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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 2nd 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 1st 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