Programme

Ist Part

1 - Homogeneous electron gas in a neutralizing background (jellium model).
Zero order approximation: Sommerfeld theory. Coulombian interaction as perturbation. Hartree-Fock theory for electron gas. Distribution Definition of correlation energy.

2- Second quantization: Fock space for bosons and fermions. Creation and destruction operators. Field operators. Operators in second quantization. The jellium model in second quantization, first order perturbation and comparison with the Hartree-Fock results. Beyond the first order approximation the many body perturbative theory.

3 - The Green Functions for the Electron Gas. Green functions for non-interacting electrons. Lehmann representation.
Perturbative development for Green's functions. Dyson equation. Self-energy.

4 - Polarization propagator. Polarization diagrams. Proper polarization.
Correlation energy in terms of polarization propagator. Random phase approximation (RPA). Dielectric function in RPA. High-density limit, Thomas-Fermi screen. Limit to large wavelengths, plasma oscillations.

5- Introduction to the Density Functional theory for the solid state physics.

2nd Part

1 - The phenomenon of superfluity. The phase diagram of He4. The superfluid phase of liquid helium.
The theory of the two fluids. Landau's theory: critical speed, rotons and phonons.
Bogoliubov's theory of interacting bosons.
Hydrodynamics and vorticity. Vortices as liquid helium excitations.
Recent achievements of Bose-Einstein condensation.

2 - The phenomenon of superconductivity. Zero resistance, Meissner Effect, Critical Magnetic Field,
Specific heat. Analogies with the phenomenon of superfluidity. London equation. Thermodynamic considerations.
Superconductors of the first type and of the second type.

3 - Microscopic theory of superconductivity. Electron-phonon interaction. Attractive interaction between electrons. Cooper pairs. Bardeen-Cooper-Schrieffer's theory (BCS): fundamental state, definition of energy gap. Excited states. Calculation at finite temperature. Quantization of magnetic flux.

4 - Phenomenological theory of Ginzburg-Landau. Landau theory of phase transitions. Free energy of superconductors.
Ginzburg-Landau equations and relation with the London equation.

5 - Symmetry breaking and transition from normal to superconducting state.

Core Documentation

A.L.FETTER, J.D.WALECKA "QUANTUM THEORY OF MANY PARTICLES"
G.GROSSO, G.PASTORI-PARRAVICINI "SOLID STATE PHYSICS"

Type of delivery of the course

In the first part of the course the field theory formalism is introduced in the non relativistic limit and applied to the electron gas. The perturbative theory requires analytical developments of a certain difficulty, for this reason the use of the blackboard is preferred as it allows us to show the subsequent steps in a slower way with the possibility for students to request clarifications at each step of the developments. A similar methodology is required in the second part of the course dedicated to the behavior of matter in the limit of temperatures that tend to absolute zero.

Type of evaluation

The final exam tends to understand whether the student has achieved a reasonable understanding of the topics. On the one hand, it is necessary to check that he has a general understanding of the various parts of the course and that he knows how to connect them together. For this reason, the oral examination consists partly of questions aimed at ascertaining these aspects without going into the details of formalism. The student then presents a topic of his choice to be developed in detail. This allows to ascertain how much the student is able to manage the techniques of Quantum Mechanics in the formulation necessary for the study of systems with many electrons.