Real Hopf Insulator

Abstract

Establishing the fundamental relation between the homotopy invariants and the band topology of Hamiltonians has played a critical role in the recent development of topological phase research. In this work, we establish the homotopy invariant and the related band topology of three-dimensional (3D) real-valued Hamiltonians with two occupied and two unoccupied bands. Such a real Hamiltonian generally appears in $\mathcal{P}\mathcal{T}$ symmetric spinless fermion systems where $\mathcal{P}$ and $\mathcal{T}$ indicate the inversion and time-reversal symmetries, respectively. We show that the 3D band topology of the system is characterized by two independent Hopf invariants when the lower-dimensional band topology is trivial. Thus, the corresponding 3D band insulator with nonzero Hopf invariants can be called a real Hopf insulator (RHI). In sharp contrast to all the other topological insulators discovered up to now, the topological invariants of RHI can be defined only when the fixed number of both the occupied and unoccupied states are simultaneously considered. Thus, the RHI belongs to the category of delicate topological insulators proposed recently. We show that finite-size systems with slab geometry support surface states with nonzero Chern numbers in a $\mathcal{P}\mathcal{T}$-symmetric manner independent of the Fermi level position, and establish the bulk-boundary correspondence. We also discuss the bulk-boundary correspondence of rotation symmetric RHIs using the returning Thouless pump.

arXiv comment34 pages, 10 figures, 1 table

References in corpus (45)

• Topological Insulators
• Maximally-localized generalized Wannier functions for composite energy bands
• Classification of topological quantum matter with symmetries
• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Topological quantum chemistry
• Topological nodal line semimetals with and without spin-orbital coupling
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Topological classification of crystalline insulators through band structure combinatorics
• 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
• Magnetic Topological Quantum Chemistry
• Topological classification with additional symmetries from Clifford algebras
• Building Blocks of Topological Quantum Chemistry: Elementary Band Representations
• 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
• Non-Abelian reciprocal braiding of Weyl points and its manifestation in $\text{ZrTe}$
• Topological surface states in three-dimensional magnetic insulators
• Topological Euler class as a dynamical observable in optical lattices
• Observation of non-Abelian topological semimetals and their phase transitions
• $PT$ Symmetric Real Dirac Fermions and Semimetals
• Geometric approach to fragile topology beyond symmetry indicators
• Disconnected Elementary Band Representations, Fragile Topology, and Wilson Loops as Topological Indices: An Example on the Triangular Lattice
• Edge modes in band topological insulators
• Boundary criticality of $PT$-invariant topology and second-order nodal-line semimetals
• Fragile Topology Protected by Inversion Symmetry: Diagnosis, Bulk-Boundary Correspondence, and Wilson Loop
• Hopf Insulators and Their Topologically Protected Surface States
• Stiefel-Whitney classes and topological phases in band theory
• Non-abelian topology of nodal-line rings in $\mathcal{P}\mathcal{T}$-symmetric systems
• Fragile Phases As Affine Monoids: Classification and Material Examples
• Hopf characterization of two-dimensional Floquet topological insulators
• Symmetry Protected Topological Hopf Insulator and its Generalizations
• Topological Hopf-Chern Insulators and the Hopf Superconductor
• Surface theorem for the Chern-Simons axion coupling
• Multicellularity of delicate topological insulators
• Adiabatic Pumping of Chern-Simons Axion Coupling
• Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond
• Teleportation of Berry curvature on the surface of a Hopf insulator
• Realizing Hopf Insulators in Dipolar Spin Systems
• Classification of crystalline insulators without symmetry indicators: atomic and fragile topological phases in twofold rotation symmetric systems
• $N$-band Hopf insulator
• No-go theorem for topological insulators and sure-fire recipe for Chern insulators
• Floquet Engineering Ultracold Polar Molecules to Simulate Topological Insulators
• ${\mathbb{Z}}_2$ Spin Hopf Insulator: Helical Hinge States and Returning Thouless Pump
• Quantized surface magnetism and higher-order topology: Application to the Hopf insulator

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• Flat Bands in Three-dimensional Lattice Models with Non-trivial Hopf Index
• Representation-protected topology of spin-singlet $s$-wave superconductors
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