This is an introductory course on topological states of matter. It will start by covering the basics of topological band theory and its use to describe topological insulators (TI)s. We will then introduce low-energy effective models of TIs in two and three dimensions and discuss possible physical realisations.  A TI is insulating in the bulk but exhibits gapless excitations at a boundary with a trivial insulator (like the vacuum). We will explicitly derive these boundary states and discuss their physical ramifications. Finally, if time permits, we will cover higher-order TIs.   

The outline of the course is the following:

•    A reminder of Berry’s phase and the Berry’s phase in Block bands;
•    Topological band theory;
•    The Integer Quantum Hall effect as a topological state of matter;
•    2D TIs - The BHZ model and the Spin Hall effect;
•    3D TI – Effective Hamiltonian, surface states, etc;
•    Nanostructures of TIs (quantum dots, nanowires);
•    Higher order TIs and their boundary excitations.

 Bibliography

[1]    X.-L. Qi, S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
[2]    M. Z. Hasan, C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
[3]    D. Xiao, M.-C. Chang, Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
[4]    B. Xie, H.-X. Wang, X. Zhang, P. Zhan, et al. Nat. Rev. Phys. 3, 520 (2021).