Week 1 | |
Monday, January 26 | Introductory lecture; Read Ch. 1 (Quantum Mechanics) and Ch. 2 (Single-Particle States in Solids) on your own as a refresher. |
Wednesday, January 28 | Ch. 3 Second Quantization
3.1 N-particle wave function; 3.2 Fermi and Bose exchange symmetry; Properly symmetrized basis set; Occupations numbers; Fermions: Slater determinant; 3.4.1 Creation and annihilation operators |
Friday, January 30 | Matrix elements of one-body (3.5) and two-body (3.7) operators for fermions (A.1) |
Week 2 | |
Monday, February 2 | Bosons: 3.4.2 Creation and annihilation operators;
A.2 Matrix elements of one-body and two-body operators;
HW 1 issued (Second Quantization) |
Wednesday, February 4 | 3.11 Field operators Psi(r); Second quantization in coordinate and momentum representations |
Friday, February 6 | Discussion of HW 1: Tight-binding models on 1D, square, and
hexagonal lattices, graphene;
Spin Hamiltonian for a ferromagnet, the Holstein-Primakoff transformation, and spin waves (magnons) |
Week 3 | |
Monday, February 9 | 3.8, 3.9 Coulomb interaction between electrons in momentum representation
Ch. 4 The Electron Gas The direct and exchange contributions to the electron gas energy HW 2 issued (The Electron Gas) |
Wednesday, February 11 | 4.1 Cancellation of the direct Coulomb energy with positive
background in the jellium model
4.3 Negative exchange contribution to the Coulomb energy |
Friday, February 13 | Problem 4.3: Real-space avoidance by fermions as
the origin of the exchange contribution;
Variational minimization of the total energy; 4.2 The r_{s} parameter; High- and low-density limits; Wigner crystal; Problems 4.4 and 4.5: The two-dimensional electron gas (2DEG) |
Week 4 | |
Monday, February 16 | Ch. 5 A Brief Review of Statistical Mechanics
5.3.3 The grand canonical ensemble; 5.4 The statistical operator and density matrix; 5.5 Fermi and Bose distribution functions HW 3 issued (Statistical Mechanics) |
Wednesday, February 18 | Reduced one-particle and two-particle density matrices
(Problems 4.2 and 4.3)
Ch. 6 Real-Time Green's and Correlation Functions 6.1 A plethora of functions |
Friday, February 20 | 6.6 Linear response theory; Kubo formula; Retarded correlation function |
Week 5 | |
Monday, February 23 | 6.5 Green's function of a noninteracting system;
Analytical properties in the complex plane of omega
HW 4 issued (Real-Time Correlation Functions) |
Wednesday, February 25 | 6.4 Spectral representation; 6.4.3 Retarded correlation function;
6.4.4 Correlation function;
Real and imaginary parts of the correlations functions |
Friday, February 27 | 6.4.4 Fluctuation-dissipation theorem; Problem 6.11 Kramers-Kronig relations |
Week 6 | |
Monday, March 2 | No lecture: Late campus opening because of ice
HW 5 issued (Real-Time Correlation Functions) |
Wednesday, March 4 | 6.7 Density-density correlation function, the Lindhard function |
Friday, March 6 | No lecture: Campus closed because of snow |
Week 7 | |
Monday, March 9 |
Problem 2 of HW 5: continuum of electron-hole excitations in 3D, 2D, and 1D;
Neutron scattering;
6.9 Paramagnetic susceptibility of a noninteracting electron gas HW 6 issued (Applications of Real-Time Green's Functions) |
Wednesday, March 11 | 6.10 Equation of motion; The hierarchical chain of correlation functions; 6.12 A localized state coupled to a continuum of states |
Friday, March 13 | Ch. 7 Applications of Real-Time Green's Functions
7.1 Green's function for a single-atom Hubbard model; upper and lower Hubbard bands; 7.2 Anderson's impurity model; Mean-field approximation; Models of magnetism for one impurity, localized spins 1/2, and itinerant Hubbard/Stoner model (Problems 2, 3, and 4 in HW 6). Do Problem 1 about tunneling (7.3) on your own. |
Week 8 | |
Monday, March 23 | Ch. 8 Imaginary-Time Green's and Correlations Functions
8.1, 8.2, 8.3 Periodicity in Matsubara time, even and odd Matsubara frequencies; 8.4 Spectral representation, analytical continuation, and relation to real-time functions HW 7 issued (Imaginary-Time Green's Functions) |
Wednesday, March 25 | 8.5 Green's function for noninteracting particles
8.7.1 The interaction picture |
Friday, March 27 | 8.7.2 The U-operator; Time-ordered perturbation expansion;
