Program
The course will begin by introducing the basic theory of Wannier functions and using it to discuss fundamental quantum-geometrical and topological properties of electronic structure. We will cover modern theories of electric polarization, orbital magnetization, and the insulating state, along with key related phenomena such as the quantum anomalous Hall effect, all based on quantum-geometrical concepts like Berry phases and curvatures. Following this, we will move on to the topological properties of electronic structure. Topics will include quantum anomalous and quantum spin Hall insulators in two dimensions, and related topological invariants such as the Chern number and the Z2 invariant.
Theoretical lectures (using the blackboard and some supporting slides) will be complemented by hands-on sessions for numerical simulations with tight-binding models and density-functional theory.
Guided tutorials and homework will involve numerical experiments in Python and simulations with PythTB, Quantum ESPRESSO, and Wannier90.
The final oral exam will consist in the discussion of a numerical project and some theoretical questions.
Note: This course serves as a preparatory course for the Advanced Course “Advanced Topological Materials,” which I will offer immediately after, in Springr 2026.
Outline of the course program
- Wannier functions and Berry phases
- Hands-on: Wannier interpolation with Quantum ESPRESSO and Wannier90
- Electric polarization
- Hands-on: ab initio calculations of ferroelectric polarization with Quantum ESPRESSO
- Theory of the insulating state, quantum metric and orbital magnetization
- Introduction to topological insulators
- Hands-on tutorial on calculating band structure and Chern numbers for the Haldane model
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
- R. Resta’s Lecture Notes on Geometry and Topology in Electronic Structure Theory (http://www-dft.ts.infn.it/~resta/gtse/draft.pdf)
- A. Marrazzo, S. Beck, R. R. Margine, N. Marzari, A. A. Mostofi, J. Qiao, I. Souza, S. S. Tsirkin, J. R. Yates, G. Pizzi, Wannier-function software ecosystem for materials simulations, Rev. Mod. Phys. 96, 045008 (2024)
- Scientific articles that will be cited in class