Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics

Abstract

This monograph offers an overview on the topological invariants in fermionic topological insulators from the complex classes. Tools from K-theory and non-commutative geometry are used to define bulk and boundary invariants, to establish the bulk-boundary correspondence and to link the invariants to physical observables.

Monograph in Springer Series in Mathematical Physics Studies, see ISBN below. Correction of a few remaining typos. ISBN 978-3-319-29350-9, eBook ISBN 978-3-319-29351-6, (Springer, 2016)

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• Topological Crystalline Insulators
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• Topological Phononic Crystals with One-Way Elastic Edge Waves
• Topological Defects and Gapless Modes in Insulators and Superconductors
• Topological mechanics of gyroscopic metamaterials
• Trajectory of Anomalous Hall Effect toward the Quantized State in a Ferromagnetic Topological Insulator
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• Observation of Topological Phase Transitions in Photonic Quasicrystals
• Topological modes bound to dislocations in mechanical metamaterials
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• The Noncommutative Geometry of Graph $C^{\ast }$-Algebras I: The Index Theorem
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Cited by in corpus (158)

• Non-Hermitian Physics
• Observation of bulk boundary correspondence breakdown in topolectrical circuits
• Reciprocal skin effect and its realization in a topolectrical circuit
• Topological Classification of Non-Hermitian Hamiltonians
• Skin effect and winding number in disordered non-Hermitian systems
• Observation of topological edge modes in a quasi-periodic acoustic waveguide
• Boundary criticality of $PT$-invariant topology and second-order nodal-line semimetals
• Experimental Demonstration of Dynamic Topological Pumping Across Incommensurate Bilayered Acoustic Metamaterials
• The 6D quantum Hall effect and 3D topological pumps
• Theory and Experimental Investigation of the Quantum Valley Hall Effect
• Controlled topological phases and bulk-edge correspondence
• Topological Entanglement-Spectrum Crossing in Quench Dynamics
• A generalization of Bloch's theorem for arbitrary boundary conditions: Theory
• Anomalous bulk-edge correspondence in continuous media
• Acoustic Su-Schrieffer-Heeger lattice: Direct mapping of acoustic waveguides to Su-Schrieffer-Heeger model
• Dynamical Majorana Edge Modes in a Broad Class of Topological Mechanical Systems
• Bulk-Edge correspondence for two-dimensional Floquet topological insulators
• Real-space representation of winding number for one-dimensional chiral-symmetric topological insulator
• Topology Classification from Chiral Symmetry: Chiral Phase Index and Spin Correlations in Graphene Nanoribbons
• Self-induced topological transition in phononic crystals by nonlinearity management
• A many-body index for quantum charge transport
• Topological Edge Modes by Smart Patterning
• Non-Commutative Chern Numbers for Generic Aperiodic Discrete Systems
• Electric Field Tunable Topological Phases in Graphene Nanoribbons
• Computation of topological phase diagram of disordered Pb${}_{1-x}$Sn${}_x$Te using the kernel polynomial method
• The Nature of Topological Protection in Spin and Valley Hall Insulators
• A noncommutative framework for topological insulators
• Topological invariants and corner states for Hamiltonians on a three-dimensional lattice
• Disorder-induced topological phase transition in a 1D mechanical system
• On the Bott index of unitary matrices on a finite torus
• Topology in shallow-water waves: a violation of bulk-edge correspondence
• Notes on topological insulators
• Topological Gaps by Twisting
• Bulk-boundary correspondence for disordered free-fermion topological phases
• Coarse geometry and topological phases
• Topological edge states for disordered bosonic systems
• Topological characterization of classical waves: the topological origin of magnetostatic surface spin waves
• Connecting the Many-Body Chern Number to Luttinger's Theorem through Středa's Formula
• Topological Langmuir-cyclotron wave
• Generalization of Bloch's theorem for arbitrary boundary conditions: Interfaces and topological surface band structure
• Topological waves guided by a glide-reflection symmetric crystal interface
• Global topology of Weyl semimetals and Fermi arcs
• Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases
