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Tentative plan

Note that this plan is subject to change.

Hors d'oeuvre

Mean-field theories

Hedin's equations: the general formalism of Green's functions

References:

  • Hedin, L. and S. O. Lundqvist. “Effects of Electron-Electron and Electron-Phonon Interactions on the One-Electron States of Solids”. in Solid State Physics 23 (1969) 1.
  • refer to Solyom's book, Jishi book of diagrammatics, Bruus, advanced Kittel & Ziman for better connection to solid state theory.
  • Pine’s “elementary excitations in solids”

Goals:

  • understanding Green's function techniques in depth
  • obtaining the know-how to analyse and physically interpret the results, such as spectral functions, self-energy, response functions, vertex corrections, etc.
  • produce self-consistent diagrammatics
  • understanding adiabatic continuity and Gell-Mann--Low theorem
  • understanding the many-body results also in terms of the earlier wave-function approach, based on the single-particle picture; e.g., Hartree-Fock approximation, random phase approximation (RPA) and plasmons, Overhauser’s spin waves [Pine’s “elementary excitations in solids"]
  • learn how to relate theory to experiments, and understand and interpret experimental results: e.g., XPS, ARPES, etc. refer to Jülich lecture notes

Fundamentals of Fermi liquid theory as the paradigm of strongly interacting many-body systems

Goals:
– deriving the properties of a Fermi liquid
– Luttinger theorem
– topological view of a Fermi liquid

References:
refer to AGD book,
Mudry (esp. chp on topological Fermi liquid theory),
Martin's book

Interacting bosonic systems:

Goals:
– quantum many-body understanding of electromagnetism: photons
– quantum optics fundamentals
– Bose-Einstein condensate and anomalous Green functions

References:
AGD book, Mudry, Jishi

Transport properties

Goals:
– Linear response theory
– Meir-Wingreen formula
– Boltzmann equation
– Landauer-Bütticker formalism

References:
Bruus
Rammer
papers

Functional-integral formalism fundamentals

– Hubbard-Stratonovich transformation

Effective action formalism and conserving approximations

References:
Baym and Kadanoff paper
chapter by Bickers
Effective action formalism -> Bergerson's (?) review
Stefanucci and van Leuween: Luttinger-Ward functionals
BCS theory of Hassler and Morawetz

Magnetism (esp. itinerant magnetism) and Kondo effect

Stoner model
Anderson model
Giamarchi lecture notes
Bell Lab guys on Kondo
Nozieres paper on X-ray problem
Kondo's review

Non-equilibrium many-body theory

Goals:
Keldysh technique
refer to Jülich lecture notes
Master eq approach and Born-Markov approximation

Breakdown of quantum phases and quantum phase transitions (beyond mean-field)

Goals:
– Renormalization group (RG) techniques
Kopietz book: Functional renormalization group
paper by Shankar on fermionic renormaliztion
paper by Hertz and Millis

Exotic quantum phases

Goals:
– Breakdown of Fermi liquids and non-Fermi liquid theory
low-dimensional systems:
1D: Tomonaga-Luttinger liquid (Abelian Bosonization) -> Bruus, review by Brazilian J. Physics
2D: topological phases: Kosterlitz-Thouless transition