Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond

Abstract

Pontrjagin's seminal topological classification of two-band Hamiltonians in three momentum dimensions is hereby enriched with the inclusion of a crystallographic rotational symmetry. The enrichment is attributed to a new topological invariant which quantifies a $2\pi$-quantized change in the Berry-Zak phase between a pair of rotation-invariant lines in the bulk, three-dimensional Brillouin zone; because this change is reversed on the complementary section of the Brillouin zone, we refer to this new invariant as a returning Thouless pump (RTP). We find that the RTP is associated to anomalous values for the angular momentum of surface states, which guarantees metallic in-gap states for open boundary condition with sharply terminated hoppings; more generally for arbitrarily terminated hoppings, surface states are characterized by Berry-Zak phases that are quantized to a rational multiple of $2\pi$. The RTP adds to the family of topological invariants (the Hopf and Chern numbers) that are known to classify two-band Hamiltonians in Wigner-Dyson symmetry class A. Of these, the RTP and Hopf invariants are delicate, meaning that they can be trivialized by adding a particular trivial band to either the valence or the conduction subspace. Not all trivial band additions will nullify the RTP invariant, which allows its generalization beyond two-band Hamiltonians to arbitrarily many bands; such generalization is a hallmark of symmetry-protected delicate topology.

arXiv commentv2: added summary of key terms and notions in Appendix B; small edits in the text

References in corpus (73)

• Quantum Spin Hall Effect in Graphene
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Berry Phase Effects on Electronic Properties
• Maximally-localized generalized Wannier functions for composite energy bands
• Electronic Structure of Pyrochlore Iridates: From Topological Dirac Metal to Mott Insulator
• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Quantized Electric Multipole Insulators
• Periodic table for topological insulators and superconductors
• Topological nodal semimetals
• Higher-Order Topological Insulators
• Topological quantum chemistry
• Chern semi-metal and Quantized Anomalous Hall Effect in HgCr2Se4
• Topological Crystalline Insulators in the SnTe Material Class
• 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
• Catalogue of Topological Electronic Materials
• 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
• Inversion Symmetric Topological Insulators
• Topology of crystalline insulators and superconductors
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• Hourglass Fermions
• Fragile Topology and Wannier Obstructions
• Stability of Fermi Surfaces and K-Theory
• Double crystallographic groups and their representations on the Bilbao Crystallographic Server
• Structure and Topology of Band Structures in the 1651 Magnetic Space Groups
• Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry
• Non-Abelian band topology in noninteracting metals
• Twisted equivariant matter
• Stationary ring solitons in field theory - knots and vortons
• Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems
• Non-Abelian reciprocal braiding of Weyl points and its manifestation in $\text{ZrTe}$
• Topological surface states in three-dimensional magnetic insulators
• An Introduction to Topological Insulators
• Bloch state tomography using Wilson lines
• Topological Order and Absence of Band Insulators at Integer Filling in Non-Symmorphic Crystals
• Tuning phase transition between quantum spin Hall and ordinary insulating phases
• Triviality of Bloch and Bloch-Dirac bundles
• Geometric approach to fragile topology beyond symmetry indicators
• Topological Crystalline Materials - General Formulation, Module Structure, and Wallpaper Groups -
• Electronic Structures and Surface States of Topological Insulator Bi${}_{1-x}$Sb${}_x$
• Topological Insulators from Group Cohomology
• Diversity of knot solitons in liquid crystals manifested by linking of preimages in torons and hopfions
• 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
• Compactly-supported Wannier functions and algebraic $K$-theory
• Hopf Insulators and Their Topologically Protected Surface States
• Non-abelian topology of nodal-line rings in $\mathcal{P}\mathcal{T}$-symmetric systems
• Modern semiclassical theory of magnetic transport and breakdown
• Topo-fermiology
• Topological Bloch oscillations
• Pseudospin Vortex Ring with a Nodal Line in Three Dimensions
• Crystallographic splitting theorem for band representations and fragile topological photonic crystals
• Symmetry Protected Topological Hopf Insulator and its Generalizations
• Topological Hopf-Chern Insulators and the Hopf Superconductor
• Topological acoustic triple point
• Conversion rules for Weyl points and nodal lines in topological media
• Multicellularity of delicate topological insulators
• Classification of "Real" Bloch-bundles: Topological quantum systems of type AI
• Filling-Enforced Quantum Band Insulators in Spin-Orbit Coupled Crystals
• Teleportation of Berry curvature on the surface of a Hopf insulator
• Realizing Hopf Insulators in Dipolar Spin Systems
• Geometric and nongeometric contributions to the surface anomalous Hall conductivity
• $N$-band Hopf insulator
• No-go theorem for topological insulators and sure-fire recipe for Chern insulators
• Bulk-and-edge to corner correspondence
• Floquet Engineering Ultracold Polar Molecules to Simulate Topological Insulators
• Non-Compact Atomic Insulators
• Quantized surface magnetism and higher-order topology: Application to the Hopf insulator
• The FKMM-invariant in low dimension

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• ${\mathbb{Z}}_2$ Spin Hopf Insulator: Helical Hinge States and Returning Thouless Pump
• Extrinsic and Intrinsic Nonlinear Hall Effects across Berry-Dipole Transitions
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• Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology
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• Berry-dipole Semimetals
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• Scattering theory of delicate topological insulators
• Chern dartboard insulator: sub-Brillouin zone topology and skyrmion multipoles
• Representation-protected topology of spin-singlet $s$-wave superconductors
• Flat Bands in Three-dimensional Lattice Models with Non-trivial Hopf Index
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• Non-Hermitian Hopf insulators
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