Recommended books:

Abrikosov, Gorkov, and Dzyaloshinsky: Methods of Quantum Field Theory in Statistical Physics

Fetter and Walecka: Quantum Theory of Many-Particles Systems

Bruus and Flensberg: Many-Body Theory in Condensed Matter Physics.

Topics:

Lecture 1 (5th September): Brief review of zero-temperature Green's function method. The idea of Imaginary time evolution. The modified Heisenberg picture. Field operators in imaginary time. Thermal Green's functions. Basic properties. Density, expectation value of one-body operators. Obtaining the interaction and the total internal energy from the thermal Green's function. Obtaining the grand-canonical thermodynamical potential, thermodynamic properties.

Problem Set 1.

Lecture 2 (12th September): Imaginary time periodicity of Mastrubara Green's functions. Fourier series, Mastubara frequencies. Free Green's functions for particles. The modified interaction picture. The imaginary time evolution in the modified interaction picture.

Problem Set 2.

Lecture 3 (19h September): The pertubative series of the grand-canonical thermodynamical potential and the Green's function. The finite temperature Wick's theorem. Feynman rules in real space.

Problem Set 3.

No lecture on the 26th September.

Lecture 4 (3rd  October): The linked cluster theorem. The Feynman rules in momentum space and Mastubara frequencies. Calculation of frequency sums. The Dyson equation.

Problem Set 4.

Lecture 5 (10th October): The direct expansion of the grand-canonical thermodynamical potential.

Lecture 6 (17th October): Thermodynamics of the Coulomb-interacting gas: ring diagrams in the irreducibe self-energy, the bare polarization bubble, the grand-canonical thermodynamical potential by coupling constant integration. The classical limit (ionized classical gas).

Problem Set 5.

Lecture 7 (24th October): Real-time Green's functions. Lehmann representation, the spectral function. Analytical properties, sum rule. Analytic contunuation of Matsubara Green's funcions.

Lecture 8 (31st October): Linear response theory at finite temperature. The density-density correlation function. Application to the classical gas of charged particles. The screening of a charged impurity.

Lecture 9 (7th November): Plasma oscillations at finite temperature. Landau damping. The plasmon dispersion in three dimensions.

Lecture 10 (21st November): Superconductivity. Anomalous correlation functions. Weak-coupling superconductivity. The Gorkov equations.

Lecture 11 (28th November): The solution of the Gorkov equation in the absence of a magnetic field. The gap equation. Universal predictions for weak coupling superconductivity.

Problem Set 6.

Lecture 12 (5th December): Strong coupling superconductivity (Eliashberg theory).