Teachers: Christophe Mora (MPQ, Université de Paris), Leonardo Mazza (LPTMS, Université Paris-Saclay)
This course discusses topological phenomena in condensed matter systems. The topics covered in the lectures span from the topology of band structures to the notion of topological order in strongly interacting models. We browse various models with topological features: Chern and topological insulators, topological superconductors with Majorana fermions, the integer and fractional quantum Hall effects, the toric code. We also discuss the spectacular phenomenology resulting from topological properties, from robust edge states to fractionalized excitations, such as Majorana fermions or anyons.
The tutorials are corrected each by a group of two students (possibly three if all slots are taken). Gives additional points to the final exam.
Lectures
Lecture 1: Hamiltonian classification
Lecture 2: Berry phase and Chern number
Lecture 3: the (integer) Quantum Hall Effect
Lecture 4: SSH model, Chern insulator and Graphene
Lecture 5: Quantum Spin Hall, Z2 index and Weyl semimetals
Lecture 6: Introduction to topological order and the toric code
Lecture 7: Fractional quantum Hall effect and Laughlin wavefunction
Lecture 8: Topological order, anyons and the fractional conductivity
Lecture 9: Topological order and entanglement
Lecture 10: Revision on topological order -- Recap slides
Tutorials
TD1 The Su-Schrieffer-Heeger (SSH) model
TD2 Bound state at domain wall and protected edge states
TD3 Chern insulators
TD4 Time-reversal invariant topological insulator
TD5 Majorana fermions in Rashba nanowires
TD6 Topological order vs. spontaneous symmetry breaking (for students preparing this TD, please focus only on parts 1-3)
TD9 Fractional charge and statistics
Bibliography
Lectures notes by Adolfo Grushin, and the corresponding website
A comprehensive (and inspiring) review on the quantum Hall effect by D. Tong : The quantum Hall effect
A nice review by Asbóth, János et Oroszlány : A Short Course on Topological Insulators
Book by B.A. Bernevig : Topogical insulators and topological superconductors, another one by B. Kotetes : Topological insulators
Lectures notes by David Carpentier
Review article by J.-N. Fuchs and J. Cayssol, Topological and geometrical aspects of band theory
Introductory presentation slides (January 17, 2022): here