Module title: Introduction to Quantum Field Theory for Condensed Matter

Module convenor: Professor Igor Lerner (Birmingham)

Module timetable: 1/02/2023 Wednesdays 14:00-16:00

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.

The (tentative) titles of the first 10 are below. After this, 3-4 lectures on one of the two more advanced topics: either 1D Luttinger liquid, or 2D disordered systems

Lectures

Lecture 1 - Green function as a functional integral

Lecture 2 - Diagrammatic expansion for systems with pair interactions

Lecture 3 -Diagrammatic summation. Dyson equation.

Lecture 4 - Polarisation operator. Plasmons

Lecture 5 -Hubbard-Stratonovich transformation

Lecture 6 - - Temperature technique

Lecture 7 - QFT for superconductivity 1

Lecture 8 - Ginzburg-Landau functional.

Lecture 9 - Phase transitions in spin systems

Lecture 10 - Disordered systems; diagrammatics

Lecture 11-Replica method. Nonlinear s model.

Lecture 12-Using s model; perturbation theory and renormalisation group