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

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.

[1]  D. Thouless,et al.  Quantization of particle transport , 1983 .

[2]  S. Shen.  Topological Insulators: Dirac Equation in Condensed Matters , 2013 .

[3]  Macroscopic polarization from electronic wave functions , 1999, cond-mat/9903216.

[4]  B. Holstein.  The adiabatic theorem and Berry’s phase , 1989 .

[5]  Xiao-Liang Qi,et al.  Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors , 2005, cond-mat/0505308.

[6]  Masateru Taniguchi,et al.  Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum , 2022 .

[7]  A. Akhmerov,et al.  Scattering theory of topological insulators and superconductors , 2011, 1106.6351.

[8]  J. C. Budich,et al.  From the adiabatic theorem of quantum mechanics to topological states of matter , 2012, 1210.6672.

[9]  Frank Wilczek,et al.  Appearance of Gauge Structure in Simple Dynamical Systems , 1984 .

[10]  Quantum Mechanical Position Operator and Localization in Extended Systems , 1998, cond-mat/9810348.

[11]  L. Du,et al.  Robust helical edge transport in gated InAs/GaSb bilayers. , 2015, Physical review letters.

[12]  Shinsei Ryu,et al.  Topological insulators and superconductors: tenfold way and dimensional hierarchy , 2009, 0912.2157.

[13]  Measuring Z 2 topological invariants in optical lattices using interferometry , 2014, 1402.2434.

[14]  X. Qi,et al.  Equivalent expression of Z 2 topological invariant for band insulators using the non-Abelian Berry connection , 2011, 1101.2011.

[15]  The quantum spin Hall effect , 2011 .

[16]  M. Berry.  Quantal phase factors accompanying adiabatic changes , 1984, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[17]  Karin Schwab,et al.  The Universe In A Helium Droplet , 2016 .

[18]  Q. Xue,et al.  Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator , 2013, Science.

[19]  Yoichi Ando,et al.  Topological Insulator Materials , 2013, 1304.5693.

[20]  Zak,et al.  Berry's phase for energy bands in solids. , 1989, Physical review letters.

[21]  T. Fukui,et al.  Chern Numbers in Discretized Brillouin Zone: Efficient Method of Computing (Spin) Hall Conductances , 2005, cond-mat/0503172.

[22]  N. Marzari,et al.  Maximally-localized Wannier Functions: Theory and Applications , 2011, 1112.5411.

[23]  X. Qi,et al.  Topological insulators and superconductors , 2010, 1008.2026.

[24]  B. Bernevig.  Topological Insulators and Topological Superconductors , 2013 .

[25]  D. Vanderbilt,et al.  Smooth gauge for topological insulators , 2012, 1201.5356.

[26]  E. J. Mele,et al.  Z2 topological order and the quantum spin Hall effect. , 2005, Physical review letters.

[27]  A. Garg.  Berry phases near degeneracies: Beyond the simplest case , 2010 .

[28]  C. Kane,et al.  Topological Insulators , 2019, Electromagnetic Anisotropy and Bianisotropy.

[29]  J. Sólyom,et al.  Fundamentals of the Physics of Solids , 2007 .

[30]  E. Villaseñor.  Introduction to Quantum Mechanics , 2008, Nature.

[31]  Qian Niu,et al.  Berry phase effects on electronic properties , 2009, 0907.2021.

[32]  L. Molenkamp,et al.  Quantum Spin Hall Insulator State in HgTe Quantum Wells , 2007, Science.

[33]  C. Kane,et al.  Time Reversal Polarization and a Z 2 Adiabatic Spin Pump , 2006, cond-mat/0606336.

[34]  Chaoxing Liu,et al.  Quantum spin Hall effect in inverted type-II semiconductors. , 2008, Physical review letters.

[35]  Shou-Cheng Zhang,et al.  Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells , 2006, Science.

[36]  Joel N. Schulman,et al.  Wave Mechanics Applied to Semiconductor Heterostructures , 1991 .