Lattice gauge theories – an introduction to topological matter

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.


  1. ) 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

  2. ) 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)

  3. ) 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

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