Classification of crystalline insulators without symmetry indicators: atomic and fragile topological phases in twofold rotation symmetric systems

Abstract

Topological crystalline phases in electronic structures can be generally classified using the spatial symmetry characters of the valence bands and mapping them onto appropriate symmetry indicators. These mappings have been recently applied to identify thousands of topological electronic materials. There can exist, however, topological crystalline non-trivial phases that go beyond this paradigm: they cannot be identified using spatial symmetry labels and consequently lack any classification. In this work, we achieve the first of such classifications showcasing the paradigmatic example of two-dimensional crystals with twofold rotation symmetry. We classify the gapped phases in time-reversal invariant systems with strong spin-orbit coupling identifying a set of three ${\mathbb{Z}}_2$ topological invariants, which correspond to nested quantized partial Berry phases. By further isolating the set of atomic insulators representable in terms of exponentially localized symmetric Wannier functions, we infer the existence of topological crystalline phases of the fragile type that would be diagnosed as topologically trivial using symmetry indicators, and construct a number of microscopic models exhibiting this phase. Our work is expected to have important consequences given the central role fragile topological phases are expected to play in novel two-dimensional materials such as twisted bilayer graphene.

arXiv comment15 pages, 8 figures

References in corpus (76)

• Topological Insulators
• Topological insulators and superconductors
• Magic-angle graphene superlattices: a new platform for unconventional superconductivity
• Quantum Spin Hall Effect in Graphene
• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• Correlated Insulator Behaviour at Half-Filling in Magic Angle Graphene Superlattices
• Electronic Structure of Pyrochlore Iridates: From Topological Dirac Metal to Mott Insulator
• Topological Insulators in Three Dimensions
• Weyl and Dirac Semimetals in Three Dimensional Solids
• Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices
• Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures
• Classification of topological insulators and superconductors in three spatial dimensions
• Discovery of a Weyl Fermion Semimetal and Topological Fermi Arcs
• New directions in the pursuit of Majorana fermions in solid state systems
• Type-II Weyl Semimetals
• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Weyl Semimetal in a Topological Insulator Multilayer
• Topological Crystalline Insulators
• Quantized Electric Multipole Insulators
• Search for Majorana fermions in superconductors
• Experimental discovery of Weyl semimetal TaAs
• Higher-Order Topological Insulators
• Topological quantum chemistry
• Theoretical Discovery/Prediction: Weyl Semimetal states in the TaAs material (TaAs, NbAs, NbP, TaP) class
• Topological Crystalline Insulators in the SnTe Material Class
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• The (High Quality) Topological Materials In The World
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Topolectrical circuit realization of topological corner modes
• Reflection symmetric second-order topological insulators and superconductors
• Observation of a phononic quadrupole topological insulator
• Catalogue of Topological Electronic Materials
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• Higher-order topological insulators and semimetals on the breathing Kagome and pyrochlore lattices
• An equivalent expression of Z2 Topological Invariant for band insulators using Non-Abelian Berry's connection
• Higher-Order Topology in Bismuth
• Demonstration of a quantized microwave quadrupole insulator with topologically protected corner states
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Towards ideal topological materials: Comprehensive database searches using symmetry indicators
• Topological classification of crystalline insulators through band structure combinatorics
• Higher-order topological insulators and superconductors protected by inversion symmetry
• Quantum Hall Effect from the Topological Surface States of Strained Bulk HgTe
• Wannier representation of Z_2 topological insulators
• Second-order topological insulators and superconductors with an order-two crystalline symmetry
• Faithful Tight-binding Models and Fragile Topology of Magic-angle Bilayer Graphene
• Weyl semimetal with broken time reversal and inversion symmetries
• 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
• Spin-filtered Edge States with an Electrically Tunable Gap in a Two-Dimensional Topological Crystalline Insulator
• Quantitative mappings between symmetry and topology in solids
• Topological Zero-Energy Modes in Gapless Commensurate Aubry-André-Harper Models
• Symmetry indicators and anomalous surface states of topological crystalline insulators
• Band Topology and Linking Structure of Nodal Line Semimetals with Z2 Monopole Charges
• Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems
• Majorana Kramers pairs in higher-order topological insulators
• Stacked topological insulator built from bismuth-based graphene sheet analogues
• Higher-order topological insulators protected by inversion and rotoinversion symmetries
• An anomalous higher-order topological insulator
• Topological Crystalline Insulators and Dirac Octets in Anti-perovskites
• 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 insulators in magnetic fields: Quantum Hall effect and edge channels with non-quantized θ-term
• Experimental realization of type-II Weyl state in non-centrosymmetric TaIrTe${}_4$
• Helical Fermi arcs and surface states in time-reversal invariant Weyl semimetals
• Magnetic second-order topological insulators and semimetals
• Topological Mirror Insulators in One Dimension
• Inversion-symmetry protected chiral hinge states in stacks of doped quantum Hall layers
• Dislocation charges reveal two-dimensional topological crystalline invariants
• Topological semimetals in the SnTe material class: Nodal lines and Weyl points
• Excess charges as a probe of one-dimensional topological crystalline insulating phases
• Generic coexistence of Fermi arcs and Dirac cones on the surface of time-reversal invariant Weyl semimetals
• Topological Corner States on Kagome Lattice Based Chiral Higher-Order Topological Insulator
• Topology in time-reversal symmetric crystals

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• Geometric approach to fragile topology beyond symmetry indicators
• Fractional corner charges in spin-orbit coupled crystals
• Fragile Topology Protected by Inversion Symmetry: Diagnosis, Bulk-Boundary Correspondence, and Wilson Loop
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• Higher-order topological band structures
• Bulk-corner correspondence of time-reversal symmetric insulators: deduplicating real-space invariants
• Observation of topological Euler insulators with a trapped-ion quantum simulator
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• Dirac cones and higher-order topology in quasi-continuous media
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• Realizing and detecting Stiefel-Whitney insulators in an optical Raman lattice
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• Generalization of the Nested Wilson Loop Formalism in Topological Dirac Semimetals with Higher-order Fermi Arcs
• Homotopy invariant in time-reversal and twofold rotation symmetric systems
• Fragile topological insulators protected by rotation symmetry without spin-orbit coupling
• Wilson loop invariants and bulk-boundary correspondence in higher-order topological insulators with two anticommuting mirror symmetries