## Program

(about 16 lectures of 2 hours each)

The course will start by introducing the basic theory of Wannier functions, and use it to discuss fundamental quantum-geometrical properties of the electronic structure and topological insulators in two and three dimensions. Quantum-geometrical and topological properties, such as Berry phases and topological invariants, will be first introduced in real space through the Wannier functions formalism and then connected with the corresponding expressions and meaning in reciprocal-space. After covering quantum anomalous and quantum spin Hall insulators in two dimensions, we will discuss strong and weak three-dimensional topological insulators, axion insulators and the concept of crystalline topology. If time permits, topological semimetals (Dirac, Weyl) and topological superconductors will be introduced. Lectures will be mostly theoretical (blackboard and some slides of support), homework will include numerical experiments in Python. The final exam will consist of some theory questions and either 1) the discussion of a scientific article or 2) a small numerical project.

Tentative list of topics:

- Localized and delocalized descriptions of solids
- Basic theory of Wannier functions
- Composite and entangled bands
- Wannier function centers and Berry phases
- Wannier function spreads and the quantum metric
- Topological obstructions to Wannier functions: topological insulators
- Quantum anomalous Hall insulators and the Haldane model
- Calculating the Chern number
- Quantum spin Hall insulators and the Z2 invariant
- 3D topological insulators: weak and strong invariants
- Calculating Z2 invariants
- Chern-Simons axion coupling
- Outline of crystalline topology
- Dirac and Weyl semimetals
- Outline of topological superconductors

References used in the course:

- D. Vanderbilt,
*Berry Phases in Electronic Structure Theory*, Cambridge University Press - R. M. Martin,
*Electronic Structure (second edition)*, Cambridge University Press - B. A. Bernevig with T. Hughes,
*Topological Insulators and Topological Superconductors*, Princeton University Press - R. Resta’s Lecture Notes on
*Geometry and Topology in Electronic Structure Theory*(http://www-dft.ts.infn.it/~resta/gtse/draft.pdf) - Scientific articles that will be cited in class