# QM2 Introduction to Field Theory for Condensed Matter

## Module title: Introduction to Quantum Field Theory for Condensed Matter (QM2)

### Module convenor: Professor Igor Lerner (Birmingham)

### Module aims:

In the past few decades, the focus of research in condensed matter physics has shifted towards many-particle problems. Although historically a one-particle approach in terms of "quasi-particles", based on Landau's Fermi-liquid theory, was hugely successful for the description of electrons in metals or cold atomic Fermi gases, it utterly fails in describing "strongly-correlated" systems or even single-particle motion in a disordered potential.

The most appropriate language to describe many-body problems is that of quantum field theory. I chose for the present course a particular "dialect" of this language - the functional integral approach. It is particularly convenient for two most important tasks. First, changing variables in the functional integral allows one to find the best available "non-interacting" reference state for the system, corresponding to one or another mean-field (MF) approximation. Second, by considering fluctuations around the reference state (which play the role of low-energy elementary excitations for the system), one can find relevant corrections to the MF solution by building regular (diagrammatic) expansion. I will apply these techniques to a few condensed-matter systems, aiming mostly at illustrating the capabilities of the method rather than describing in detail physical properties of these systems.

In this introductory course, I will use the most pedestrian approach to introducing the functional integral, focusing on its applications rather than on its derivation.

### Learning objectives:

You shall understand the QFT language (or, at least, the functional-integral dialect) and shall be able to read research papers written in this language.### Syllabus

- Green's function as a functional integral: a simple derivation for non-interacting systems; a giant leap to interacting ones.
- Changing variables: Hubbard-Stratonovich transformation as a "functional bosonization"
- Gaussian approximation and perturbative expansion. Feynman diagrams
- E-h pairs and plasmons in the Coulomb gas
- Superfluidity and superconductivity. Why (and when) the Coulomb repulsion cannot beat the phonon-mediated attraction.
- The renormalisation group and (maybe) epsilon-expansion