# QM4: Quantum Hall and Many Body Effects

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

### Module convenors: Andy Schofield (Birmingham) and Nicholas d'Ambrumenil (Warwick)

The course structure is as follows:- Lecture 1 (AJS): Quantum criticality in itinerant Fermi systems: Fermi liquid theory and the Reizer singularity
- Lecture 2 (AJS):Quantum criticality in itinerant Fermi systems: The Reizer singularity
- Lecture 3 (Nd'A): Kondo Effect and Heavy Fermions
- Lecture 4 (AJS): Quantum criticality in itinerant Fermi systems: RG approach and how to correct it
- Lecture 5 (Nd'A): Fractional Quantum Hall Effect

### Module aims:

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

To understand better the background to some current 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 to characterise the order using
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.