Equilibrium and Non-Equilibrium diagrammatic
techniques
for
many-particle systems
Priv. Doz. Dr. Mikhail Kiselev
4 hours per week (in English)
This lecture course is proposed for the students starting from 7 Semester, interested in a Diploma Project in theoretical physics. This course gives an introduction to modern methods of theoretical physics (Feynman diagram technique, path integral representation, Schwinger-Keldysh formalism). The applications of equilibrium and non-equilibrium (real-time) approaches for many-particle systems will be illustrated by numerous examples (Fermi-liquid theory, Bose-Einstein Condensation, perturbation theory in mesoscopic and nano-systems etc). The “language” of Feynman diagrams is widely used in modern scientific publications and is very useful both for the visualization of results of calculations of physical quantities and the interpretation of various many-particle effects. The course will be useful for students interested in theory of low-dimensional systems and strongly correlated phenomena.
Tentative plan for the lectures:
Lecture #1. Introduction. Ideal Fermi and Bose gases. Symmetries, spin statistics and parastatistics.
Lecture #2. Second quantization for fermions and bosons.
Lecture #3. Schrödinger, Heisenberg and Farry (interaction) representations. Green’s functions for non-interacting fermions and bosons at T=0.
Lecture #4. Wick’s theorem. Feynman diagrams.
Lecture #5. Analytical properties of the Green’s functions. Spectral densities. Lehmann representation.
Lecture #6. Dyson equation. Vertices and self-energy parts. Many-particle Green’s functions. Bethe-Salpeter equation.
Lecture #7. Some applications of T=0 technique. Electron gas with Coulomb interaction. RPA. Polarization operator. Screening. Fridel oscillations.
Lecture #8. Diagrammatic technique at finite temperatures (imaginary time formalism).
Lecture #9. Coherent states for Bose and Fermi oscillators. Path integral representation. Partition function. Generating functional.
Lecture #10. Analytical properties of Green’s functions at finite temperatures. Analytical continuation from Matsubara (discrete) to real frequencies.
Lecture #11. Method of successive integration over fast and slow variables. Electron-Ion plasma. Plasmons in 3D and 2D. Landau damping.
Lecture #12. Weakly interacting Bose gas.
Bose-Einstein Condensation.
Hugenholtz-Pines theorem.
Lecture #13. Bose gas with long-range interaction. Screening. Fridel oscillations.
Lecture #14. Basic concepts of Fermi-liquid theory. Zero sound. Luttinger theorem. Ward identities.
Lecture #15. 1D conductors. Luttinger liquid. Tomonaga model.
Lecture #16. Diagrammatic approach to the theory of superconductivity. Instability in the Cooper’s channel. Summation of ladder diagrams.
Lecture #17. Gor’kov equations. Superconductors at weak electromagnetic field.
Lecture #18. Superconducting alloys. Magnetic and non-magnetic impurities. Gapless superconductivity.
Lecture #19. Basic concepts of weak localization theory.
Lecture #20. Excitonic Insulator. Charge and Spin density waves.
Lecture #21. Mahan singularity. Summation of parquet diagrams.
Lecture #22. Diagrammatic technique for spin operators. Schwinger bosons, pseudo-fermions, Majorana fermions, Hubbard operators.
Lecture #23. Electron-mediated spin exchange. RKKY interaction in 3D and 2D. Role of imperfection.
Lecture #24. Itinerant ferromagnetism. Stoner criterion. Spin waves.
Lecture #25. Kondo effect. Perturbation theory at high temperatures. Vertices and self-energy parts. Scaling properties.
Lecture #26. Kondo effect at low temperatures. Unitarity condition. Orthogonality catastrophe.
Lecture #27. Real-time (non-equilibrium) formalism. Path integral representation. Keldysh contour.
Lecture #28. Electron-electron interaction in disordered metals: Keldysh formalism.
Recommended literature: