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Many-body physics

Alex Tsyplyatyev, WS 2023/2024

Lectures: 4 hours per week, on Wednesdays 14:00-16:00 and on Fridays 12:00-14:00
Room: Phys 01.114 (Wednesdays and Fridays)
First lecture: 18.10.2023
Last lecture: 09.02.2024 (29 two-hour lectures in total, 2 weeks of Christmas vacation 22.12.2023 - 05.01.2024)

Tutorials: 2 hours every second week on Fridays 10:00-12:00
Room: Phys 01.114
Tutor: Mr. Dmytro Tarasevich/Mr. Renan Hirayama
First tutorial: 03.11.2023 (7 weeks of tutorials in total, dates are 03.11, 17.11, 01.12, 15.12, 12.01, 26.01, 09.02)

This many-body course builds on top of the traditional bachelor courses in theoretical physics and is aimed at all students starting from the 7th semester. It introduces the foundations of the quantum field theory methods needed for describing systems consisting of many interacting particles in the solid-state physics. The prerequisites for this course are all of the bachelor theory courses (VHT1-5). Knowledge of the advanced quantum mechanics is advantageous but is not strictly necessary, its relevant parts will be briefly introduced at the beginning of this course.

Announcements

The tutorial classes start on Friday the 3rd of November.

Lecture notes

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Complete lecture notes

Problem sheets

  1. Ladder operators, Their matrix representation, Conservation of particle number.
  2. Green function of an interacting system, Fourier transform of the Theta function, Spectral function.
  3. Time evolution operator in the interaction picture, Infinite sums via the residue theorem, Sokhotsky-Plemelj formula.
  4. Spectral representation of the retarded Green's function, Density of states of a many-body system, Finite-temperature Green function of a non-interacting system.
  5. Wick's theorem, Second order Feynman diagrams
  6. Self-energy of the Hubbard Model in perturbation theory
  7. Divergent perturbation theory in one dimension, Spectral function of Luttinger liquid

Master solutions

  1. Ladder operators, Their matrix representation, Conservation of particle number.
  2. Green function of an interacting system, Fourier transform of the Theta function, Spectral function.
  3. Time evolution operator in the interaction picture, Infinite sums via the residue theorem, Sokhotsky-Plemelj formula.
  4. Spectral representation of the retarded Green's function, Density of states of a many-body system, Finite-temperature Green function of a non-interacting system.
  5. Wick's theorem, Second order Feynman diagrams
  6. Self-energy of the Hubbard model in perturbation theory
  7. Divergent perturbation theory in one dimension, Spectral function of Luttinger liquid
Content of the course
  1. Introduction: the many-body problem in physics
  2. Second quantisation
  3. Common many-body models
  4. Green functions at T=0 and T>0
  5. Landau's Fermi liquid in D>1
  6. Grand potential in the diagrammatic language
  7. Random phase approximation
  8. Ladder approximation
  9. Bosonisation technique
  10. Luttinger liquid in D=1
Literature
  • A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Dover, New York, 1975. (main book for Green functions)
  • T. Giamarchi, Quantum physics in one dimension, Clarendon press, Oxford, 2010. (main book for Luttinger liquid)
  • R. D. Mattuck, A guide to Feynman diagrams in the many-body problem, Dover, Toronto, 1992.
  • G. D. Mahan, Many particle physics, Kluwer, New York, 2000.