## A Short Course on Topological Insulators: Band Structure and Edge States in One and Two DimensionsThis course-based primer provides newcomers to the field with a concise introduction to some of the core topics in the emerging field of topological insulators. 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. The present approach uses 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 discussion of simple toy models is followed by the formulation of the general arguments regarding topological insulators. 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. |

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### Contents

1 | |

2 Berry Phase Chern Number | 23 |

3 Polarization and Berry Phase | 45 |

4 Adiabatic Charge Pumping RiceMele Model | 55 |

5 Current Operator and Particle Pumping | 69 |

The QiWuZhang Model | 85 |

7 Continuum Model of Localized States at a Domain Wall | 99 |

The BernevigHughesZhang Model | 119 |

9 The Z2 Invariant of TwoDimensional Topological Insulators | 139 |

10 Electrical Conduction of Edge States | 153 |

164 | |

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### Common terms and phrases

ˇ ˇ adiabatic adiabatic deformation adiabatic pumping Berry curvature Berry flux Berry phase BHZ model Brillouin zone bulk gap bulk Hamiltonian bulk momentum-space Hamiltonian chain Chern insulators Chern number chiral symmetry conductance consider corresponding cross section cycle defined degree of freedom disorder dispersion relation domain wall edge state branches eigenvalues electrons energy eigenstates energy gap Fermi energy gauge invariant Hamiltonian OH HgTe Hilbert space hopping amplitudes integer internal degree intracell hopping inversion symmetry lattice Hamiltonian lattice models momentum number of edge number of particles number of pumped NX mD1 OH.k one-dimensional onsite potential parameter space propagating pump sequence pumped particles quantum wire quasi-adiabatic QWZ model Rice-Mele model right edge sample Schrödinger equation shown in Fig spectrum SSH model sublattice time-reversal symmetry topological invariant torus two-dimensional time-reversal invariant unit cells unitary vector Wannier center flow wavefunctions wavenumber Wilson loop winding number zero energy