More details provided in the syllabus.

Assignments

1. due Tuesday, February 2 — pdf
2. due Tuesday, February 9 — pdf (assignment) / pdf (solutions)
3. due Thursday, February 18 — pdf (assignment) / pdf (solutions)
4. due Thursday, February 25 — pdf
5. due Thursday, March 10 — pdf (assignment) / pdf (solutions)
6. due Thursday, April 13 — pdf (assignment) / pdf (solutions)

Suggested Readings

• Kittel, Chapters 6 and 7
• Ashcroft and Mermin, Chapters 1, 2, and 3
• Fetter and Walecka, Chapters 1 and 2
• Ashcroft and Mermin, Chapter 17
• Kittel, Problems 5 and 6 from Chapter 11

Lectures

1. Introduction to non-relativistic, many-body quantum mechanics; exhange statistics of fermions and bosons; field operators and their (anti)commutation relations; Example: Degenerate electron gas (part 1) — pdf
2. Many-body Hamiltonian in second-quantized form; expansion of the field operators in one-body eigenstates; Example: Degenerate electron gas (part 2) — pdf
3. Cancelling divergences that arise in charge neutral systems; mathematical regularization tricks; Fermi Sea ground state of noninteracting systems; states of definite particle number; Example: Degenerate electron gas (part 3) — pdf
4. Evaluating expectation values of bilinear and biquadratic operator strings in the Fermi Sea; first-order energy shift in perturbation theory; Example: Degenerate electron gas (part 4) — pdf
5. Scaling of kinetic energy and Coulomb interactions terms with the relative interparticle separation, \(r_s\); expansion of the ground-state energy around \(r_s = 0\); breakdown of perturbation theory; Wigner crystal in the vicinity of \(r_s = \infty\); Example: Degenerate electron gas (part 5) — pdf
6. Destabilizing the Fermi Sea; quantum phase transitions; Wigner crystal; itinerant ferromagnetism; spontaneous spin polarization; comparison of variational energies; Example: Degenerate electron gas (part 6) — pdf
7. Energy and stability of the itinerant ferromagnet; spontaneous versus induced polarization; Zeeman coupling of a free electron gas to an external magnetic field; Example: Degenerate electron gas (part 7) — pdf
8. Spin polarization induced by an applied magnetic field; Zeeman splitting of the single-particle dispersion; linear response; Pauli susceptibility — pdf
9. Classical, macroscopic, coarse-grained picture of magnetism versus the quantum, microscopic picture; Pauli exclusion and repulsive interactions as the ingredients for magnetism; spinless versus spinful fermions; Example: Itinerant ferromagnetism in a gas of spin-S fermions subject to a repulsive contact potential — pdf
10. Crystalline systems; tight-binding models and band structure; contact interactions; derivation/motiviation for the Hubbard-\(U\) term — pdf
11. In-class exercise: the trihydrogen cation
12. Two-site Hubbard model at half filling; localization of the electrons; quenching of the occupation number fluctuations and emergence of an effective spin model at large \(U\) — pdf
13. Deriving the Heisenberg model from the half-filling Hubbard model, as an expansion in \(t/U\) — pdf
14. Long range magnetic order; classical limit of the Heisenberg model; energy minimization with Lagrange multipliers and with a relaxed constraint; helical spin order — pdf
15. Correlation functions as a measure of long range order; ground state of a ferromagnetic quantum Heisenberg model; spin waves as delocalized spin-flip excitations; spin wave dispersion relation Example: Heisenberg model on the linear chain for spin-half and nearest-neighbour-only ferromagnetic exchange coupling — pdf
16. Holstein-Primakov method; bosonic operator approach to spin waves; Heisenberg model solved as a \(1/S\) expansion; spin wave contributions to the specific heat for a ferromagnet — pdf
17. Coexistince of itinerant and localized electrons in solids; periodic Anderson model; Kondo lattice model; hybridization mean field theory; heavy fermion quasiparticles — pdf
18. Kondo Lattice Model; strong- and weak-coupling limits; RKKY interaction — pdf
19. Review of quantum statistical mechanics; partition functions, free energies, and Legendre transforms; Example: the one-site Hubbard model — pdf
20. Attractive versus repulsive interactions; decomposing the Hubbard interaction in the magnetic and pairing channels; mean-field BCS Hamiltonian; Bogoliubov quasiparticles — pdf
21. Electron bound states; centre-of-mass and relative coordinates for an interacting particle pair; pairing in the presence of a Fermi Sea (the Cooper problem) — pdf
22. Electron-phonon interaction; isotope effect; phonon-mediated effective attraction between electrons; BCS theory revisited — pdf
23. Isospin representation of the BCS Hamiltonian; representing the BCS ground state as a coherent state of Cooper pairs; uncertainty relation between particle number and phase — pdf