• Complete homotopy invariants for translation invariant symmetric quantum walks on a chain
• Wannier functions and Z_2 invariants in time-reversal symmetric topological insulators
• Index theorem on chiral Landau bands for topological fermions
• Converging Periodic Boundary Conditions and Detection of Topological Gaps on Regular Hyperbolic Tessellations
• The fate of high winding number topological phases in the disordered extended Su-Schrieffer-Heeger model
• Graded quasiperiodic metamaterials perform fractal rainbow trapping
• Wave topology brought to the coast
• Embedded Topological Insulators
• The Krein-Schrödinger Formalism of Bosonic BdG and Certain Classical Systems and Their Topological Classification
• Disorder driven phase transitions in weak AIII topological insulators
• Chiral Maxwell waves in continuous media from Berry monopoles
• Material vs. structure: Topological origins of band-gap truncation resonances in periodic structures
• Topological boundary invariants for Floquet systems and quantum walks
• Generalized Connes-Chern characters in KK-theory with an application to weak invariants of topological insulators
• Characterization of edge states in perturbed honeycomb structures
• Topological Gaps in Quasi-Periodic Spin Chains: A Numerical and K-Theoretic Analysis
• Topological Insulator in an Atomic Liquid
• Edge mode manipulation through commensurate multifrequency driving
• Duality of Topological Edge States in a Mechanical Kitaev Chain
• Strain topological metamaterials
• Restoring number conservation in quadratic bosonic Hamiltonians with dualities
• An Introduction to Noncommutative Physics
• T-duality simplifies bulk-boundary correspondence: some higher dimensional cases
• Topological Solitons in Su-Schrieffer-Heeger Chain with periodic hopping modulation, domain walls and disorder
• Edge-following topological states
• Fu-Kane-Mele monopoles in semimetals
• Toeplitz operators on concave corners and topologically protected corner states
• Bulk-interface correspondences for one dimensional topological materials with inversion symmetry
• Quantum walks: Schur functions meet symmetry protected topological phases
• Chiral Flow in One-dimensional Floquet Topological Insulators
• T-duality simplifies bulk-boundary correspondence: the noncommutative case
• Topological states in disordered arrays of dielectric nanoparticles
• Topology in shallow-water waves: A spectral flow perspective
• Majorana edge magnetization in the Kitaev honeycomb model
• Revealing the boundary Weyl physics of the four-dimensional Hall effect via phason engineering in metamaterials
• Stability of entanglement-spectrum crossing in quench dynamics of one dimensional gapped free-fermion systems
• Efficient computation of the $W_3$ topological invariant and application to Floquet-Bloch systems
• Generalized Real-space Chern Number Formula and Entanglement Hamiltonian
• The Cayley transform in complex, real and graded $K$-theory
• Topological junction states and their crystalline network in chiral-symmetric systems: application to graphene nanoribbons
• Index theory and topological phases of aperiodic lattices
• Asymmetric transport computations in Dirac models of topological insulators
• Bulk-edge correspondence and the cobordism invariance of the index
• Universal edge transport in interacting Hall systems
• Flat Energy Bands within Antiphase and Twin Boundaries and at Open Edges in Topological Materials
• A semiclassical treatment of spinor topological effects in driven, inhomogeneous insulators under external electromagnetic fields
• Local disorder, topological ground state degeneracy and entanglement entropy, and discrete anyons
• Long-time relaxation dynamics of a spin coupled to a Chern insulator
• Wannier band transitions in disordered $\pi$-flux ladders
• Three dimensional two-band Floquet topological insulator with $Z_2$ index
• The Streda Formula for Floquet Systems: Topological Invariants and Quantized Anomalies from Cesaro Summation
• Wave-packet propagation in a finite topological insulator and the spectral localizer index
• Gauge invariance and relativistic effects in photon absorption and scattering by matter
• Mode-Shell correspondence, a unifying phase space theory in topological physics -- Part I: Chiral number of zero-modes
• Crystallographic bulk-edge correspondence: glide reflections and twisted mod 2 indices
• A many-body Fredholm index for ground state spaces and Abelian anyons
• On ${\mathbb{Z}}_2$-indices for ground states of fermionic chains
