Symmetries, Dimensions, and Topological Insulators: the mechanism behind the face of the Bott clock

Abstract

We provide an account of some of the mathematics of Bott periodicity and the Atiyah, Bott, Shapiro construction. We apply these ideas to understanding the twisted bundles of electron bands that underly the properties of topological insulators, spin Hall systems, and other topologically interesting materials.

RevTeX 28 pages, no figures.

References in corpus (9)

• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Discovery (theoretical prediction and experimental observation) of a large-gap topological-insulator class with spin-polarized single-Dirac-cone on the surface
• A topological Dirac insulator in a quantum spin Hall phase : Experimental observation of first strong topological insulator
• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Field Theory of Time-Reversal Invariant Insulators
• First direct observation of Spin-textures in Topological Insulators : Spin-resolved ARPES as a probe of topological quantum spin Hall effect and Berry's phase
• Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
• Classification of Topological Insulators and Superconductors
• On the $Z_2$ classification of Quantum Spin Hall Models

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• Classification of topological insulators and superconductors in the presence of reflection symmetry
• Classification of reflection symmetry protected topological semimetals and nodal superconductors
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• Topology by dissipation
• Twisted equivariant matter
• Topological invariants of Floquet systems: General formulation, special properties, and Floquet topological defects
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• Compactly-supported Wannier functions and algebraic $K$-theory
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• On the K-theoretic classification of topological phases of matter
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• Classification of Crystalline Topological Insulators and Superconductors with Point Group Symmetries
• Dimensional transmutation from non-Hermiticity
• Revealing a topological connection between stabilities of Fermi surfaces and topological insulators/superconductors
• A noncommutative framework for topological insulators
• Symmetry and topological classification of Floquet non-Hermitian systems
• Homotopy Theory of Strong and Weak Topological Insulators
• Tenfold Topology of Crystals: Unified classification of crystalline topological insulators and superconductors
• Interaction effects on the classification of crystalline topological insulators and superconductors
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• Symmetry-enforced topological band crossings in orthorhombic crystals: Classification and materials discovery
• Bott periodicity for the topological classification of gapped states of matter with reflection symmetry
• Embedded Topological Insulators
• Symmetry-protected entangling boundary zero modes in crystalline topological insulators
• Universal Finite-Size Scaling around Topological Quantum Phase Transitions
• Sinai Diffusion at Quasi-1D Topological Phase Transitions
• Global phase diagram of two-dimensional Dirac fermions in random potentials
• Equivalence of Topological Insulators and Superconductors
• Lyapunov spectra for all symmetry classes of quasi-one-dimensional disordered systems of non-interacting Fermions
• Topological invariants beyond symmetry indicators: Boundary diagnostics for twofold rotationally symmetric superconductors
• Squeezing-induced Topological Gap Opening on Bosonic Bogoliubov Excitations
• Chiral Magnetic and Vortical Effects in Higher Dimensions at Weak Coupling
• Defects in Graphene : A Topological Description
• Finite Size Scaling at the Topological Transition: Bilinear-Biquadratic Spin-1 Chain
• T-duality and the bulk-boundary correspondence
• Bott Periodicity and Realizations of Chiral Symmetry in Arbitrary Dimensions
• Elementary models of 3D topological insulators with chiral symmetry
• The product of two independent Su-Schrieffer-Heeger chains yields a two-dimensional Chern insulator
• Index of Dirac operators and classification of topological insulators
• Dimensional reduction and scattering formulation for even topological invariants
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• The $Z_2$ Classification of Dimensional Reduced Hopf Insulators
• Higher-order Topological Knots and the classification of non-Hermitian lattices under $C_n$ symmetry
• Transgression field theory at the interface of topological insulators
• Topological Tenfold Classification and the Entanglement of Two Qubits