Pure Scaling Operators at the Integer Quantum Hall Plateau Transition

If a longitudinal voltage is applied to a MOSFET subject to low temperature and a strong perpendicular magnetic field, one finds that the transversal conductivity is strictly quantized according to $\frac{ne^2}{h}$ with positive integer $n$. This is called the plateau region. At the same time, the longitudinal conductivity vanishes; this is the integer quantum Hall effect. Changing the magnetic field eventually results in a quantum phase transition between the plateaus. By a central paradigm of statistical physics this transition should be described by a conformal field theory. Starting from the critical Chalker-Coddington network model, which is a discrete model for this transition, we suggest a family of observables as discretizations of conformal primary fields. In the simplest case these reduce to the disorder average of moments of the absolute value square of stationary scattering states of the network evaluated at a specific link $l$ in the presence of a point contact $c$. In a plane network algebraic decay with the distance between $l$ and $c$ is expected. The network model is connected to the supersymmetric vertex model by a duality transformation and the suggested observables correspond to highest weight operators of the symmetry algebra $\fgl_{2n|2n}$. Numerical simulations in cylindrical and rectangular geometries provide strong evidence for the conjecture that these observables have indeed the desired properties. Finally, we address the question of the precise form of the multifractal spectrum of conformal dimensions, which is an important one to ask in view of the correct continuum field theory. The family of observables is, compared to work done in the past two decades, quite extensive and will hopefully contribute to a conclusive answer on the parabolicity of the spectrum.