Crystallographic splitting theorem for band representations and fragile topological photonic crystals

Abstract

The fundamental building blocks in band theory are band representations (BRs): bands whose infinitely-numbered Wannier functions are generated (by action of a space group) from a finite number of symmetric Wannier functions centered on a point in space. This work aims to simplify questions on a multi-rank BR by splitting it into unit-rank bands, via the following crystallographic splitting theorem: being a rank-$N$ BR is equivalent to being splittable into a finite sum of bands indexed by $\{1,2,\dots ,N\}$, such that each band is spanned by a single, analytic Bloch function of $k$, and any symmetry in the space group acts by permuting $\{1,2,\dots ,N\}$. Applying this theorem, we develop computationally efficient methods to determine whether a given energy band (of a tight-binding or Schrödinger Hamiltonian) is a BR, and, if so, how to numerically construct the corresponding symmetric Wannier functions. Thus we prove that rotation-symmetric topological insulators in class AI are fragile, meaning that the obstruction to symmetric Wannier functions is removable by addition of BRs. An implication of fragility is that its boundary states, while robustly covering the bulk energy gap in finite-rank tight-binding models, are unstable if the Hilbert space is expanded to include all symmetry-allowed representations. These fragile insulators have photonic analogs that we identify; in particular, we prove that an existing photonic crystal built by Yang et al. [Nature 565, 622 (2019)] is fragile topological with removable surface states, which disproves a widespread perception of 'topologically-protected' surface states in time-reversal-invariant, gapped photonic/phononic crystals. Our theorem is finally applied to derive various symmetry obstructions on the Wannier functions of topological insulators, and to prove their equivalence with the nontrivial holonomy of Bloch functions.

53 pages, 11 figures. We now clarify how certain band representations of finite-rank tight-binding Hamiltonians can evade the tight-binding (or 'atomic') limit. The symmetry-indicator analysis of a P3-symmetric model is corrected

References in corpus (98)

• Quantum Spin Hall Effect in Graphene
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Maximally-localized generalized Wannier functions for composite energy bands
• Electronic Structure of Pyrochlore Iridates: From Topological Dirac Metal to Mott Insulator
• Topological Insulators in Three Dimensions
• A topological Dirac insulator in a quantum spin Hall phase : Experimental observation of first strong topological insulator
• Topological invariants of time-reversal-invariant band structures
• Topological Crystalline Insulators
• First observation of Spin-Momentum Helical Locking in Bi2Se3 and Bi2Te3, demonstration of Topological-Order at 300K and a realization of topological-transport-regime
• Quantized Electric Multipole Insulators
• Photonic Analogue of Two-dimensional Topological Insulators and Helical One-Way Edge Transport in Bi-Anisotropic Metamaterials
• Periodic table for topological insulators and superconductors
• Scheme to Achieve Silicon Topological Photonics
• Higher-Order Topological Insulators
• Topological quantum chemistry
• Robust optical delay lines via topological protection
• Topological Crystalline Insulators in the SnTe Material Class
• Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals
• Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase
• The (High Quality) Topological Materials In The World
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Direct Measurement of the Zak phase in Topological Bloch Bands
• Experimental realization of a topological crystalline insulator in SnTe
• Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• An equivalent expression of Z2 Topological Invariant for band insulators using Non-Abelian Berry's connection
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Surface States of the Topological Insulator Bi_{1-x}Sb_x
• Multi-Weyl Topological Semimetals Stabilized by Point Group Symmetry
• A topological crystalline insulator (TCI) phase via topological phase transition and crystalline mirror symmetry
• Topological classification of crystalline insulators through band structure combinatorics
• Computing topological invariants without inversion symmetry
• Inversion Symmetric Topological Insulators
• All "Magic Angles" Are "Stable" Topological
• Wannier representation of Z_2 topological insulators
• Z2Pack: Numerical Implementation of Hybrid Wannier Centers for Identifying Topological Materials
• Hourglass Fermions
• Exponential localization of Wannier functions in insulators
• Faithful Tight-binding Models and Fragile Topology of Magic-angle Bilayer Graphene
• Entanglement spectrum of topological insulators and superconductors
• Fragile Topology and Wannier Obstructions
• Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle
• Time-reversal invariant realization of the Weyl semimetal phase
• Realization of a three-dimensional photonic topological insulator
• Robustness of the Spin-Chern number
• Three-Dimensional All-Dielectric Photonic Topological Insulator
• Double crystallographic groups and their representations on the Bilbao Crystallographic Server
• Structure and Topology of Band Structures in the 1651 Magnetic Space Groups
• Band Topology and Linking Structure of Nodal Line Semimetals with Z2 Monopole Charges
• Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry
• Twisted equivariant matter
• New classes of topological crystalline insulators with unpinned surface Dirac cones
• Band topology in classical waves: Wilson-loop approach to topological numbers and fragile topology
• Wannier Center Sheets in Topological Insulators
• Topological surface states in three-dimensional magnetic insulators
• Topological Order and Absence of Band Insulators at Integer Filling in Non-Symmorphic Crystals
• Bloch state tomography using Wilson lines
• Experimental discovery of nodal chains
• Z2-topology in nonsymmorphic crystalline insulators: Mobius twist in surface states
• Topology of nonsymmorphic crystalline insulators and superconductors
• Topological non-symmorphic crystalline insulators
• Experimental Discovery of the First Nonsymmorphic Topological Insulator KHgSb
• Triviality of Bloch and Bloch-Dirac bundles
• Topology of Disconnected Elementary Band Representations
• Disconnected Elementary Band Representations, Fragile Topology, and Wilson Loops as Topological Indices: An Example on the Triangular Lattice
• Topological Crystalline Materials - General Formulation, Module Structure, and Wallpaper Groups -
• Spinless Topological Insulators without Time-Reversal Symmetry
• Dynamics of Berry-phase polarization in time-dependent electric fields
• Topological Insulators from Group Cohomology
• Edge modes in band topological insulators
• Tensor network trial states for chiral topological phases in two dimensions and a no-go theorem in any dimension
• Smooth gauge for topological insulators
• Trace Index and Spectral Flow in the Entanglement Spectrum
• Compactly-supported Wannier functions and algebraic $K$-theory
• Topological states from topological crystals
• Engineering Fragile Topology in Photonic Crystals: Topological Quantum Chemistry of Light
• Berry-phase description of Topological Crystalline Insulators
• Möbius Kondo Insulators
• Symmetry representation approach to topological invariants in $C_{2z}T$-symmetric systems
• The impossibility of exactly flat non-trivial Chern bands in strictly local periodic tight binding models
• Modern semiclassical theory of magnetic transport and breakdown
• The Axion Insulator as a Pump of Fragile Topology
• Fragile Phases As Affine Monoids: Classification and Material Examples
• Topological Bloch oscillations
• Classification of topological crystalline insulators based on representation theory
• Bloch bundles, Marzari-Vanderbilt functional and maximally localized Wannier functions
• Entanglement spectrum and Wannier center flow of the Hofstadter problem
• Classification of "Real" Bloch-bundles: Topological quantum systems of type AI
• Fragile topological phases in interacting systems
• Filling-Enforced Quantum Band Insulators in Spin-Orbit Coupled Crystals
• Surface polarization and edge charges
• Smooth gauge and Wannier functions for topological band structures in arbitrary dimensions
• Gapless surface states in a lattice of coupled cavities: a photonic analog of topological crystalline insulators
• No-go theorem for topological insulators and sure-fire recipe for Chern insulators
• Gapless surface states originated from accidentally degenerate quadratic band touching in a three-dimensional tetragonal photonic crystal
• Landau quantization of nearly degenerate bands, and full symmetry classification of avoided Landau-level crossings
• Search for localized Wannier functions of topological band structures via compressed sensing
• Hybrid method of plane-wave and cylindrical-wave expansions for distributed Bragg-reflector pillars: formalism and its application to topological photonics

