| # |
Date |
Read |
Topic |
| 1. |
Jan. 24, Tuesday |
FW 1 |
Course info. Illustration of perturbation theory and "Feynman diagrams".
Second quantization. Hamiltonian and Hilbert space.
Periodic boundary conditions. Particle in the box. Many distinguishable particles in the box.
Basis of the Hilbert space and matrix elements of the Hamiltonian. Identical particles in the box.
Fermions and bosons in first quantization. Occupation numbers.
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| 2. | Jan 26, Thursday |
FW 1-2 |
Hilbert space for identical particles. Vacuum, Fock states, and Fock space.
Creation and annihilation operators. Hamiltonian in the second
quantized form. Occupation numbers. Fermi and Bose statistics. Fields.
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| 3. | Jan 31, Tuesday |
FW 1-2, part 3 |
HW1 given.
Particle density operator. Fourier transform. Tight-binding model.
Ideal Fermi gas. Fermi sea.
Fermi energy and Fermi momentum. Density of states.
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| 4. | Feb 2, Thursday |
FW 4-5 |
Ideal Fermi gas approximation:
interaction versus kinetic energy. r_s parameter. Wigner crystal.
Particle-hole pairs and quasiparticles. Specific heat of a degenerate Fermi gas.
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| 5. | Feb 7, Tuesday |
FW 6 |
Interaction representation.
Schroedinger, Heisenberg, and interaction pictures.
Perturbation theory. Chronological ordering. S-matrix.
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| 6. | Feb 9, Thursday |
FW 7 |
Adiabatic switching on. Gell-Mann and Low theorem. Ordinary perturbation theory from S-matrix.
Green's functions.
Definitions. Time ordering. Relation to observables. Expectation value of a single-particle operator.
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| 7. |
Feb 14, Tuesday |
FW 7 |
HW2 given.
The average potential energy of interactions. The exact ground state energy.
Green's function of free Fermi gas. The number of particles from the Green's function.
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| 8. |
Feb 16, Thursday |
FW 7 |
Green's function of free Fermi gas from the equation of motion.
Analytic structure of Green's functions. Plemelj formulas.
Translational invariance. Lehmann representation.
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| 9. |
Feb 21, Tuesday |
FW 7 |
Spectral functions A and B.
Advanced and retarded Green's functions.
Analytic structure of Green's functions. Asymptotics at omega->infinity.
Particle and hole propagators. Poles of Green's functions. Physical meaning of poles.
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| 10. |
Feb 23, Thursday |
FW 7, 8 |
Energy and lifetime of quasiparticles. Wick's theorem.
Perturbation theory for Green's functions. Green's functions in interaction representation.
Expectation values over the ground state.
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| 11. |
Feb 28, Tuesday |
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NO CLASS. MARCH MEETING (will be rescheduled)
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| 12. |
Mar 1, Thursday |
FW 8, 9 |
HW3 given.
Time ordering. Normal ordering.
Contraction. Contraction withing normal ordering.
Wick's theorem. Examples.
Feynman diagrams. First order diagrams in coordinate representation.
Bubble diagram.
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| |
Mar 6, Tuesday |
|
NO CLASS (will be rescheduled)
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| 13. |
Mar 8, Thursday |
FW 9 |
Cancellation of disconnected diagrams. Topologically distinct connected diagrams.
Feynman rules in coordinate space. Examples: first order.
Examples: second order. Feynman diagrams in momentum space.
Feynman rules in momentum space. Example: First order diagrams.
First order diagrams.
|
| 14. |
Mar 13, Tuesday |
FW 9 |
Dyson's equations.
Self-energy. Proper self-energy. Integral equations and summation of geometric series.
Proper energy in the first order. Corrections to the dispersion of particles.
|
| 15. |
Mar 14, Wednesday |
FW 9, 46 |
5:20-6:40pm, B-131
Effective interaction. Polarization insertions. Proper polarization insertion.
Vertex parts. Proper vertex parts. Skeleton diagrams. Dyson's equations.
|
| 16. |
Mar 15, Thursday |
FW 10 |
HW4 given.
Hartree-Fock approximation.
Fock's approach following Landau-Lifshitz III. Variational approach. Slater determinants.
Hartree-Fock equations. The energy of the ground state. Hartree and Fock diagrams. Hartree-Fock equations on G.
Conventional Hartree-Fock equations from diagrams.
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| 17. |
Mar 20, Tuesday |
FW 10, 11 |
Hartree-Fock equations on G.
Conventional Hartree-Fock equations from diagrams. HF for translationally invariant systems.
Imperfect Fermi gas.
Singular potentials and gas parameter. Scattering theory. Differential cross section.
Scattering amplitude. Integral equation for the wave function in momentum space.
