1 Introduction: second quantisation, interacting electrons, the Hubbard model and itsdescendants
2 Quantum Ising model in transverse magnetic field: exact solution via Jordan Wigner, Fourier and Bogoliubov transform. Quantum phase transitions and criticality. Order an disorder. Duality. Excitations and domain walls. Entanglement entropy: area laws and logarithmic divergence.
3 Half-integer spin chains: Heisenberg antiferromagnets, Lieb-Schultz-Mattis theorem, order and disorder, Goldstone-bosons, Mermin-Wagner theorem, exact solution via coordinate Bethe ansatz.
4 Integer spin chains: Haldane’s conjecture, Affleck-Kennedy-Tasaki-Lieb model, introduction to MPS (Matrix Product States) and tensor networks. Gapless edge modes and symmetry protected topological order.
5 Topological classification of free fermion systems: periodic table of topological insulators and superconductors, Su-Schriefer-Heeger model and Kitaev’s quantum wire: topological.
degeneracy and majorana edge modes.
6 Spin models in higher dimensions, spin liquids, gauge theories and Kitaev's toric code
1 model, topological order and anyons
There will also be a group project, which can be chosen as either a literature review (e.g.
quantum hall effect, Levin-Wen string net models, topological insulators, entanglement
renormalization for critical systems, entanglement entropy in conformal field theory, …) or
(density matrix renormalization group algorithm, tensor renormalization group, …).