• The bulk-dislocation correspondence for weak topological insulators on screw-dislocated lattices
• Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Determinants with Parametric Dependence
• Machine learning of phases and structures for model systems in physics
• T-duality and the bulk-boundary correspondence
• A groupoid approach to interacting fermions
• On spectral flow and Fermi arcs
• Application of semifinite index theory to weak topological phases
• Some remarks on $K_0$ of noncommutative tori
• The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects
• Topological Field Theories of Three-Dimensional Rotation Symmetric Insulators: Coupling Curvature and Electromagnetism
• 'Real' gerbes and Dirac cones of topological insulators
• Spectral decimation of a self-similar version of almost Mathieu-type operators
• Topological pump of $SU(Q)$ quantum chain and Diophantine equation
• Entanglement and particle fluctuations of one-dimensional chiral topological insulators
• Classification of topological invariants related to corner states
• Index Theory of Chiral Unitaries and Split-Step Quantum Walks
• An index theorem for quarter-plane Toeplitz operators via extended symbols and gapped Invariants related to corner states
• Toeplitz extensions in noncommutative topology and mathematical physics
• Bulk-Boundary Correspondence for Topological Insulators with Quantized Magneto-Electric Effect
• The cohomology invariant for class DIII topological insulators
• The spectral Flow of a family of Toeplitz operators
• Absolutely Continuous Edge Spectrum of Hall Insulators on the Lattice
• The product of two independent Su-Schrieffer-Heeger chains yields a two-dimensional Chern insulator
• Interacting Chern Insulator in Infinite Spatial Dimensions
• A linear algebra-based approach to understanding the relation between the winding number and zero-energy edge states
• Mode-Shell correspondence, a unifying phase space theory in topological physics -- part II: Higher-dimensional spectral invariants
• From orbital magnetism to bulk-edge correspondence
• Topological Anderson insulators by latent symmetry
• An elementary proof and detailed investigation of the bulk-boundary correspondence in the generic two-band model of Chern insulators
• Tensor Theory for Higher Dimensional Chern Insulators with Large Chern Numbers
• Topological Anderson insulator phases in one dimensional quasi-periodic mechanical SSH chains
• Classifying bulk-edge anomalies in the Dirac Hamiltonian
• Selective branching, quenching, and converting of topological modes
• Dimensional reduction and scattering formulation for even topological invariants
• Eigenvalue crossings in Floquet topological systems
• C*-framework for higher-order bulk-boundary correspondences
• Topological insulators and geometry of vector bundles
• Boundary conditions and violations of bulk-edge correspondence in a hydrodynamic model
• Classification of unitary operators by local generatability
• Gap labeling theorem for multilayer thin film heterostructures
• Floquet Gauge Pumps as Sensors for Spectral Degeneracies Protected by Symmetry or Topology
• Topology of Fermi Surfaces and anomaly inflows
• Topological Insulators and the Kane-Mele Invariant: Obstruction and Localisation Theory
• Scattering theory of topologically protected edge transport
• Statistical Topology -- Distribution and Density Correlations of Winding Numbers in Chiral Systems
• A ${\mathbb{Z}}_2$-Topological Index for Quasi-Free Fermions
• Fractional Quantum Numbers, Complex Orbifolds and Noncommutative Geometry
• Crystallographic Interacting Topological Phases and Equivariant Cohomology: To assume or not to assume
• On Choosing a Physically Meaningful Topological Classification for Non-Hermitian Systems and the Issue of Diagonalizability
• Bulk-boundary correspondence in topological two-dimensional non-Hermitian systems: Toeplitz operators and singular values
• A Universal Chern Model on Arbitrary Triangulations
• Exact sequence between real and complex bivariant K theories and application to the Z2 pairing
• Topological junctions for one-dimensional systems
• A $C^{\ast }$-algebraic view on the interaction of real- and reciprocal space topology in skyrmion crystals
• Spectral Sequence Computation of Higher Twisted $K$-Groups of $SU(n)$
• From Bopp Shifts to Toroidal Shadows: K-Theoretic Gap Labels in Noncommutative Quantum Mechanics
• Sources of quantized excitations via dichotomic topological cycles
• Homological bulk-edge correspondence for Weyl semimetal