Università Roma Tre

First part: Electron gas.
I-1 – Second quantization. Systems of identical particles. Multi-particle Hamiltonian. Operators for bosons and fermions. Field operators.

I-2 – Electron gas in an uniform neutralizing background (jellium model).
Zero-order approximation: theory of Sommerfeld. Coulomb interaction as a perturbation. Hartree-Fock theory for the electron gas.
Definition of the correlation energy. Single particle theory for electrons in solids. Hohemberg and Kohn theorem, functional density theory.

I-3 - Green functions for Fermions.
Schroedinger representation. Heisenberg representation. Representation of
interaction. Time-ordered product of operators. Adiabatic perturbation theory. Theorem of Gell-Mann and Low. Definition of Green functions. Observable in terms of Green functions. Propagator of free electrons. Lehmann representation. Analytic properties of the Green functions.
Wick's theorem. Analysis in diagrams of the perturbation theory. Feynman diagrams.

I-4 - Perturbation theory for the electron gas. Self-energy and Dyson equation. Linear response theory. Polarization propagator. Polarization diagrams. Correlation energy in terms of the polarization propagator. Random Phase Approximation (RPA). Dielectric function in RPA. Limit at high density and Thomas-Fermi screening. Limit at large wavelengths, plasma oscillations.

Second part: States of matter at low temperatures.

II-1 The phenomenon of superfluidity. Bose-Einstein condensation. Liquid helium in the superfluid phase. Theory of the two fluids. Landau theory: critical velocity, phonons and rotons. Vorticity. Bogoliubov theory for interacting bosons. Recent realization of the BEC.

II-2 - The phenomenon of superconductivity. Transition to zero resistivity, Meissner effect, critical magnetic field, specific heat. Analogies with the phenomenon of superfluidity. London equations. Thermodynamic considerations. Superconductors of the first type and
of the second type.

II-3 - Microscopic theory of superconduttivity. Electron-phonon interaction. Attractive interaction between electrons. Cooper pairs. Theory of Bardeen-Cooper-Schrieffer (BCS). The ground state, the definition of the energy gap. Excited states. Calculation at finite temperature. Bogoliubov theory: equations in the presence of external fields. Quantizationof the magnetic flux.

II-4 - Phenomenological Ginzburg-Landau theory. Landau theory of phase transitions.
Free energy of the superconductor, Ginzburg-Landau equations and relation
with the London equations. Ginzburg parameter. Symmetry breaking and transition from normal to superconductivity state.

A. L. Fetter and J. D.Walecka, Quantum theory of many-particle systems.
G. Grosso and G. Pastori-Parravicini, Solid State Physics