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 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, related topological invariants, as well as strong and weak three-dimensional topological insulators.

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 exam will consist of theoretical questions and a small numerical project.

Note: This course serves as a preparatory course for the Advanced Course “Advanced Topological Materials,” which I will offer immediately after, in Spring 2025.

Tentative list of topics:

• Basic theory of Wannier functions 
  • Localized and delocalized descriptions of solids
  • Composite and entangled bands
  • Wannier interpolation
  • Gauge-freedom and gauge-invariance
  • Hands-on: calculating Wannier functions
• Berry phases and curvatures
  • Wannier function centers and Berry phases
  • Berry phases, connection, curvatures
  • Gauge transformations and parallel transport
  • The Chern theorem
  • Adiabatic dynamics
  • Quantization of charge transport
  • Hands-on: calculating Berry phases
• Electric polarization
  • Berry-phase theory of electric polarization
  • Polarization as a lattice and polarization differences
  • Surface charge theorem
  • Hands-on: ab initio calculations of ferroelectric polarization
• The insulating state
  • Wannier function spreads and the quantum metric
  • Insulators and metals: a ground-state perspective
  • Hands-on: calculating the localization length
• Orbital magnetization and anomalous Hall conductivity
  • Anomalous Hall effect: intrinsic and extrinsic contributions
  • Orbital magnetization
  • Hands-on: AHC from Wannier function interpolation
• Topological obstructions to Wannier functions: topological insulators
  • The TKNN invariant
  • Quantum anomalous Hall insulators and Chern numbers
  • The Haldane model
  • Quantum spin Hall insulators and the Z2 invariant
  • The Kane-Mele and Bernevig-Hughes-Zhang models 
  • 3D topological insulators: weak and strong invariants
  • The Fu-Kane-Mele model
  • Hands-on: Calculating the Chern number
  • Hands-on: Calculating the Z2 invariant
  • Hands-on: Simulating a 3D topological insulator with DFT

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, The Wannier-Functions Software Ecosystem for Materials Simulations, arXiv:2312.10769 (2023)
• Scientific articles that will be cited in class

NB: Lecture time schedule is tentative.