Quantum Magnetism Approaches to Strongly Correlated Electrons
  Problems of strongly interacting electrons can be greatly simplified by reducing them to effective quantum spin models. The initial step is renormalization of the Hamiltonian into a lower energy subspace. The positive and negative U Hubbard models are explicitely transformed into the Heisenberg and -x-xz models respectively. Basic tools of quantum magnetism are introduced and used: spin coherent states path integral, spin wave theory, and continuum theory of rotators.The last lecture concerns pseudospin approaches to superconductivity and superfluidity. The SO(3) rotator theory for the -x-xz model describes a charge density wave to superconductor transition. Analogously, Zhang's SO(5) rotator theory describes the antiferromagnet to d-wave superconductor transition in high Tc cuprates. Finally the Magnus force on the two dimensional vortices and their momentum, are derived from the Berry phase of the spin path integral. 
A. Auerbach (Lecture notes compiled by F. Berruto and L. Capriotti)