3,943 research outputs found
Edge Currents for Quantum Hall Systems, II. Two-Edge, Bounded and Unbounded Geometries
Devices exhibiting the integer quantum Hall effect can be modeled by
one-electron Schroedinger operators describing the planar motion of an electron
in a perpendicular, constant magnetic field, and under the influence of an
electrostatic potential. The electron motion is confined to bounded or
unbounded subsets of the plane by confining potential barriers. The edges of
the confining potential barriers create edge currents. This is the second of
two papers in which we review recent progress and prove explicit lower bounds
on the edge currents associated with one- and two-edge geometries. In this
paper, we study various unbounded and bounded, two-edge geometries with soft
and hard confining potentials. These two-edge geometries describe the electron
confined to unbounded regions in the plane, such as a strip, or to bounded
regions, such as a finite length cylinder. We prove that the edge currents are
stable under various perturbations, provided they are suitably small relative
to the magnetic field strength, including perturbations by random potentials.
The existence of, and the estimates on, the edge currents are independent of
the spectral type of the operator.Comment: 57 page
Smoothness of Correlations in the Anderson Model at Strong Disorder
We study the higher-order correlation functions of covariant families of
observables associated with random Schr\"odinger operators on the lattice in
the strong disorder regime. We prove that if the distribution of the random
variables has a density analytic in a strip about the real axis, then these
correlation functions are analytic functions of the energy outside of the
planes corresponding to coincident energies. In particular, this implies the
analyticity of the density of states, and of the current-current correlation
function outside of the diagonal. Consequently, this proves that the
current-current correlation function has an analytic density outside of the
diagonal at strong disorder
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