|
PHZ 7428 - Fall 2004
Topics Covered
The subject matter of the course is defined by the content of the lectures.
The main topics are listed below in the order they were covered.
Each topic is cross-referenced to the most closely related section(s) of
Abrikosov, Gorkov, and Dzyaloshinski ("AGD"), Fetter and Walecka ("FW"),
Mahan ("Mah"), and/or Mattuck ("Mat").
Note that AGD and FW have sections numbered consecutively from the beginning
of the book, rather than from the start of each chapter.
For example, FW Ch. 3 consists of Secs. 6-9.
The table below lists section numbers, not chapter numbers.
1. Introductory Review
| 1.1.
| Statistical mechanics of ideal gases
(FW 4, 5): Grand canonical ensemble for a noninteracting gas of ideal particles
in the number representation.
|
| 1.2.
| Ideal Fermi gas
(FW 5): Fermi-Dirac distribution, properties of a free electron gas,
effect of a periodic potential.
|
| 1.3.
| Ideal Bose gas
(FW 5): Deferred until later.
|
| 1.4.
| Second quantization
(FW 1, 2; AGD 3): Creation and annihilation operators, field operators,
second quantized form of observable operators.
|
2. Green's Functions
| 2.1.
| Single-particle Green's functions: Definitions and
uses
(FW 7; AGD 7.1, 10.3):
Definition of time-ordered Green's function,
calculation of one-particle properties and total energy.
|
| 2.2.
| Green's functions at T = 0
(FW 7; AGD 7.1):
Form for noninteracting electron gas, some general features.
|
| 2.3.
| Spectral representation
(FW 7, 31; AGD 7.2, 17):
Lehmann representation, meaning of poles, the spectral function,
retarded and advanced Green's functions.
|
| 2.4.
| Spectral properties of infinite systems
(Mat Appendix H; [1]): discrete vs continuous energy levels,
analytic properties of Green's function, quasiparticles.
|
| 2.6.
| Temperature Green's functions
(FW 23, 31; AGD 11,17):
Definition of temperature (imaginary time) Green's function,
periodicity, Fourier decomposition, form for noninteracting electron gas,
connection to real-time Green's functions, calculation of one-particle
properties and total energy.
|
3. Perturbation Theory I: Potential Scattering and Disorder
| 3.1.
| Introduction:
Specification of the impurity-scattering problem.
|
| 3.2.
| Perturbative expansion of the single-particle
Green's function
([2]):
Equation of motion for the temperature Green's function, iterative solution,
perturbative expansion, diagrammatic representations in the imaginary time
and imaginary-frequncy domains.
|
| 3.3.
| Disorder averaging
(AGD 39.2; Mat 3.5; Mah 4.1.E; [3]):
Term-by-term averaging of the Green's function expansion in reciprocal
space, diagrammatic representation, limits of low impurity concentration and
weak scattering, reducible and irreducible diagrams.
|
| 3.4.
| Dyson's equation
(AGD 39.2; Mat 3.5; Mah 4.1.E; [3]):
Diagrammatic derivation of Dyson's equation,
calculation of the dosorder-averaged self-energy in the Born limit,
quasiparticle beahvior, qualitative features of higher-order self-energy
diagrams, self-consistent Born approximation, t-matrix approximation.
|
| 3.5.
| Linear response and the Kubo formula:
Response coefficients and the fluctuation-dissipation theorem,
density-density correlations and two-particle Green's functions.
|
| 3.6.
| Kubo formula for the electrical
conductivity
(Mah 3.8):
Finite-temperature conductivity as a current-current correlation function.
|
| 3.7.
| Electrical conductivity in the presence of
impurities
(Mah 7.1.B):
Disorder-averaged current-current correlation function as a sum of ladder
diagrams (Born limit), contour-integral technique for perfoming Matsubara sums,
evaluation of the zero- and one-rung d.c. conductivity diagrams,
the quasiparticle and transport lifetimes.
|
| 3.8.
| Weak localization
([4]):
Maximally crossed current-current correlation diagrams, the diffuson and
cooperon propgators, quantum-mechanical corrections to the conductivity
in one, two, and three dimensions, coherent back-scattering due to
time-reversed paths.