8.7.3 Green's function in terms of the U-operator;
8.7.4 Perturbation expansion of the imaginary-time Green's function |
Week 9 | |
Monday, March 30 | 8.8 Wick's theorem
HW 8 issued (Diagrammatic Techniques) |
Wednesday, April 1 | 8.9 Case study: first-order interaction; 8.10 Cancellation of disconnected diagrams |
Friday, April 3 | Ch. 9 Diagrammatic Techniques
9.1 Second-order perturbation;
Feynman rules in momentum-frequency (9.2) and coordinate (9.4) spaces; Summation over omega_n using contour intergration HW 9 issued (Electron Gas: A Diagrammatic Approach) |
Week 10 | |
Monday, April 6 | Factor (-1) for fermion loops; Cancellation of 1/2 and 1/n!;
9.5 Self energy and Dyson's equation; 9.6 Energy shift and the lifetime of excitations Skip 9.7 Time-ordered diagrams; 9.8 Dzyaloshinski's rules |
Wednesday, April 8 | No lecture: Victor Yakovenko is away |
Friday, April 10 | Ch. 10 Electron Gas: A Diagrammatic Approach
10.4 Classification of diagrams according to the degree of divergence Victor Yakovenko is away, substituted by Prof. Jay Sau |
Week 11 | |
Monday, April 13 | 10.6 Summations of the ring diagrams;
10.7 Screened Coulomb interaction;
10.8 Collective electronic density fluctuations
Victor Yakovenko is away, substituted by Prof. Jay Sau |
Wednesday, April 15 | 10.10 Dielectric function; 10.10.1 Thomas-Fermi model;
10.11 Plasmons and Landau damping; 10.11.1 Plasmons; 10.11.2 Landau damping Skip 10.12 Dielectric function of graphene |
Friday, April 17 | Ch. 11 Phonons, Photons, and Electrons
11.1 Lattice vibrations in 1D; 11.2 1D diatomic lattice; acoustic and optical phonons; 11.3 Phonons in 3D; 11.4 Phonon statistics; Victor Yakovenko is away, substituted by Prof. Jay Sau |
Week 12 | |
Monday, April 20 | 11.5 Electron-phonon interaction: rigid-ion approximation
HW 10 issued (Phonons, Photons, and Electrons) |
Wednesday, April 22 | 11.7 Phonon Green's function; 11.8 Free-phonon Green's function;
11.9 Feynman rules for the electron-phonon interaction; 11.10 Electron self energy |
Friday, April 24 | Phonon self energy; Peierls instability in 1D; Charge-density waves;
The Su-Schrieffer-Heeger model of polyacetylene (CH)_{n} |
Week 13 | |
Monday, April 27 | 11.11 Quantization of the electromagnetic field;
11.12 Electron-photon interaction; Gauge invariance
Skip 11.13 Light scattering by crystals; 11.14 Raman scattering in insulators |
Wednesday, April 29 | Ch. 12 Superconductivity
12.1 Properties of superconductors; 12.2 The London equation; the Meissner effect; the London penetration depth HW 11 issued (Superconductivity) |
Friday, May 1 | 12.3 Effective electron-electron attraction mediated by phonons; Summation of the Cooper ladder diagrams |
Week 14 | |
Monday, May 4 | Divergence of superconducting susceptibility at the
transition temperature T_{c} (Problem 1 of Homework 11);
Mean-field description of the superconducting state at T<T_{c} |
Wednesday, May 6 | The Bogoliubov-de Gennes equations; Diagonalization of the
2x2 Hamiltonian by a unitary transformation to the
Bogoliubov operators; The BCS theory; Energy gap in the spectrum of quasiparticles (Problems 2 and 3 of Homework 11) |
Friday, May 8 | Energy minimization with respect to Delta;
Energy gain at T=0;
Delta_{0} at T=0 and the universal ratio
Delta_{0}/T_{c};
General equation for Delta(T); The free energy F(Delta,T) and spontaneous symmetry breaking (Problem 3 of Homework 11) |
Week 15 | |
Monday, May 11 | 12.7 Green's function approach to superconductivity;
12.9 The Nambu formalism;
12.10 Response to a weak (electro)magnetic field |
Final Exam | |
Monday, May 18, 8-10 am
update point |
Make-up lecture for the snow days
in lieu of final exam
Gauge-invariant expression for supercurrent; vortices in superconductors; magnetic flux quantization; type II superconductors; mixed state; lower and upper critical magnetic fields; Ch. 13 Nonequilibrium Green's Functions 13.2 Schroedinger, Heisenberg, and interaction pictures; 13.3 The malady and the remedy; Diagram technique at T=0; 13.4 Contour-ordered Green's function; 13.5 Kadanoff-Baym and Keldysh contours; 13.6 Dyson's equation; 13.7 Langreth rules; 13.8 Keldysh equations |
Last updated May 18, 2015