Many Body Theory 2

The course presents the theory of many particles in thermal equilibrium, with applications to: particles in disordered potential, superconductivity, superfluidity. The main topics are: the electron-phonon interaction and the Cooper pairing, review of the gran-canonical formalism, imaginary-time evolution, thermal T-ordered and retarded Green functions, expansion with Matsubara frequencies, KMS property, equations of motion, Wick's theorem, Feynman diagrams, Lehmann representation, linear response, evaluatiuon of the thermodynamic potential, particles in a random potential (conductivity and T-matrix). Thermodynamics of superconductivity, Ginzburg-Landau equations. BCS model. Superfluidity (phenomenology and Bogoliubov's theory)

Basics of elasticity theory. Origin of the electron-phonon interaction, attractive regime, Cooper pairing.
Basics of thermodynamics in gran-canonical ensemble. Perturbative evaluation of the potential.
Knowledge of the interaction picture and T-exp of propagator in imaginary time.
Thermal Green functions. Motivate the distinction among fermionic and bosonic frequencies.
Apply Wick's theorem to the evaluation of correlators. Deduce the reduction formula.
Evaluate the analytic expression of a Feynman diagram in x and k space.
Evaluate the Lehmann representation for retarded and T-ordered Green functions.
Evaluate the necessary formulae for linear response.
Basics of thermodynamics of supeconductors. Deduce the Ginzburg-Landau equations. Reproduce Abrikosov's evaluation to classify type I and type II superconductors. Know the order of magnitude of critical fields and temperatures, typical lengths.
Write and motivate the BCS Hamiltonian. Knowledge of the matrix formalism by Nambu and Gorkov for Green functions.
Obtain and discuss the gap equation for a homogeneous superconductor.
Basics of phenomenology of superfluid Helium, and the theory by Bogoliubov.