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25
Charge Deficiency, Charge Transport and Comparison of Dimensions
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1994
"... We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. ..."
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Cited by 41 (0 self)
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We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. We apply the relative index to counting the change in the number of electrons below the Fermi energy of certain quantum systems and interpret it as the charge deficiency. We study the relation of the charge deficiency with the notion of adiabatic charge transport that arises from the consideration of the adiabatic curvature. It is shown that, under a certain covariance, (homogeneity), condition the two are related. The relative index is related to Bellissard's theory of the Integer Hall effect. For Landau Hamiltonians the relative index is computed explicitly for all Landau levels.
P.D.: Landau Hamiltonians with random potentials: localization and the density of states
 Commun. Math. Phys
, 1996
"... We prove the existence of localized states at the edges of the bands for the twodimensional Landau Hamiltonian with a random potential, of arbitrary disorder, provided that the magnetic field is sufficiently large. The corresponding eigenfunctions decay exponentially with the magnetic field and dis ..."
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Cited by 35 (9 self)
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We prove the existence of localized states at the edges of the bands for the twodimensional Landau Hamiltonian with a random potential, of arbitrary disorder, provided that the magnetic field is sufficiently large. The corresponding eigenfunctions decay exponentially with the magnetic field and distance. We also prove that the integrated density of states is Lipschitz continuous away from the Landau energies. The proof relies on a Wegner estimate for the finitearea magnetic Hamiltonians with random potentials and exponential decay estimates for the finitearea Green’s functions. The proof of the decay estimates for the Green’s functions uses fundamental results from twodimensional bond percolation theory. KeyWords: Landau Hamiltonians, random operators, localization. Number of figures: 4
Linear Response Theory for Magnetic Schrödinger Operators in Disordered Media
, 2004
"... We justify the linear response theory for an ergodic Schrödinger operator with magnetic field within the noninteracting particle approximation, and derive a Kubo formula for the electric conductivity tensor. To achieve that, we construct suitable normed spaces of measurable covariant operators whe ..."
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Cited by 31 (14 self)
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We justify the linear response theory for an ergodic Schrödinger operator with magnetic field within the noninteracting particle approximation, and derive a Kubo formula for the electric conductivity tensor. To achieve that, we construct suitable normed spaces of measurable covariant operators where the Liouville equation can be solved uniquely. If the Fermi level falls into a region of localization, we recover the wellknown KuboStreda formula for the quantum Hall conductivity at zero temperature.
Dynamical delocalization in random Landau Hamiltonians
, 2004
"... We prove the existence of dynamical delocalization for random Landau Hamiltonians near each Landau level. Since typically there is dynamical localization at the edges of each disorderedbroadened Landau band, this implies the existence of at least one dynamical mobility edge at each Landau band, n ..."
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Cited by 27 (8 self)
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We prove the existence of dynamical delocalization for random Landau Hamiltonians near each Landau level. Since typically there is dynamical localization at the edges of each disorderedbroadened Landau band, this implies the existence of at least one dynamical mobility edge at each Landau band, namely a boundary point between the localization and delocalization regimes, which we prove to converge to the corresponding Landau level as either the magnetic field or the disorder goes to zero.
Edge current channels and Chern numbers in the integer quantum Hall effect
, 2000
"... A quantization theorem for the edge currents is proven for discrete magnetic halfplane operators. Hence the edge channel number is a valid concept also in presence of a disordered potential. Under a gap condition on the corresponding planar model, this quantum number is shown to be equal to the quan ..."
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Cited by 25 (12 self)
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A quantization theorem for the edge currents is proven for discrete magnetic halfplane operators. Hence the edge channel number is a valid concept also in presence of a disordered potential. Under a gap condition on the corresponding planar model, this quantum number is shown to be equal to the quantized Hall conductivity as given by the KuboChern formula. For the proof of this equality, we consider an exact sequence of C algebras (the Toeplitz extension) linking the halfplane and the planar problem, and use a duality theorem for the pairings of Kgroups with cyclic cohomology. 1 Introduction In quantum Hall effect (QHE) experiments, one observes the quantization of the Hall conductance of an effectively twodimensional semiconductor in units of the universal constant e 2 =h [35, 45]. As the Hall conductance is a macroscopic quantity, this effect is of completely different nature than any quantization in atomic physics resulting from BohrSommerfeld rules. Although also a pur...
Boundary maps for C ∗ crossed product with R with an application to the quantum Hall effect
 Commun. Math. Phys
"... The boundary map in Ktheory arising from the WienerHopf extension of a crossed product algebra with R is the ConnesThom isomorphism. In this article the Wiener Hopf extension is combined with the Heisenberg group algebra to provide an elementary construction of a corresponding map on higher trace ..."
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Cited by 10 (7 self)
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The boundary map in Ktheory arising from the WienerHopf extension of a crossed product algebra with R is the ConnesThom isomorphism. In this article the Wiener Hopf extension is combined with the Heisenberg group algebra to provide an elementary construction of a corresponding map on higher traces (and cyclic cohomology). It then follows directly from a noncommutative Stokes theorem that this map is dual w.r.t. Connes ’ pairing of cyclic cohomology with Ktheory. As an application, we prove equality of quantized bulk and edge conductivities for the integer quantum Hall effect described by continuous magnetic Schrödinger operators. 1 Motivation and main result In a commonly used approach to study aperiodic solids, particles in the bulk of the medium are described by covariant families of oneparticle Schrödinger operators {Hω}ω∈Ω where Ω is the probability space of configurations furnished with an ergodic action of space translations. Crossed product algebras provide a natural framework for such families [Be86]. In particular their bounded functions are represented by elements of a C ∗crossed product, the socalled
Quantization of the Hall conductance and delocalization in ergodic Landau Hamiltonians
"... Abstract. We prove quantization of the Hall conductance for continuous ergodic Landau Hamiltonians under a condition on the decay of the Fermi projections. This condition and continuity of the integrated density of states are shown to imply continuity of the Hall conductance. In addition, we prove t ..."
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Cited by 7 (3 self)
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Abstract. We prove quantization of the Hall conductance for continuous ergodic Landau Hamiltonians under a condition on the decay of the Fermi projections. This condition and continuity of the integrated density of states are shown to imply continuity of the Hall conductance. In addition, we prove the existence of delocalization near each Landau level for these twodimensional Hamiltonians. More precisely, we prove that for some ergodic Landau Hamiltonians there exists an energy E near each Landau level where a “localization length ” diverges. For the AndersonLandau Hamiltonian we also obtain a transition between dynamical localization and dynamical delocalization in the Landau bands, with a minimal rate of transport, even in cases when the spectral gaps are closed.