Quantum Hall transitions: an exact theory based on conformal restriction

Details Quantum Hall transitions: an exact theory based on conformal restriction Quantum Hall transitions: an exact theory based on conformal restriction

2012-02-21

arxiv.org/abs/1202.4573v1

Physics

Condensed Matter

Disordered Systems and Neural Networks

Scientific paper

We revisit the problem of the plateau transition in the integer quantum Hall effect. Here we develop a novel analytical approach for this transition, and for other 2D disordered systems, based on the theory of `{\it conformal restriction}'. This theory was recently developed within the context of the `{\it Schramm-Loewner Evolution}' [SLE]. Observables elucidating the connection with the plateau transition include {\it point-contact conductances} [PCCs] between points on the boundary of the sample, described within the Chalker-Coddington model for the transition. We show that the disorder-averaged PCCs are characterized by a {\it classical} probability distribution for objects in the plane, occuring with positive statistical weights, that satisfy the crucial `restriction property' with respect to changes in the shape of the sample with {\it absorbing boundaries} - physically, these are boundaries connected to {\it ideal leads}. At the transition point these geometrical objects ({\it pictures}) become fractals. Upon combining this `restriction property' with the expected conformal invariance at the transition point, we employ the mathematical theory of `conformal restriction measures' to relate the disorder-averaged PCCs to correlation functions of (Virasoro) primary operators in a conformal field theory (of central charge $c=0$). We show how this can be used to calculate these functions in a number of geometries with various boundary conditions. Our results are equally applicable to a number of other critical disordered electronic systems in two spatial dimension, including for example the spin quantum Hall effect, the thermal metal phase in symmetry class D, and classical diffusion in two dimensions in a perpendicular magnetic field. For most of these systems we also predict exact values of critical exponents related to the spatial behavior of various disorder-averaged PCCs.

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