### by M. Dalmonte (2 credits, type C)

**Idea of the course:** this is a course on topological matter and gauge theories, with emphasis on lattice aspects and connections to quantum information.

**Pre-requisites:** quantum mechanics, basics of group theory, statistical mechanics, atomic physics. No previous knowledge of topology, field theory or gauge theory is required (the course is construed to be self-consistent).

**Material:** notes will be shared in advanced. Some of the topics are also either covered by review articles, or by books I will suggest during lectures.

### Modules:

- )
**Basics [4 hours]**- An overview of the classification of quantum matter
- Quantum mechanical phases and the Berry phase
- Basics of topology: Gauss-Bonnet theorem, Chern numbers, homotopy

- )
**Selected phenomenology of some simple lattice models [4 hours]**- The integer quantum Hall effect on a lattice: Hofstadter model / transport and topology
- Topological superconductors: Kitaev chain / entanglement and topology
- Fractionalization of quantum numbers: an introduction via examples (Valence bond liquids, AKLT chain)

- )
**Lattice gauge theories and spin liquids [8 hours]**- Introduction to lattice gauge theories in their Hamiltonian formulation
- Confinement, deconfinement, and their diagnostics
- Ising lattice gauge theory: topological order as deconfinement
- Deconding topological spin liquids: topological entanglement entropy and Ising-gauge duality
- Beyond Wilson’s lattice gauge theory: Quantum link models and quantum dimers
- Quantum simulation of gauge theories: the Schwinger model
- Quantum simulation of gauge theories: topological field theories on the lattice