An Introduction to Topological Insulators

Abstract

Electronic bands in crystals are described by an ensemble of Bloch wave functions indexed by momenta defined in the first Brillouin Zone, and their associated energies. In an insulator, an energy gap around the chemical potential separates valence bands from conduction bands. The ensemble of valence bands is then a well defined object, which can possess non-trivial or twisted topological properties. In the case of a twisted topology, the insulator is called a topological insulator. We introduce this notion of topological order in insulators as an obstruction to define the Bloch wave functions over the whole Brillouin Zone using a single phase convention. Several simple historical models displaying a topological order in dimension two are considered. Various expressions of the corresponding topological index are finally discussed.

46 pages, 29 figures. This papers aims to be a pedagogical review on topological insulators. It was written for the topical issue of "Comptes Rendus de l'Académie des Sciences - Physique" devoted to topological insulators and Dirac matter

References in corpus (12)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Topological Insulators with Inversion Symmetry
• Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator
• Topological Field Theory of Time-Reversal Invariant Insulators
• Topological Crystalline Insulators
• Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
• Nonlocal edge state transport in the quantum spin Hall state
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• Wannier representation of Z_2 topological insulators
• On the $Z_2$ classification of Quantum Spin Hall Models
• Many-body generalization of the Z2 topological invariant for the quantum spin Hall effect

Cited by in corpus (63)

• Classification of topological quantum matter with symmetries
• Colloquium: Topological Band Theory
• Topology of crystalline insulators and superconductors
• Topological active matter
• Odd Viscosity and Odd Elasticity
• Topological Fano Resonances
• Non-Hermitian band topology and skin modes in active elastic media
• Topological and geometrical aspects of band theory
• Exploring nonlinear topological states of matter with exciton-polaritons: Edge solitons in kagome lattice
• Robustness of topologically protected edge states in quantum walk experiments with neutral atoms
• Valley qubit in gated MoS${}_2$ monolayer quantum dot
• Topology and Observables of the Non-Hermitian Chern Insulator
• Continuous bulk and interface description of topological insulators
• Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond
• Chiral Hinge Magnons in Second-Order Topological Magnon Insulators
• Kagome Lattice from Exciton-Polariton Perspective
• Tunable edge states and their robustness towards disorder
• Floquet Engineering of Haldane Chern Insulators and Chiral bosonic phase transitions
• Characterizing topological phases by quantum quenches: A general theory
• ${\mathbb{Z}}_2$ invariants of topological insulators as geometric obstructions
• Construction and properties of a topological index for periodically driven time-reversal invariant 2D crystals
• Topological magnetic excitations
• Scaling Theory of ${\mathbb{Z}}_2$ Topological Invariants
• Dirac synchronization is rhythmic and explosive
• Wannier functions and Z_2 invariants in time-reversal symmetric topological insulators
• Topological Microlaser with A non-Hermitian Topological Bulk
• Pressure-induced phase transition in Bi${}_2$Se${}_3$ at 3 GPa: electronic topological transition or not?
• Wave topology brought to the coast
• Anomalous heat flow in 8-$Pmmn$ borophene with tilted Dirac cones
• T-Duality of Topological Insulators
• Topological Polaritons in a Quantum Spin Hall Cavity
• Dynamically characterizing topological phases by high-order topological charges
• Existence and computation of generalized Wannier functions for non-periodic systems in two dimensions and higher
• Anyonic Topological Order in Twisted Equivariant Differential (TED) K-Theory
• Topological Corner States in Graphene by Bulk and Edge Engineering
• Constructing Landau framework for topological order: Quantum chains and ladders
• Asymmetric transport computations in Dirac models of topological insulators
• Chiral Edge Modes in Helmholtz-Onsager Vortex Systems
• Measurement of the topological Chern number by continuous probing of a qubit subject to a slowly varying Hamiltonian
• Four-point functions of gravitons and conserved currents of CFT in momentum space: testing the nonlocal action with the TTJJ
• Anomalous Lorenz number in massive and tilted Dirac systems
• Topological study of a Bogoliubov-de Gennes system of pseudo spin-$1/2$ bosons with conserved magnetization in a honeycomb lattice
• Kramers' doublet ground state in topological Kondo insulators
• Exactly solvable model of topological insulator realized on spin-1/2 lattice
• Anomalous quantum Hall effect induced by disorder in topological insulators
• Anomalous photo-thermal effects in multi-layered semi-Dirac black phosphorus
• Factors affecting the topological Hall effect in strongly correlated layered magnets: spin of the magnetic atoms, polar and azimuthal angle subtended by the spin texture
• Two-atom-thin topological crystalline insulators lacking out of plane inversion symmetry
• Index of Dirac operators and classification of topological insulators
• Dynamical Gap Generation in Topological Insulators
• Transversal transport of magnons in a modified Lieb lattice
• A Quantized Interband Topological Index in Two-Dimensional Systems
• Spin Geometry and Some Applications
• Topological insulators and geometry of vector bundles
• A Brillouin torus decomposition for two-dimensional topological insulators
• Quantum Field Theory and its Anomalies for Topological Matter
• The integration problem for principal connections
• Chern insulators in two and three dimensions: A global perspective
• Particle-hole symmetry broken solutions in graphene nanoribbons: a multi-orbital, mean-field perspective
• Quantum geometric tensor in systems with fractional band dispersion
• Kaluza-Klein Reduction of the 6 Dimensional \\ Dirac Equation on ${\mathbb{S}}^3\cong SU(2)$ and \\ Non-abelian Topological Insulators
• Phase transitions of the dimerized Kane-Mele model with/without the strong interaction
• Code CFTs and Topological Matter