Physics 682: Quantum Magnetism


Y - K. Yosida
W - Robert White
A - Assa Auerbach
F - Eduardo Fradkin
T - Alexei Tsvelik
GNT - Gogolin, Nersesyan, Tsvelik
B - Baxter
LLV - Landau, Lifshitz, v. 5
C - Cardy
P - Polyakov

# Date Read Topic
1. Tuesday, January 27 qm-notes,
W 1.1-1.3
1. Introduction. Class info. Brief history of magnetism. Introduction. Bohr-van Leeuwen theorem. Magnetic materials. Simple estimates. Weakness of dipole interaction.
2. Thursday, January 29 W 2.1-2.2, 3.1; Y 1 2. Noninteracting moments. Dirac Hamiltonian. Magnetism of atoms and molecules. Langevin diamagnetism and Van Vleck paramagnetism. Langevin paramagnetism. Curie law. Spin-orbit coupling. Lande factor.
3. Tuesday, February 3 W 3.1-3.3, 4.1 Quantum paramagnetism. Brillouin function. Rare earth ions. Crystal field splitting. Quenching of the orbital angular momentum. Transition metal ions. Nuclear paramagnetism. Magnetism of conduction electrons. 3. Classical orders. Ferromagnetism. Weiss molecular field approximation.
4. Thursday, February 5 W 2.2.8, Y 4 HW 1 given. Exchange interactions. Direct ferro- and antiferromagnetic exchange. Hubbard model on two lattice sites. Superexchange. Double exchange.
5. Tuesday, February 10 W 2.3, 4.1 Spin Hamiltonian. Single-ion, exchange, and Lande tensor anisotropies. Static susceptibility in RPA approximation.
6. Thursday, February 12 B 2
C 2
Ferromagnetism, antiferromagnetism, helimagnetism, and ferrimagnetism. Deficiency of mean field approximation (1d Ising). Effective (Gibbs) potential.
7. Tuesday, February 17 Read Effective (Gibbs) potential. Spontaneous symmetry breaking. T=0. Classical ground states. Spin flop transition.
8. Thursday, February 19 Read HW 2 given. Easy plane triangular antiferromagnet. Degeneracy of classical ground states: Kagome lattice. Order from disorder. Static susceptibility and spin-spin correlation function. Low temperatures. Spin waves.
9. Tuesday, February 24 LLV XIV
C 3
Restoration of broken symmetry by spin waves in one and two dimensions. Gaussian integrals. 4. Landau theory of continuous phase transitions. Critical indices: &beta, &gamma, &delta, &alpha, &nu, &zeta, &epsilon, &mu. Validity of mean field (Landau) theory. Ginzburg-Levanyuk criterion.
10. Thursday, February 26 LLV XIV
C 3
B 2
Scaling relations between critical indices. Scaling hypothesis. 5. Solvable classical models. One-dimensional Ising model, transfer matrices. Correlation functions in 1d Ising model.
11. Tuesday, March 2 B 2
P 3
1d Ising. Critical behavior at T=0, kinks, effective quantum problem. Two-dimensional Ising model. Free energy of domain wall at low temperature. High temperature expansion.
12. Thursday, March 4 B 6 HW 3 given. Low temperature expansion. Kramers-Wannier duality. Hamiltonian formulation of strongly anisotropic Ising model. Disorder parameter.
13. Tuesday, March 9 Read 2d Ising model as free fermions. Critical behavior of 2d Ising model. Classical XY model in 2d. The absence of long range order. Quasi long range order. High temperature expansion.
14. Thursday, March 11 Read Vortices. BKT transition. Interaction of vortices. Many vortex states: Coulomb gas problem. Classical XY model in 2d. Renormalization of fugacity and dielectric constant in Coulomb gas. BKT phase transition. Universal jump of stiffness.
15. Tuesday, March 16 Read Topological order. Critical behavior. 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.
16. Thursday, March 18 Read Large N limit. 1/N expansion. Mermin-Wagner theorem. Upper and lower critical dimensions. 6. Topological defects and textures. Topological primer: homotopy groups.
17. Tuesday, March 23 Read HW 4 given. 7. Single Quantum Spin. Single spin problem. SU(2) group and its representations. Holstein-Primakoff representation of spins. Schwinger boson representation of spins.
Thursday, March 25 Read NO CLASS. MARCH MEETING.
18. Tuesday, March 30 Read Multiplication of representations of an SU(2). Schwinger bosons and CP^1 representation. Path integral for a single spin. Wess-Zumino term.
19. Thursday, April 1 Read Spin precession from Wess-Zumino term. Wess-Zumino term and magnetic monopole. 8. Spin waves. Ferromagnet by Holstein-Primakoff representation: 1/S expansion. Spin waves. Quantum vs. classical energy scales (JS vs. JS^2).
20. Tuesday, April 13 Read Antiferromagnet by Holstein-Primakoff: spin waves, zero mode fluctuations. Possibility of quantum disordered states. Ferromagnet by path integral. Spin waves and Landau-Lifshitz equation.
21. Thursday, April 15 Read HW 5 given. Antiferromagnet by path integral. Spin waves. 9. Nonlinear sigma model. Correspondence between quantum model at T=0 in d-spatial dimensions and classical model at finite temperature T in d+1 dimensions.
22. Tuesday, April 20 Read Topological term is absent in dimesnions d>1. In d=1 topological term is theta-term. Effects of finite temperature. Quantum phase transitions.
23. Thursday, April 22 Read 10. Spin Chains. 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. Nonlinear sigma model with theta term. Vortex (meron) interference.
24. Tuesday, April 27 Read Lieb-Schulz-Mattis theorem. AKLT models and Valence Bond states. Phase diagram of spin 1 chains.
25. Thursday, April 29 Read Spin 1 chains. Boundary spin 1/2 in spin 1 chains: WZW from theta-term. Majumdar-Ghosh Hamiltonian versus Bethe ansatz results. Resonating valence bonds. AKLT states in 2d - valence bond solids.
26.Tuesday, May 4 Read Frustrated 2d QAFM: NLSM, columnar dimerized state and phase diagram. Confinement of spinons in VBS. RVB short and long range. 11. Spin Liquids. Gauge theory of AFM. Flux and uniform states. Confinement of spinons due to gauge field fluctuations. Remarks on relevance for high Tc superconductivity.
27.Thursday, May 6 Read Presentations:
1. Sebastian: Spinons versus magnons. O(N) versus CPM-1 approach to spin models.
2. Fabio: Quantum Phase Transitions.
28.Thursday, May 6 Read Presentations:
1. Ryan: Kalmeyer-Laughlin spin liquid.
2. Ivar: Topological versus "spin-wave" transitions in statistical mechanics. XY and Ising models, and 3d classical Heisenberg.