Lecture notes

Here is the lecture notes on TI, which is partly based on the lecture notes given by Janos Asboth, Laszlo Oroszlany, and Andras Palyi, Here is the resource, and you can also download a copy here.

• Lecture 1 : 1-d SSH model
  • The Su-Schrieffer–Heeger (SSH) model
  • Tight-binding method: first quantization
  • Second quantization
  • Review
  • Energy band and topology
  • Winding number
  • Chiral symmetry
  • Review 2
  • Topological transition
  • Winding number v.s. Zak phase
• Lecture 2 : Berry Phase and Chern number
  • Berry Phase review
  • Berry curvature
  • Berry curvature from perturbation theory
  • Benchmark: Spin-1/2
• Lecture 3 : Chern Insulator
  • Menu
  • Quantum Hall Insulator
  • Edge states with Quantization of the Hall conductivity
  • Half BHZ model
  • Chern number using \(\mathbf{h}\)
  • Chern number as a Skyrmion number
  • Low-energy Hamiltonian
  • Chern number as an obstruction
  • Haldane Model

Other good Books and review papers on this subject:

• Topological Insulators and Topological Superconductors by Bernevig, B..
• Topological Insulators: Dirac Equation in Condensed Matters by Shun-Qing Shen.
• 13. Fruchart and D. Carpentier, Comptes Rendus Physique 14, 779 (2013).
• 13. Kohmoto, Annals of Physics 160, 343 (1985).
• 4. Xiao, M.-C. Chang, and Q. Niu, Reviews of Modern Physics 82, 1959 (2010).
• 25. Mokrousov and F. Freimuth, arXiv:1407.2847 [cond-Mat] (2014).

More geometrical staffs at this one Geometry, Topology and Physics, by Nakahara.