# QM4: Quantum Hall and Many Body Effects

## Module title: Quantum Hall and Many Body Effects (QM4)

### Module convenors: Andy Schofield and Nicholas d'Ambrumenil

The course structure is as follows:- Lecture 1 (AJS): Quantum criticality in itinerant Fermi systems: Fermi liquid theory and the Reizer singularity
- Lecture 2 (Nd'A): Kondo Effect and Heavy Fermions
- Lecture 3 (Nd'A): Quantum Hall Effect
- Lecture 4 (AJS): Quantum criticality in itinerant Fermi systems: SCR and Hertz Millis theory
- Lecture 5 (AJS): Quantum criticality in itinerant Fermi systems: What is going wrong?

### Module aims:

To explore quantum phenomena in interacting many-particle systems.### Learning objectives:

At the end of the module you should understand better the background to some current major issues in condensed matter physics.### Syllabus

Kondo Effect

The Anderson impurity, solution for the ground state in the large N limit. The Kondo resonance. Scattering theory and phase shifts: elastic scattering rate, Friedel sum rule, Wigner delay time. The quasiparticle interpretation valid for T \ll T_K and unitarity limit. Kondo lattices and the idea of the coherence temperature.

Quantum Hall Effect

Modulation doped heterostructures and the measurements. Zeros in wavefunctions in a magnetic field. Landau levels (LL) and projection onto a LL. Laughlin's wavefunction and quasiholes. Composite fermions and the singular gauge transformation. Effective mass as a parameter to fit to experiment/exact diagonalization studies. Wavefunctions in higher LL's, the pfaffian. Mention of degeneracy and non-abelian statistics.

Quantum Criticality in Itinerant Fermi Systems

Derivation of the effective (Hertz-Millis) action for a system close to a zero temperature ordering transition
(quantum critical point).
Introduction of the slowly varying field characterising the order via
the Hubbard-Stratonovic transformation. Integrating
out the fermions, expansion in the coupling to fermions. Importance of the term linear in frequency and its relation
to relaxation . Comparison with standard Landau theory of phase transitions at non-zero temperature.
Scaling analysis of the action and identification of the temperature-controlled and
order-controlled parts of the phase diagram. Relation of Tc to the fourth order term.
Temperature-dependence of heat capacity and resistivity, comparison to experiments.