| 1. | Thursday, January 25 | First meeting. Introduction. Bohr-van Leeuwen theorem. Weakness of dipole interaction. |
| 2. | Tuesday, January 30 | Exchange. Direct ferro- and antiferromagnetic exchange, superexchange. |
| 3. | Thursday, February 1 | Double exchange. Spin Hamiltonian. Transition metal ions. Spin-orbit coupling. Crystal field. Single-ion, exchange, and Lande tensor anisotropies. |
| 4. | Tuesday, February 6 | Classical spins. Curie Law. Brillouin function. Weiss molecular field approximation. Ferromagnetism. |
| 5. | Thursday, February 8 | Effective (Gibbs) potential. Spontaneous symmetry breaking. Static susceptibility in RPA approximation. |
| 6. | Tuesday, February 13 | Ferromagnetism, antiferromagnetism, helimagnetism, and ferrimagnetism. |
| 7. | Thursday, February 15 | T=0. Classical ground states. Spin flop transition. Easy plane triangular antiferromagnet. Degeneracy of classical ground states: Kagome lattice. Static susceptibility and spin-spin correlation function. Low temperatures. Classical spin wave theory for ferromagnet. |
| 8. | Tuesday, February 20 | Classical spin wave theory for ferromagnet. "Thermodynamic spin waves" in antiferromagnet. Dynamics of "classical" spins. Spin wave spectrum in antiferromagnet. |
| 9. | Thursday, February 22 | Spin waves for arbitrary classical ground state. Example of triangular antiferromagnet. Dynamic susceptibility. Causality, Kramers-Kronig relation, dissipation. |
| 10. | Tuesday, February 27 | Fluctuation-dissipation theorem. Landau mean field theory of phase transitions. Validity of mean field. Fluctuation region. Critical indices. |
| 11. | Thursday, March 1 | Scaling relations between critical indices. Scaling hypothesis. Upper critical dimension. |
| 12. | Thursday, March 8 | Magnetic measurements. Susceptibility, resonances, and neutron scattering. One-dimensional Ising model, transfer matrices. |
| 13. | Tuesday, March 13 | One-dimensional Ising model. Correlation functions, critical behavior at T=0, kinks, effective quantum problem. Two-dimensional Ising model. |
| 14. | Tuesday, March 27 | Two-dimensional Ising model. High temperature expansion. Low temperature expansion. Kramers-Wannier duality. Hamiltonian formulation of strongly anisotropic Ising model. |
| 15. | Thursday, March 29 | Disorder parameter. 2d Ising model as free fermions. Critical behavior of 2d Ising model. Classical XY model in 2d. The absence of long range order. |
| 16. | Tuesday, April 3 | Classical XY model in 2d. Quasi long range order. High temperature expansion. Vortices. BKT transition. Interaction of vortices. Many vortex states: Coulomb gas problem. |
| 17. | Thursday, April 5 | Classical XY model in 2d. Renormalization of fugacity and dielectric constant in Coulomb gas. BKT phase transition. Critical behavior. Universal jump of stiffness. Topological order. |
| 18. | Tuesday, April 10 | Other applications of 2d XY model: superfluidity of films and melting of 2d crystals. Classical 2d Heisenberg model. O(3) nonlinear sigma-model in 2d. The absence of LRO and renormalization group. O(N) nonlinear sigma-model. Large N limit. 1/N expansion. |
| 19. | Thursday, April 12 | Topological defects and textures. Topological primer: homotopy groups. Single spin problem. SU(2) group and its representations. |
| 20. | Tuesday, April 17 | Holstein-Primakoff representation of spins. Schwinger boson representation of spins. Path integral for single spin. Wess-Zumino term. |
| 21. | Thursday, April 19 | Wess-Zumino term and magnetic monopole. Ferromagnet by Holstein-Primakoff representation: spin waves. Antiferromagnet: spin waves. |
| 22. | Tuesday, April 24 | Antiferromagnet. Zero mode fluctuations. Possibility of quantum disordered states. Ferromagnet by path integral. Spin waves and Landau-Lifshitz equation. |
| 23. | Thursday, April 26 | Antiferromagnet by path integral. Nonlinear sigma model. Correspondence between quantum model at T=0 in d-spacial dimensions and classical model at finite temperature T in d+1 dimensions. Topological term is absent in dimesnions d>1. In d=1 topological term is theta-term. |
| 24. | Friday, April 27 | Effects of finite temperature. Quantum phase transitions. 1d Quantum antiferromagnet. Difference between integer and half-integer spins for NLSM. Example of theta-term in quantum mechanics: a particle on a circle. Degeneracy of the ground state due to the theta-term. |
| 25. | Tuesday, May 1 | Lieb-Schulz-Mattis theorem. Nonlinear sigma model with theta term. Vortex (meron) interference. AKLT models and Valence Bond states. Spin 1 chains. |
| 26. | Thursday, May 3 |
Presentations: 1. Topological order. Boundary states of spin chains. 2. Geometric phase associated with instantons for two-dimensional quantum Heisenberg model. |
| 27. | Monday, May 7 | Majumdar-Ghosh Hamiltonian versus Bethe ansatz. Resonating valence bonds. AKLT states in 2d - valence bond solids. Frustrated 2d QAFM: NLSM, columnar dimerized state and phase diagram. Confinement of spinons in VBS. RVB short and long range. Gauge theory of AFM. Flux and uniform states. Confinement of spinons due to gauge field fluctuations. Remarks on relevance for high Tc superconductivity. |
| 28. | Tuesday, May 8 |
Presentations: 1. Dimerized versus Haldane phase of spin ladders. 2. Topological versus "spin-wave" transitions in statistical mechanics. XY and Ising models, and 3d classical Heisenberg. |