# QM5: Quantum Field Theory and Luttinger Liquids

## Module title: Quantum Field Theory and Luttinger Liquids(QM5)

### Module convenor: Prof. 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.

The second part of the course is about strongly correlated one-dimensional (1D) systems. Although 1D systems were of great interest for theorists for more than 60 years (with main tools for their description built in 1980's), the experimental realisations were relatively scarce until the end of 1980's (polymers and organic compounds). But since then we witnessed a real burst in 1D experiments: quantum wires, carbon nanotubes, Josephson junction arrays, edge states in Quantum Hall systems, cold atomic Bose or Fermi gases in 1D optical traps. On top of this, properties of many quasi-1D systems (like organic superconductors, various ladder and spin compounds) are mostly defined by their 1D excitations.

In one-dimensional systems both interaction and disorder are crucial: in contrast to higher dimensions, particles cannot avoid each other and are strongly localized even by a small disorder. This makes low-energy elementary excitations in 1D systems very special: they practically never resemble the underlying particles. Landau's Fermi-liquid theory never works for fermions in 1D: instead, the interacting particles form the Luttinger liquid of "plasmons" or "phonons". It appears possible to describe it quite thoroughly – without disorder, one can build an exact description of a strongly interacting 1D system, something which is sadly impossible in higher dimensions. The key to this is a so called “bosonisation” which will be our main QFT tool for dealing with 1D systems.

### 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
- Interacting 1D systems – a total failure of the perturbative expansion.
- Spin-charge separation.
- Functional and operator bosonisation – reducing a quartic Hamiltonian to an exactly solvable quadratic one.
- The one-impurity problem – how a single obstacle kills metallic conductivity in 1D.