Localisation in a Quantum Hall Wire

Abstract

The high precision of quantization of the Hall conductance of a 2–dimensional electron system in a strong magnetic field is known to be due to the binding of electrons to localised states in the bulk of the system. Thereby, a change of electron density does not result in a change of the Hall conductance. The transition between Hall–plateaus is understood as a delocalisation transition, when the chemical potential is in the middle of a disorder broadened Landau–band, En. There, a critical state does exist. It is believed, and supported by analytical, numerical and experimental studies, that this transition is in an extended system of second order, characterised by a diverging localisation length with universal exponent ν, ξ ∼ (μ − En)−ν . This exponent is found to be close to ν = 2.3, for spin split Landau levels, although there are experiments and numerical simulations which do not fit into the single parameter scaling scheme. The conformal field theory of the critical point is believed to belong to a supersymmetric manifold, and candidates have been suggested. The derivation of this critical theory is expected to follow from a nonperturbative analysis of the Hamiltonian of disordered electrons by means of the supersymmetry method. Integrating out the modes with wave lengths shorter than the elastic mean free path le, this theory reduces to a nonlinear sigma model (NLSM), which depends only on two parameters, the bare longitudinal conductivity σxx, and the Hall conductivityσxy, as given by self consistent Born approximation (SCBA). There are strong indications, based on a perturbative renormalization group analysis, that this supersymmetric field theory is critical in two dimensions, whenever the Hall conductivity is a halfinteger multiple of the conductance quantum e/h. 12) A mapping on the thermodynamics of super spin chains has supported this conjecture, generalising the Lieb, Schultz, Mattis theorem for the existence of quantum criticality to super spin chains. Exact information is so far only available when the NLSM is restricted to its compact sector, by mapping the problem on the thermodynamics of antiferromagnetic spin chains. This theory is critical when the ”spin” quantum number, as given by the Hall conductance σxy, has a half integer value. Meanwhile, it is of interest to use the available nonperturbative field theory to calculate measurable quantities other than the universal exponent ν. Exact calculations are possible for quasi-1–dimensional wires, in which