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| 18. |
Mar 21, Wednesday |
FW 11 |
5:20-6:40pm, B-131
Integral equation for the scattering amplitude.
Scattering of hard spheres. Born approximation. Ladder diagrams.
First order in (k_F a) from Born approximation. Replacing Born scattering length by an exact one.
Effective two-particle interaction in the medium. Bethe-Salpiter equation.
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| 19. |
Mar 22, Thursday |
FW 11 |
Galitskii's integral equations. Solution of Galitskii's integral equations through the order (k_F a)^2.
Proper self-energy to this order.
Physical quantities: spectrum renormalization, effective mass, lifetime of quasiparticles,
chemical potential, ground state energy, sound velocity.
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| 20. |
Mar 27, Tuesday |
FW 3, 12 |
HW5 given.
Degenerate electron gas. Jellium model.
Kinetic and potential energy and length scales.
Expansion in r_s. Ground state energy calculation in the first order in e^2. Fermi surface integrals.
Divergencies in the second and higher orders. Specifics of a long range interaction.
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| 21. |
Mar 29, Thursday |
FW 12 |
Reordering of perturbation series. Ring diagrams. Divergence of the effective mass in HF approximation.
Density-density correlation function and the total polarization insertion.
Proper polarization in RPA approximation. Evaluation of \Pi_0(q).
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Apr 3, Tuesday |
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No classes, Spring recess.
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Apr 5, Thursday |
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No classes, Spring recess.
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| |
Apr 10, Tuesday |
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NO CLASS (will be rescheduled)
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| 22. |
Apr 12, Thursday |
FW 12 |
Evaluation of \Pi_0(q). Real part. Imaginary part. Creation of particle-hole pair and the structure of
Im\Pi_0(\omega,q) in a frequency-momentum plane. Useful limits of \Pi_0.
Correlation energy in terms of \Pi(\omega,q). Correlation energy in RPA approxiamation.
Effective interaction in RPA. Static limit.
Thomas-Fermi screening. Friedel oscillations.
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| 23. |
Apr 17, Tuesday |
FW 13 |
HW6 given.
Linear response and collective modes. Introduction.
General theory of linear response. External perturbation.
The retarded density correlation function and generalized susceptibility.
Retarded and chronological correlation functions.
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| 24. |
Apr 18, Wednesday |
FW 14 |
5:20-6:40pm, B-131
Screening in electron gas. Yukawa potential. Friedel oscillations at large distances.
2k_F singularities. Plasma oscillations. Poisson, Euler, and continuity equations for motion of the
classical plasma. Linearization of equations. Solution of linearized equations. Classical plasma frequency.
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| 25. |
Apr 19, Thursday |
FW 15, 16 |
Plasma oscillations in electron gas.
Collective modes: damping and dispersion. Plasmon dispersion in the limit of long wavelengths.
Sound waves in "ideal" Fermi gas (first sound). Zero sound in an imperfect Fermi gas. The zero sound speed.
Comparison with ordinary sound. Physical picture of zero sound.
Linear response for conductivity.
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| 26. |
Apr 24, Tuesday |
AGD 39.2 |
Linear response for conductivity: correction to Kubo formula.
Conductivity of an ideal Fermi gas. Damping in Green's function. Naive bubble with damped Green's
functions. Drude formula. Wrong scattering time. Gauge invariance and ladder diagrams
for polarization operator. Transport scattering time and correct Drude formula.
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| 27. |
Apr 26, Thursday |
FW 23, 24 |
Field theory at T>0. Grand canonical ensemble.
Real time Green's functions. Matsubara time and Matsubara Green's functions.
Observables: single-particle operators. Energy and theromodynamic potential. Ideal gas.
Perturbation theory. Connected diagrams.
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| 28. |
May 1, Tuesday |
FW 24 |
HW7 given.
(Anti)periodicity in imaginary time. Wick's theorem.
Feynman rules. Noninteracting Green's function in momentum representation.
Diagrams and Dyson's equations. Frequency sums.
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| 29. |
May 3, Thursday |
FW 24, AGD |
Real time temperature Green's functions. Lehmann representation and spectral function. Analytical properties.
Matsubara susceptibility. The theorem about analytical continuation for Matsubara susceptibility.
Instabilities in particle-hole and particle-particle channels. Logarithmic ivergence of 2k_F susceptibility
in particle-hole channel for nested Fermi surfaces. CDW, SDW, and SC instabilities.
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Parquet diagrams in 1D.
Bosonization. Right and left densities.
Commutator of right densities. Schwinger anomalous term. Bosonization of free fermions: Hamiltonian.
Interactions: g-ology. Renormalization of velocity.
Free boson field theory. Vertex operators and bosonization of fermion fields. Spin-charge separation.
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