# QFT for condensed matter

**Module title: Introduction to Quantum Field Theory for Condensed Matter**

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

**Module timetable: TBA**

**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.

**Syllabus I**

- 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
- The renormalisation group (RG) and 4-ε dimensional expansion
- 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 second part of the course (partly to be given in the 2^{nd} semester) will be **either** on the QFT for Luttinger liquid (1D strongly interacting systems), **or** QFT for disordered systems – to be decided by the students vote by the end of the 1^{st} part.

**Lecture notes**

Lecture 1 - Green's function as a functional integral

Lecture 2 - Pair interaction. Hubbard - Stratonovich transformation

Lecture 3 - Polarisation operator

Lecture 4 - Diagrammatic expansion

Lecture 5 - Diagrammatic summation. Dyson equation

Lecture 6 - QFT for weakly interacting bosons

Lecture 7 - QFT for superconductivity 1.

Lecture 8 - Superconductivity. Ginzburg-Landau functional.

Lecture 9 - Fluctuations. BKT transition

Lecture 10 - Linearisation - Luttinger liquid. Bosonisation