A Short Course on Topological Insulators: Band-structure topology and edge states in one and two dimensions

Abstract

This course-based primer provides newcomers to the field with a concise introduction to some of the core topics in the emerging field of topological band insulators in one and two dimensions. The aim is to provide a basic understanding of edge states, bulk topological invariants, and of the bulk--boundary correspondence with as simple mathematical tools as possible. We use noninteracting lattice models of topological insulators, building gradually on these to arrive from the simplest one-dimensional case (the Su-Schrieffer-Heeger model for polyacetylene) to two-dimensional time-reversal invariant topological insulators (the Bernevig-Hughes-Zhang model for HgTe). In each case the model is introduced first and then its properties are discussed and subsequently generalized. The only prerequisite for the reader is a working knowledge in quantum mechanics, the relevant solid state physics background is provided as part of this self-contained text, which is complemented by end-of-chapter problems.

Submitted to Lect. Notes Phys

References in corpus (4)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• Measuring Z2 topological invariants in optical lattices using interferometry

Cited by in corpus (58)

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• Interplay between long range hopping and disorder in topological systems
• Su-Schrieffer-Heeger Model Inspired Acoustic Interface States and Edge States
• Topological nodal points in two coupled SSH chains
• Topological Zak Phase in Strongly-Coupled LC Circuits
• An elementary rigorous proof of bulk-boundary correspondence in the generalized Su-Schrieffer-Heeger model
• Demonstration of topological wireless power transfer
• Room temperature topological polariton laser in an organic lattice
• Topological phases of a dimerized Fermi-Hubbard model for semiconductor nano-lattices
• Symmetry-protected topological phases in lattice gauge theories: topological QED${}_2$
• Universalities of thermodynamic signatures in topological phases
• Longitudinal and anomalous Hall conductivity of a general two-band model
• Disorder-induced Chern insulator in Harper--Hofstadter--Hatsugai model
• Topological edge states in periodically-driven trapped-ion chains
• Driven quantum many-body systems and out-of-equilibrium topology
• Symmetry-protected quantization of complex Berry phases in non-Hermitian many-body systems
• Topological phase transitions at finite temperature
• Strong-field-driven dynamics and high-harmonic generation in interacting 1D systems
• Geometry and Topology Tango in Ordered and Amorphous Chiral Matter
• Reservoir-assisted symmetry breaking and coalesced zero-energy modes in an open PT-symmetric Su-Schrieffer-Heeger model
• Synthetically Non-Hermitian Nonlinear Wave-like Behavior in a Topological Mechanical Metamaterial
• Generalized lattice Wilson-Dirac fermions in (1+1) dimensions for atomic quantum simulation and topological phases
• Edge mode manipulation through commensurate multifrequency driving
• Topological phase transitions in tilted optical lattices
• Quench Dynamics in 1D Optomechanical Arrays
• Are quantum spin Hall edge modes more resilient to disorder, sample geometry and inelastic scattering than quantum Hall edge modes?
• Topological Protection and Control of Quantum Markovianity
• First-Principles Demonstration of Non-adiabatic Thouless Pumping of Electrons in A Molecular System
• Uniaxial Strain Induced Topological Phase Transition in Bismuth-Tellurohalide-Graphene Heterostructures
• Bulk Localised Transport States in Infinite and Finite Quasicrystals via Magnetic Aperiodicity
• Calculating the polarization in bi-partite lattice models: application to an extended Su-Schrieffer-Heeger model
• Competition of trivial and topological phases in patterned graphene based heterostructures
• Reentrant Localization Transitions in a Topological Anderson Insulator: A Study of a Generalized Su-Schrieffer-Heeger Quasicrystal
• Flat bands and $Z_2$ topological phases in a non-Abelian kagome lattice
• Coalescing Majorana edge modes in non-Hermitian PT-symmetric Kitaev chain
• Ferromagnetism in armchair graphene nanoribbon heterostructures
• Universal Landau-Zener regimes in dynamical topological phase transitions
• Effects of correlations on phase diagrams of the two-dimensional Su-Schrieffer-Heeger model with the longer-range hoppings
• Software implementation for calculating the Chern and $Z_2$ topological invariants with WIEN2k all-electron density functional package
• Berry curvature, semiclassical electron dynamics, and topological materials: Lecture notes for Introduction to Solid State Physics
• 4$\pi$ and 8$\pi$ dual Josephson effects induced by symmetry defects
• Resonant edge-site pumping of polaritonic Su-Schrieffer-Heeger lattices
• Topological Phase Diagram of BiTeX--Graphene Hybrid Structures
• Thermal Uhlmann phase in a locally driven two-spin system
• On the complex behaviour of the density in composite quantum systems
• Spectrum of localized states in fermionic chains with defect and adiabatic charge pumping
• Topological edge states in bowtie ladders with different cutting edges
• Disordered contacts can localize helical edge electrons
• Disordered contacts can localize chiral edge electrons
• Internal wave crystals
• From orthogonal link to phase vortex in generalized dynamical Hopf insulators
• Topological edge states and disorder robustness in one-dimensional off-diagonal mosaic lattices
• Particle-hole symmetry broken solutions in graphene nanoribbons: a multi-orbital, mean-field perspective
• Biot-Savart law in quantum matter
• Giant Enhancement of Nonlinear Harmonic Generation in a Silicon Topological Photonic Crystal Nanocavity Chain