of a Filling-Enforced Quantum Band Insulator
• Accelerated Discovery of Topological Conductors for Nanoscale Interconnects
• One-dimensional topology and topolectrics of nonsymmorphic Kramers degenerate systems
• Non-self-averaging topological Anderson insulator
• Proposal for realizing and probing topological crystalline insulators in optical lattices
• Graphyne as a second-order and real Chern topological insulator in two dimensions
• Wilson loop invariants and bulk-boundary correspondence in higher-order topological insulators with two anticommuting mirror symmetries
• Systematic study for two-dimensional $Z_2$ topological phase transitions at high-symmetry points in all layer groups
• Efficient prediction of topological superlattice bands with spin-orbit coupling
• Bulk-boundary correspondence in topological two-dimensional non-Hermitian systems: Toeplitz operators and singular values
• Orbital magnetization reveals multiband topology
• Competitive Orders in Altermagnetic Chiral Magnons
• Symmetry-Based Real-Space Framework for Realizing Flat Bands and Unveiling Nodal-Line Touchings
• Glide-resolved photoemission spectroscopy: Measuring topological invariants in nonsymmorphic space groups
• Crystalline Weyl semimetal phase in Quantum Spin Hall systems under magnetic fields
• The mixing-spacetime symmetry in the Floquet-Bloch band theory
• Exploiting $\vartheta -$functions for the identification of topological materials
• Squeezing Quantum States in Three-Dimensional Twisted Crystals
• Three-dimensional spinless Euler insulators with rotational symmetry
• Projected branes as platforms for crystalline, superconducting, and higher-order topological phases
• Symmetry-protected topological wire in a topological vacuum
• Dynamics of 2D topological quadrupole insulator and Chern insulator induced by real-space topological changes
• Detection of 2D SPT Order with Partial Symmetries
• Non-unique connection between bulk topological invariants and surface physics
• Reduced Wannier representation for topological bands
• Physical Pictures for Quasisymmetry in Crystals
• Topological property of graphene with triangular array of nanoholes
• Gapped spinful phases obtained via Gutzwiller projections of Euler states
• Effect of superconductivity by Nb and V substitution in kagome CaPd5