A Short Course on Topological Insulators: Band Structure and Edge States in One and Two Dimensions

2015/09/08 by János K. Asbóth, László Oroszlány, András Pályi · 3 voices · 46 citations
Materials Science · Physics and Astronomy · #Graphene research and applications #Quantum many-body systems #Topological Materials and Phenomena #cond-mat.mes-hall #cond-mat.other

paper · pdf · doi:10.1007/978-3-319-25607-8

openalex publication_date 2016/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Quantal phase factors accompanying adiabatic changes
• Berry phase effects on electronic properties

Cited by

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• Topological Vibration Analysis of Elastic Lattices via Bloch Sphere Mapping
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• Skyrmionic polarization textures in structured dielectric planar media
• Why is topology hard to learn?
• Topological transitions controlled by the interaction range
• Tunable quantum metric and band topology in bilayer Dirac models
• Constructing qubit edge states by inverse-designing the electromagnetic environment
• Topological edge states and pumping in a chain of coupled superconducting qubits
• Unifying framework for non-Hermitian and Hermitian topology in driven-dissipative systems
• Bulk-edge coulping induced by a moving impurity
• The Su-Schrieffer-Heeger model on a one-dimensional lattice: Analytical wave functions of topological edge states
• High-dimensional topological photonic entanglement
• Quantized topological transport mediated by the long-range couplings
• -Continuum limit of bipartite lattices -- The SSH model
• Topological Phase Diagram of Generalized SSH Models with Interactions
• Impurity-Induced Interference at a Topological Boundary in an Infinite SSH Heterojunction
• Superstrong Dynamics and Directional Emission of a Giant Atom in a Structured Bath
• Topological characterization of phase transitions and critical edge states in one-dimensional non-Hermitian systems with sublattice symmetry
• On the Electronic Contribution to Crystalline Diffraction Patterns
• Evolution of quasiparticle edge states with Hubbard interaction in Rice-Mele chain
• Entanglement entropy as a probe of topological phase transitions
• Full counting statistics and first-passage times in quantum Markovian processes: Ensemble relations, metastability, and fluctuation theorems
• High-root topological edge-state bands
• A non-Hermitian Su-Schrieffer-Heeger model with the energy levels of free parafermions
• Survival and Detection of Symmetry-Protected Topology in Loop Quenches
• Non-Abelian Charge Transport in Three-Flavor Gauge Semimetal Model with Braiding Majoranas
• Topological band insulators without translational symmetry
• Dynamical axion fields coupled with one-dimensional spinless fermions
• Localized states and skin effect around non-Hermitian impurities in tight-binding models
• Counting topological interface modes using simplicial characteristic classes
• Fragmented exceptional points and their bulk and edge realizations in lattice models
• Identification and Properties of Topological States in the Bulk of Quasicrystals
• Majorana-like oscillation for edge states in one-dimensional topological chain with dissipative couplings
• Beyond symmetry protection: Robust feedback-enforced edge states in non-Hermitian stacked quantum spin Hall systems
• Projectively topological exceptional points in non-Hermitian Rice-Mele model
• Topological phases and Edge states in an exactly solvable Gamma matrix model
• Fibonacci Waveguide Quantum Electrodynamics
• Observation of generic U(m) non-Abelian holonomy in photonics
• Wilson Line and Disorder Invariants of Topological One-Dimensional Multiband Models
• Phase-Space Topology in a Single-Atom Synthetic Dimension
• Universal phase correction for quantum state transfer in one-dimensional topological spin chains
• Exploring Topological Effects in Thin-Film X-Ray Cavities
• Compact and programmable large-scale optical processor in free space
• Homological bulk-edge correspondence for Weyl semimetal
• Dynamically stable topological edge states in an extended Su-Schrieffer-Heeger ladder with balanced perturbation

Discussions

• I am a physics undergraduate self-teaching topological band theory through this textbook: http://arxiv.org/abs/1509.02295 I'm trying to understand the geometric/topological meaning of the Chern number [stackexchange, 6 points]
• This is largely a follow up to this question from a few years ago, with regard to Asboth's "A short introduction to Topological insulators" He defines the topological invariant $N_A - N_B$ , where $N_ [stackexchange, 2 points]
• I am working through the discrete formulation of topological invariants on a lattice, specifically from "A Short Course on Topological Insulators" by Asbóth et al. ( https://arxiv.org/abs/1509.02295 ) [stackexchange, 1 points]

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