|
4. Perturbation Theory II: Interactions
| 4.1.
| The Imaginary-Time Interaction Picture
(AGD 12.1; FW 24):
The interaction picture propagator S, expansion of S in powers
of H1, expansion of temperature Green's functions in
powers of H1.
|
| 4.2.
| Wick's Theorem
(AGD 12.2; FW 24):
Statement and proof of Wick's theorem for thermal expection values of
imaginary-time-ordered products of Heisenberg-picture operators.
|
| 4.3.
| Feynman Diagrams in Position Space
(AGD 8.3, 13; FW 9, 25):
Diagrams corresponding to terms in the Wick's theorem expansion of the
single-particle Greens' function, connected and disconnected diagrams,
cancellation of disconnected diagrams, restriction to topologically
inequivalent diagrams, treatment of equal-time Green's functions, spin,
and signs arising from Wick contractions, summary of Feynman rules.
|
| 4.4.
| Feynman Diagrams in Fourier Space
(AGD 14; FW 25):
Fourier expansion of Green's functions and interaction potentials for
spatially homogeneous systems, conservation rules at vertices, treatment
of equal-time Green's functions, summary of Feynman rules.
|
| 4.5.
| Dyson's Equation
(FW 26):
Diagrammatic derivation of Dyson's equation, calculation of lowest-order
self-energy diagrams for interacting fermions.
|
| 4.6.
| Hartree-Fock Approximation
(FW 10, 27):
First-order self-energy approximation (direct and exchange diagrams),
self-consistent Hartree-Fock approximation.
|
5. The Interacting Electron Gas
| 5.1.
| Introduction
(Mah 1.2; FW 3):
Jellium model, cancellation of the uniform component of the
electron-electron interaction.
|
| 5.2.
| Hartree-Fock Approximation
(Mah 5.1):
First-order exchange self-energy, divergence of Fermi velocity.
|
| 5.3.
| Linear screening
(Mah 5.4, 5.5; FW 14, 30, 12):
Macroscopic definition of the longitudial dielectric function and the
polarizability, Kubo formula for the polarization function, diagrammatic
expansion of the polarization function, irreducible polarization function,
the bare polarization bubble (the Lindhard function), dissipative vs
non-dissipative screening in the noninteracting electron gas.
|
| 5.4.
| Random phase approximation
(Mah 5.5, FW 12):
Identification of leading polarization diagrams in the limit of long
wavelengths and high electron density, the RPA dielectric function,
the Thomas-Fermi dielctric function, Friedel oscillations, plasmons,
Landau damping, local field corrections to RPA.
|
| 5.5.
| Equilibrium properties beyond Hartree-Fock
(Mah 5.8):
The screened electron-electron interaction, electron self-energy and
quasiparticle lifetime, quasiparticle renormalization factor, effective mass.
electron-electron interaction.
|
6. Electrons and Phonons
| 6.1.
| Introduction:
Two types of effects of ions on electrons in metals: the periodic potential
and scattering by lattice vibrations.
|
| 6.2.
| Phonons
(Mah 1.1; FW 44):
Ionic Hamiltonian in the harmonic approximation, phonon excitations.
|
| 6.3.
| Phonon Green's functions
(Mah 2.3, 3.2):
Different conventions for phonon Green's functions, bare Green's functions
(time-ordered, retarded, advanced, and Matsubara).
|
| 6.4.
| Electron-phonon interaction
(Mah 1.3, 6.4; FW 45-47):
The electron-phonon Hamiltonian term.
|
| 6.5.
| Cooper instability
|
| 6.6.
| Green's function description of superconductivity
|
References
| [1]
| S. Doniach and E. H. Sondheimer, Green's Functions in Solid State
Physics (W. A. Benjamin, 1974), Sec. 5.1.
[Available at the UF Science Library.]
|
| [2]
| Ibid., Sec 4.2, 4.3.
|
| [3]
| Ibid., Sec 5.2, 5.3.
|
| [4]
| P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
[online access]
|
|
