## Diagrammatic methods

### Yu. Makhlin

The goal of this course is to introduce the students to the modern methods of condensed-matter theory and at the same time to discuss a number of specific physical phenomena. In the class we consuder problems, which illustrate various methods, based on the diagram technique for the Green functions. These methods are applied to modern problems in many-particle physics, including Fermi-liquid theory, disordered systems, superconductivity.

### Syllabus

1. Quasiparticles. Elementary excitations in a quantum Fermi liquid. Second quantization. Canonical transformations.
2. Green functions of Fermi and Bose systems at zero temperature. Interaction representation. Chronological ordering. Green function of a macroscopic system. Physical meaning of the poles. Analytical properties of the Green function.
3. Basic principles of diagrammatics. Interacting particles. Wick theorem. Feynman diagrams for various interactions. Diagram technique in coordinate and momentum space. Block diagram summation. Dyson equation. Vertex diagrams. Two-particle Green function. Bethe-Salpiter equation.
4. Ideal Fermi gas. Green function for an ideal Fermi gas. Electrons at the Fermi surface. Electron-hole symmetry. Kubo formula.
5. Electron-phonon interaction. Hamiltonian of the electron-phonon interaction. Fröhlich Hamiltonian. Phonon Green function. Migdal theory. Quasiparticle lifetime. Renormalization of the electron spectrum. Absence of the vertex renormalization. Renormalization of the sound velocity. Peierls instability.
6. Diagram technique at finite temperatures. Matsubara time. Matsubara Green function. Discrete frequencies. Feynman rules for Matsubara diagrammatics. Method of analytical continuaion.
7. Fermi-liquid theory. Quasiparticles. Landau functional. Kinetic equation. Collective excitations. Properties of the vertex diagrams at low momentum transfer.
8. Electrons in a random potential. Averaging over disorder. Diagram technique for disorder averaging. Diagrams without self-intersection. Averaging of response fuinctions. Conductivity of the electron gas. Diffusion equation. Quantum correction to the conductivity.
9. Microscopic theory of superconductivity. Diagram technique for the BCS theory. Scattering in the Cooper channel, Cooper ladder. Green function for a superconductor. Basic equations for a superconductor. Superconductor in an electromagnetic field.

### Recommended texts

1. A. A. Abrikosov, L. P. Gor'kov, I. E. Dzyaloshinskii, Methods of Quantum Field Theory in Statistical Physics, Prentice-Hall, 1963
2. E. M. Lifshitz, L. P. Pitaevskii, Statistical Physics, Part 2, Pergamon, 1980
3. L. S. Levitov and A. V. Shytov, Green's functions. Theory and practice (in Russian), Fizmatlit, Moscow, 2003.
4. G.D.Mahan, Many-particle Physics, N.Y., Plenum Press, 1990.