Cited by in corpus (40)

• Topological acoustics
• High-throughput calculations of magnetic topological materials
• Real Space Invariants: Twisted Bulk-Boundary Correspondence of Fragile Topology
• Twisted bilayer graphene. II. Stable symmetry anomaly
• Geometric approach to fragile topology beyond symmetry indicators
• Experimental characterization of fragile topology in an acoustic metamaterial
• Fragile topology and flat-band superconductivity in the strong-coupling regime
• Band Representations and Topological Quantum Chemistry
• Fragile Topology Protected by Inversion Symmetry: Diagnosis, Bulk-Boundary Correspondence, and Wilson Loop
• Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates
• Spin-Resolved Topology and Partial Axion Angles in Three-Dimensional Insulators
• Topological photonics: fundamental concepts, recent developments, and future directions
• Multi-gap topology and non-Abelian braiding of phonons from first principles
• Multicellularity of delicate topological insulators
• Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond
• Fragile topology in line-graph lattices with two, three, or four gapped flat bands
• Teleportation of Berry curvature on the surface of a Hopf insulator
• General construction of flat bands with and without band crossings based on wave function singularity
• Cubic 3D Chern photonic insulators with orientable large Chern vectors
• Flat bands with band crossings enforced by symmetry representation
• Absence of topological protection of the interface states in ${\mathbb{Z}}_2$ photonic crystals
• Non-Compact Atomic Insulators
• Wannier Function Methods for Topological Modes in 1D Photonic Crystals
• Universal higher-order bulk-boundary correspondence of triple nodal points
• Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology
• Electronic structure and spin-lattice relaxation in superconducting vortex states on the kagome lattice near van Hove filling
• Dynamical Fragile Topology in Floquet Crystals
• Anderson Critical Metal Phase in Trivial States Protected by Average Magnetic Crystalline Symmetry
• Protected Gapless Edge States In Trivial Topology
• Topological gapless points in superconductors: From the viewpoint of symmetry
• Fractal Surface States in Three-Dimensional Topological Quasicrystals
• The Future of the Correlated Electron Problem
• Omnidirectional domain wall modes protected by fragile topological states
• Intrinsic (Axion) Statistical Topological Insulator
• Representation-protected topology of spin-singlet $s$-wave superconductors
• Selective branching, quenching, and converting of topological modes
• Fragile topology in nodal-line semimetal superconductors
• Fragile topological insulators protected by rotation symmetry without spin-orbit coupling
• Dynamical Symmetry Indicators for Floquet Crystals
• Three-dimensional spinless Euler insulators with rotational symmetry