Many body theory 2

A.Y. 2020/2021
Overall hours
Learning objectives
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)
Expected learning outcomes
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.
Course syllabus and organization

Single session

Lesson period
First semester
The lectures will be delivered online via ZOOM, according to the timetable. The modulus begins just after the end of the first modulus (approximately mid-November).
The recorded lesson and the notes will be published daily in ARIEL. The program should not vary significantly, as well as the exam (online).
Course syllabus
Homogeneous elastic media. Phonons. Debye cutoff. Phonon-electron interaction.
Gran canonical formalism. Ideal gases, Bose Einstein condensation. Hartree-Fock at finite T. Temperature Green functions. KMS property and Matsubara frequencies.
Wick's theorem and Feynman's rules at finite T. Self energy, polarization and Dyson's equations. Lehmann's expansion and retarded functions. Linear response. Applications: RPA, screening, plasma oscillations, Debye Huckel's state equation. Particle in a random potential, T matrix, Ohm's law. Superconductivity: phenomenology, thermodynamics, London's equations, Ginzburg Landau theory, surface energy and type I,II superconductors, Bogoliubov De Gennes theory, BCS, Nambu Gorkov equations.
Superfluidity: phenomenology, vortices, phonon modes.
Prerequisites for admission
It is strongly advised to attend the first module. Necessary background: second-quantisation, basics of statistical mechanics and thermodynamics (ideal gases, gran canonical ensemble), of mathematical methods (complex integral, Fourier series and integral, distributions, Gamma and Zeta functions), of quantum mechanics (identical particles, Heisenberg and interaction pictures, symmetries).
Teaching methods
Lesson with blackboard. Normally, in the end of the course, we visit the laboratory for superconductivity at LASA.
Teaching Resources
Assessment methods and Criteria
Oral exam of about 1H, in agreed date. Before, the student solves some exercises freely chosen among proposed ones, or from the textbook. After the presentation of a
subject of choice, there are questions about the program, with evaluation of order of magnitude of some quantity, of a Feynman diagram, connection with other courses (structure of matter, quantum mechanics).
Lessons: 42 hours