Eldad Bettelheim, Ilya A. Gruzberg, Andreas W. W. Ludwig
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.
View original:
http://arxiv.org/abs/1202.4573
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