**Introduction to topological matter (spring 2021)**

### Outline of the course

- General ideas
- Recap of tight-binding Hamiltonians in second quantization
- Unitary and non-unitary symmetries
- 10 fold way

- 1D topological phases:
- The SSH chain
- The Kitaev wire

- 2D topological phases:
- Chern Insulators and the quantum spin-Hall effect
- Quantized Hall and spin-Hall conductivity and material realizations

- 3D topological phases
- 3D-topological insulators
- Experimental signatures and material realizations
- ARPES

- Topological Semimetals:
- What is a Weyl semimetal?
- Experimental signatures and material realizations
- ARPES: topological surface states
- Transport phenomena: negative magnetoresistance

- Other phases (if time permits):
- Crystalline topological insulators
- Fragile phases and delicate topological phases
- Topological superconducting phases

- Interacting topological phases (if time permits):
- Entanglement entropy and topological entanglement entropy
- Long range vs short range entangled phases.

**References and notes:**

**Video Lessons:** Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 (first hour missing, sorry!), Lecture 6, Lecture 7, Lecture 8

**General reviews:**

- Non-technical review on topological phases by Joel Moore (UC Berkeley)
- Hasan and Kane RMP (2010): A relatively light review on topological insulators
- Qi and Zhang RMP (2011): A bit more in depth and theoretical oriented review on topological insulators
- Armitage RMP (2018): Review on topological semimetals
- A short course on topological insulators, (2015) by J. K. Asbóth, L. Oroszlány, A. Pályi
- Lecture notes on topological crystalline insulators
- Check out this Delft website (https://topocondmat.org/) with plenty of examples
- Jean Dalibards lecture notes (in french)

**Books**

- There are by now several books on the subject. Some I willl be drawing from are the one by B. A. Bernevig, the one by P. Kotetes, or this one that will come out soon by R. Moessner and J. E. Moore
- D. Vanderbilt has also a book on Berry phases and polarization which is quite good.
- Many of the concepts that people use today were popularized by G. Volovik in his famous book

**Chapter 1: Basics** **and Symmetries**

(*) I am not sure who is the author of these notes, please contact me if you see this, and I will give you credit.

- This review by Ludwig is relatively clear in differentiating how symmetries act in second and first quantized notation
- Chiu, Teo, Schnyder, Ryu (2016) This is a review focused on symmetries and topological phases.
- Symmetry indicators of topology (2020) by H . C. Po. A more modern perspective on crystal symmetry and topology.

**Chapter 2: ** Examples of 1D topological phases: SSH, Rice-Mele and Kitaev wire

- Charlie Kane’s notes on topological phases have a discussion on the topology of the SSH model
- You can find a good summary of the types of SSH chain by R. Verresen in stack exchange here
- There is a great discussion on the properties of the SSH model in this review (Section II C)
- Jason Alicea’s review of topological superconductors has an excellent intro to Kitaev’s wire.

**Chapter 3:** Chern insulators and Quantum Spin-Hall insulators

- The first Chern insulator: Haldane honeycomb lattice model 1988
- This is a good (but long) review of many relevant topological insulators with concrete models. Chern Insulators are well explained too.
- Here you can find a derivation of the Chern number/Hall conductivity for generic 2 band Hamiltonians

**Chapter 4:** 3D Time-reversal invariant topological insulators in class AIII

- Same ref as above: This is a good (but long) review of all topological insulators with concrete models. 3D-TIs are explained too.