Graduate Course: Introduction to Integrable Models

Integrable models are completely solvable quantum many-body systems used in condensed matter physics. The spin chain which mimics a one-dimensional magnet is a prominent example. Apart from being interesting and beautiful in their own right, integrable models are closely connected to other fields of physics such as supersymmetric gauge theories and string theory and are therefore also relevant to the high energy physicist. We will discuss the very intuitive and conceptually appealing coordinate Bethe ansatz in order to diagonalize the Hamiltonian of the XXX spin chain and then move on to the more abstract but more powerful algebraic Bethe ansatz. Towards the end of the course, I will point out the connections to supersymmetric gauge theories.

The course will be largely self-contained, but basic notions of quantum mechanics (e.g. spin) and some familiarity with group theory are useful.

• Introduction to integrable spin chains
• Coordinate Bethe ansatz (Example: XXX spin chain)
• Introduction to the algebraic Bethe ansatz
• Algebraic Bethe ansatz for general spin
• Relations to supersymmetric gauge theories

• Spin Chains and Bethe Ansatz:
  • Gaudin: La fonction d'onde de Bethe
  • Bethe: Zur Theorie der Metalle
  • arXiv:cond-mat/9809162
  • arXiv:hep-th/9605187
• Supersymmetric gauge theory:
  • Hori et al.: Mirror Symmetry (Ch. 12, 15)
• Relations between the two:
  • arXiv:0901.4744, arXiv:0901.4748
  • arXiv:1005.4445

• Lecture Notes (lectures 1-5, 8)
• Lecture 6 (from Hori et al., Mirror Symmetry)
• Lecture 7 (from Hori et al., Mirror Symmetry)