Titles and Abstracts

TITLES AND ABSTRACTS


V. Bazhanov
 Quantum Brownian motion in a periodic potential and conformal field theory

J. Bellissard
The Non Commutative Geometry of the Integer Quantum Hall Effect

1. The Non Commutative Brillouin Zone
Main physical facts about the IQHE: localisation and topological invariants. The need for a new mathematical tool: the earlier works (Laughlin, Thouless et al, Avron et al.). The Non Commutative language: non commutative topological spaces and Non Commutative Manifolds. The Non Commutative Brillouin Zone. Calculus.

2. The Four Traces Way
The Kubo formula and Non Commutative Chern classes. The four traces: ordinary, trace per unit volume, graded trace, Dixmier's trace. Fredholm modules and the 1st Connes formula. The 2nd Connes formula and localization. Quantization of the Hall conductivity, Fredholm index, the relative index and the Laughlin argument. Existence of plateaux.

3. Why are the Plateaux so Flat ?
Localization theory within the Non Commutative set up. Transport theory. The Relaxation Time Approximation (RTA). Application to the QHE: success and failure of the RTA. Beyond the RTA: Mott's variable range hopping conductivity. Mathematical formalism. Flatness of plateaux and accuracy of the IQHE. Conclusion: possible extensions to the FQHE.

P. Bouwknegt
The fractional quantum Hall effect and conformal field theory

This will be a review talk on the applications of (two-dimensional) conformal field theory to the study of fractional quantum Hall liquids. In particular we will review the works of Wen (cf. Adv. Phys. 44 (1995) 405) and Moore and Read (Nucl. Phys. B360 (1991) 362).

A. Cappelli
 W-Infinity Symmetry in the Quantum Hall Effect

I first describe the W-infinity symmetry of the incompressible Hall fluids and its consequences for the dynamics of the edge excitations in the quantum Hall effect. Next, I present the W-infinity minimal models, which we have proposed for describing the edge excitations of the hierarchical Hall states. Finally, I discuss the evidences in support of these models.

L.-H. Chim
Exclusion Statistics and Fermionic Character Formulas

In this talk I will review Haldane's notion of exclusion statistics and its application to the quantum Hall effect. I will also discuss a connection between this notion and fermionic character formulas arising in conformal field theory.

P. Forrester
 Correlation functions and Jack Polynomials

K. Hannabuss
 The Quantum Hall Effect in non-commutative hyperbolic space

The non-commutative geometric treatment of the integer quantum Hall effect in Euclidean space given by Bellissard has an analogue for the hyperbolic unit disc or upper half plane, with a discrete group action, which will be discussed in this talk.

A. Hassell
 On the ground state energy of the fractional quantum Hall effect

Talk based on recent work of J. Xia [Commun. Math. Phys. 204 (1999) 189-206].

P. Jarvis
 S-function determinant formulae, tableaux decompositions and the fermion-boson correspondence

Talk based on work with Angele Hamele and Ming Yung [Journ. of Mathematical and Computer Modelling 26 (1997) 149-159].

D. Neilson
 Destabilization of the 2D conducting phase by an in-plane magnetic field

A mechanism is proposed for the recently reported destabilization by an in-plane magnetic field of the conducting phase of low density electrons in 2D. We apply a many-body conserving approach we have developed to the the fully spin polarized correlated 2D electron system. Our approach is based on the memory function formalism and takes into account both disorder and exchange-correlation effects. We show that spin polarization significantly favors localization because of the enhancement of the exchange-correlations. A key outcome is that the conducting phase for the fully spin polarized system is significantly suppressed. The in-plane magnetic field needed to stabilize the fully spin polarized state lies in the range 0.1 H 1 T, depending on the carrier density. We determine the metal-insulator phase diagram for the unpolarized and fully polarized systems, and obtain good agreement with the experimentally observed dependence of the critical magnetic field on the carrier density.

I. Raeburn
 Modelling graphs in C*-algebras

A survey of Cuntz and Cuntz-Krieger type algebras.

D. Robinson
 On periodic systems

Periodic systems arise naturally when considering evolution equations on manifolds with a compact direction. A review of some of the relevant topics and recent results will be given.

K. Schoutens
 1. The fractional quantum Hall effect

The opening lecture of the workshop will be a colloquium-style review of the various quantum Hall effects: integer, fractional and non-abelian. I shall review the basic phenomenology and discuss some aspects of the theory, with special emphasis on three proto-typical quantum Hall states: an integer quantum Hall state at filling fraction nu=1, Laughlin's fractional quantum Hall state at nu=1/3 and a non-abelian quantum Hall state at nu=1/2.

2. Non-abelian quantum Hall states

The experimental observation of a quantum Hall effect at nu=5/2 has prompted the study of non-abelian quantum Hall states. Characteristic features are (i) the quasi-hole excitations satisfy non-abelian (exchange or exclusion) statistics and (ii) the wave functions have a BCS-type form which reveals a fundamental pairing (or clustering) of the constituent electrons. In this talk I present a novel class of non-abelian quantum Hall states, and discuss the exclusion statistics of their fundamental excitations. The resulting K-matrix structure reveals a fundamental duality between the non-abelian statistics and the pairing physics. [Based on work with E. Ardonne, P. Bouwknegt and S. Guruswamy.]

M. Simmons
 Metal-insulator transitions in two dimensions

The existence of the Quantum Hall effect is direct evidence for the presence of both localised and extended states in two-dimensional systems. It is still however unclear what happens to these extended states as B->0. In 1979 the single parameter scaling theory of localisation stated that any amount of disorder in a 2D system prevents the existence of extended states in the absence of a magnetic field at T=0. In this talk I will discuss weak localisation experiments that supported this theory and more recent results which have re-opened the question of metallic behaviour at B=0 in 2D. I will review the growing body of experimental evidence in high quality 2D systems, where the interaction strength is known to be strong, which supports the existence of a metallic state. In particular I will discuss recent results that we have obtained on extremely low disorder two dimensional hole gases in the high quality GaAs material system. Using a series of samples we have investigated the effects of disorder and interaction strength on the metallic state, which have important implications for the low temperature nature of this state.

T. Sunada
 Long time asymptotics for the transition probability of a random walk on a crystal lattice

E. Tosatti
 How are the zero field 2D metal-insulator and the Quantum Hall-insulator transitions connected?

In this very informal lecture, I shall try to address possible connections between the 2D insulator-metal transition at B_perp = 0, discussed in detail in the two previous contributions by Neilson and by Simmons, and the insulator-Quantum Hall state transitions which are found as a function of B_perp. Data by the Princeton group do show that these two apparently unrelated kinds of transition, one attributed to the quantum melting of a disordered Wigner glass, the other to crossing a half filling delocalized state at the end of Quantum Hall plateau, can be continuously deformed into one another, strongly suggesting a deeper common origin. I will discuss in what sense one could envisage quantum melting in the Quantum Hall system at nu = 1/2. The electronic two and three body short range correlation functions are extracted from small cluster calculations and their nature shown to change across nu = 1/2. The crucial role here is played by electron-hole symmetry, requiring hole correlations above nu = 1/2 to be identical to electron correlations below, and viceversa for electrons. This introduces a frustration precisely at nu = 1/2, where electron and hole short-range correlations cannot be simultaneously the same, as electron-hole symmetry would dictate. Hence a quantum melting, or at least a quantum disorder point, must necessarily take place at nu = 1/2.