# (Symmetry-Protected) Topological Order

Dr Elio König (MPI, Stuttgart)|

This introductory course offers a discussion of physical topological phenomena and the possibility to learn basic mathematical and field-theoretical techniques. Based on archetypical examples we will introduce the concept of symmetry-protected topological phases and of topological order.

The lecture course will be supplemented by exercise sheets. Solutions will be made available, informal or formal discussions about the problem sheets upon request.

Prerequisites are quantum mechanics and statistical mechanics. Ideally, the students have also taken introductory (master-level) courses in quantum field theory before.

This course will be presented in frames of Summer School 2024.

Exercise 1 Exercise 2 Exercise 3

Program

1. Introduction: What is this all about?

2. 1D Transverse field Ising model and Kitaev chain: concepts of symmetry protected topological phases

2.1 Transverse eld Ising model 101

2.2 Mapping to the Kitaev chain. Band topology

2.3 Symmetries, Symmetry fractionalization and ground state degeneracy

2.4 More about the symmetries in Kitaev like chains

3. Spin rotational invariant spin chains: More symmetry protected topological phases and a topological proof

3.1 Antiferromagnetic Spin-1 chains and the AKLT phase

3.2 Fundamental theorems in quantum magnetism: Topology as a "principle of

proof"

4. 2D Transverse field Ising model, Z2 gauge theory and toric code: Introduction to topological order

4.1 Z2 lattice gauge theory, Kramers-Wannier duality in 2D and the toric code

as the simplest quantum spin liquid

4.2 Solution of the Toric Code

5. Fermi surface reconstruction without symmetry breaking: A topological loophole to Luttinger's theorem

5.1 Fermions Coupled to the Toric Code

5.2 Discussion of different types of topological order

5.3 Luttinger-Oshikawa Theorem

5.4 Fermionic correlators and orthogonal metal

6. Quantum Spin liquids: Topological order in frustrated magnetism

6.1 The RVB solution

6.2 Kitaev's honeycomb model

6.3 Kitaev 16-fold way

7. Topological field theories

7.1 Path integral of a spin

7.2 General summary of topological terms

7.3 Antiferromagnetic spin chains: Haldane's conjecture from eld theory perspective

7.4 Spin-1 chain with open boundary conditions

7.5 Yet another denition of symmetry protected topological order

8. Exercise sheets and solutions thereof

8.1 Exercise about the transverse field clock chain

8.2 Exercise about the ZN toric code

8.3 Recap-exercise: Feynman path integral and bosonic coherent states

8.4 Exercise about topological field theories

On June 20, 9.00, the lecture "Zero is not nothing" will be also presented