Fragile Phases As Affine Monoids: Classification and Material Examples

Abstract

Topological phases in electronic structures contain a new type of topology, called fragile, which can arise, for example, when an Elementary Band Representation (Atomic Limit Band) splits into a particular set of bands. We obtain, for the first time, a complete classification of the fragile topological phases which can be diagnosed by symmetry eigenvalues, to find an incredibly rich structure which far surpasses that of stable/strong topological states. We find and enumerate all hundreds of thousands of different fragile topological phases diagnosed by symmetry eigenvalues (available at http://www.cryst.ehu.es/html/doc/FragileRoots.pdf), and link the mathematical structure of these phases to that of Affine Monoids in mathematics. Furthermore, we predict and calculate, for the first time, (hundred of realistic) materials where fragile topological bands appear, and show-case the very best ones.

Accepted in PRX. Complete supplementary tables (979 pages) containing all the fragile criteria and fragile roots are available at http://www.cryst.ehu.es/html/doc/FragileRoots.pdf

References in corpus (25)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Topological Insulators with Inversion Symmetry
• Discovery (theoretical prediction and experimental observation) of a large-gap topological-insulator class with spin-polarized single-Dirac-cone on the surface
• Tuning superconductivity in twisted bilayer graphene
• Topological Crystalline Insulators
• Weyl semimetal phase in non-centrosymmetric transition metal monophosphides
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Reflection symmetric second-order topological insulators and superconductors
• Higher-order topological insulators and semimetals on the breathing Kagome and pyrochlore lattices
• Classification of stable three-dimensional Dirac semimetals with nontrivial topology
• Origin of Magic Angles in Twisted Bilayer Graphene
• Surface States of the Topological Insulator Bi_{1-x}Sb_x
• Topological Semimetals
• The space group classification of topological band insulators
• Topology of crystalline insulators and superconductors
• All "Magic Angles" Are "Stable" Topological
• Double crystallographic groups and their representations on the Bilbao Crystallographic Server
• Building Blocks of Topological Quantum Chemistry: Elementary Band Representations
• New classes of topological crystalline insulators with unpinned surface Dirac cones
• Graph Theory Data for Topological Quantum Chemistry
• Engineering Fragile Topology in Photonic Crystals: Topological Quantum Chemistry of Light
• Fragile topological phases in interacting systems

Cited by in corpus (51)

• Progress and prospects in magnetic topological materials
• All Topological Bands of All Nonmagnetic Stoichiometric Materials
• Real Space Invariants: Twisted Bulk-Boundary Correspondence of Fragile Topology
• 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
• Superfluid weight bounds from symmetry and quantum geometry in flat bands
• Topological correspondence between magnetic space group representations
• Hofstadter Topology: Non-crystalline Topological Materials at High Flux
• Catalogue of topological phonon materials
• Crystallographic splitting theorem for band representations and fragile topological photonic crystals
• Multicellularity of delicate topological insulators
• Filling anomaly for general 2D and 3D $C_4$ symmetric lattices
• Subdimensional topologies, indicators and higher order phases
• Homotopy, Symmetry, and Non-Hermitian Band Topology
• 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
• Tunable Fragile Topology in Floquet Systems
• Classification of Dirac points with higher-order Fermi arcs
• Location and topology of the fundamental gap in photonic crystals
• Symmetry Analysis of Anomalous Floquet Topological Phases
• Topological continuum charges of acoustic phonons in 2D
• Flat bands with band crossings enforced by symmetry representation
• The hybrid-order topology of weak topological insulators
• Landau Levels of the Euler Class Topology
• Non-Compact Atomic Insulators
• Interacting Topological Quantum Chemistry in 2D: Many-body Real Space Invariants
• Two-dimensional weak-type topological insulators in inversion symmetric crystals
• Correlated Fragile Topology: a Parton Approach
• From Atomic Semimetal to Topological Nontrivial Insulator
• Real Hopf Insulator
• Symmetry indicators in commensurate magnetic flux
• Generalized Fermion Doubling Theorems: Classification of 2D Nodal Systems in Terms of Wallpaper Groups
• Projected spin texture as a bulk indicator of fragile topology
• 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
• Simulating Floquet topological phases in static systems
• Dynamical Fragile Topology in Floquet Crystals
• Protected Gapless Edge States In Trivial Topology
• Kagome and honeycomb flat bands in moiré graphene
• Classification of Classical Spin Liquids: Topological Quantum Chemistry and Crystalline Symmetry
• Fractional corner charges in threefold-symmetric two-dimensional materials with fragile topology
• Two-dimensional orbital-obstructed insulators with higher-order band topology
• Representation-protected topology of spin-singlet $s$-wave superconductors
• Fragile topology in nodal-line semimetal superconductors
• Mott transition and magnetism in a fragile topological insulator
• Hilbert band complexes and their applications
• Fragile topological insulators protected by rotation symmetry without spin-orbit coupling
• Nodal higher-order topological superconductivity from C6-symmetric Dirac semimetals
• Three-dimensional spinless Euler insulators with rotational symmetry