Topological phases in solids: emergence of Dirac, Weyl, and Majorana fermions
Abstract:
The discovery of topological band insulators in 2005-2007 started the "topological era" of condensed matter physics. These solid state systems are bulk-insulating and feature topologically protected gapless edge or surface states which realize a two-dimensional Dirac theory. The topological protection enforced by time-reversal symmetry makes them robust against disorder. Similar physics is realized in certain superconductors which can even lead to the appearance of Majorana bound states. All these systems have in common, that they feature a bulk-gap.
Recently, it was shown that in three dimensions protected surface states can arise even in gapless systems, so-called "Weyl semimetals". These are systems where two non-degenerate bands accidentally touch at isolated points in the Brillouin zone.
In this lecture, I will first provide all the necessary "tools" to investigate and understand these topological phenomena occurring in solids. Starting with basic concepts such as the Berry-phase and how time-reversal symmetry acts on Bloch states, we will derive from scratch the physics of graphene, topological insulators and superconductors. Eventually we will discuss the main features of Weyl semi-metals, their surface states and the chiral anomaly. At the end, an overview is given about the zoo of exotic phases which emerge when topological phases come "into conflict" with strong electron-electron interactions.
Prior knowledge in elementary quantum mechanics and basic concepts of solid state physics is required (e.g. the concepts of momentum space, Brillouin zone, and Bloch states). Knowledge of second quantization is helpful but not obligatory.
Outline:
- 1) Berry phase and first Chern number
- 2) Time-reversal symmetry and Kramer's theorem
- 3) Graphene
- 4) The massive Dirac-Hamiltonian and Chern insulators
- 5) Z2 topological insulators
- 6) p-wave superconductivity and Majorana fermions
- 7) Gapless topological states of matter: Weyl semimetals
- 8) Strongly interacting topological insulators
