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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 16 (9 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 11 (5 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.
F.: Edge and Impurity Effects on Quantization of Hall Currents
 Commun. Math. Phys
"... Abstract: We consider the edge Hall conductance and show it is invariant under perturbations located in a strip along the edge (decaying perturbations far from the edge are also allowed). This enables us to prove for the edge conductances a general sum rule relating currents due to the presence of t ..."
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Cited by 8 (1 self)
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Abstract: We consider the edge Hall conductance and show it is invariant under perturbations located in a strip along the edge (decaying perturbations far from the edge are also allowed). This enables us to prove for the edge conductances a general sum rule relating currents due to the presence of two different media located respectively on the left and on the right half plane. As a particular interesting case we put forward a general quantization formula for the difference of edge Hall conductances in semiinfinite samples with and without a confining wall. It implies in particular that the edge Hall conductance takes its ideal quantized value under a gap condition for the bulk Hamiltonian, or under some localization properties for a random bulk Hamiltonian (provided one first regularizes the conductance; we shall discuss this regularization issue). Our quantization formula also shows that deviations from the ideal value occurs if a semi infinite distribution of impurity potentials is repulsive enough to produce currentcarrying surface states on its boundary. 1.
Surface States and Spectra
, 2000
"... Let Z d+1 + = Z d Z+,letH 0 be the discrete Laplacian on the Hilbert space l 2 (Z d+1 + ) with a Dirichlet boundary condition, and let V be a potential supported on the boundary #Z d+1 + .We introduce the notions of surface states and surface spectrum of the operator H = H 0 + V and ex ..."
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Cited by 6 (1 self)
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Let Z d+1 + = Z d Z+,letH 0 be the discrete Laplacian on the Hilbert space l 2 (Z d+1 + ) with a Dirichlet boundary condition, and let V be a potential supported on the boundary #Z d+1 + .We introduce the notions of surface states and surface spectrum of the operator H = H 0 + V and explore their properties. Our main result is that if the potential V is random and if the disorder is either large or small enough, then in dimension two H has no surface spectrum on #(H 0 ) with probability one. To prove this result we combine AizenmanMolchanov theory with techniques of scattering theory.
Fractional Moment Estimates for Random Unitary
 Band Matrices”, Lett. Math. Phys
, 2005
"... We consider unitary analogs of d−dimensional Anderson models on l 2 (Z d) defined by the product Uω = DωS where S is a deterministic unitary and Dω is a diagonal matrix of i.i.d. random phases. The operator S is an absolutely continuous band matrix which depends on parameters controlling the size of ..."
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Cited by 5 (2 self)
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We consider unitary analogs of d−dimensional Anderson models on l 2 (Z d) defined by the product Uω = DωS where S is a deterministic unitary and Dω is a diagonal matrix of i.i.d. random phases. The operator S is an absolutely continuous band matrix which depends on parameters controlling the size of its offdiagonal elements. We adapt the method of AizenmanMolchanov to get exponential estimates on fractional moments of the matrix elements of Uω(Uω − z) −1, provided the distribution of phases is absolutely continuous and the parameters correspond to small offdiagonal elements of S. Such estimates imply almost sure localization for Uω. AMS classification numbers: 81Q05; 47B80 Keywords: Fractional Moment Method, Unitary Operators, Localization.
Contribution à la théorie mathématique du transport quantique dans les systèmes de Hall
, 2011
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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 2 (2 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.
DELOCALIZATION FOR RANDOM LANDAU HAMILTONIANS WITH UNBOUNDED RANDOM VARIABLES
, 2009
"... In this note we prove the existence of a localization/delocalization transition for Landau Hamiltonians randomly perturbed by an electric potential with unbounded amplitude. In particular, with probability one, no Landau gaps survive as the random potential is turned on; the gaps close, filling up ..."
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In this note we prove the existence of a localization/delocalization transition for Landau Hamiltonians randomly perturbed by an electric potential with unbounded amplitude. In particular, with probability one, no Landau gaps survive as the random potential is turned on; the gaps close, filling up partly with localized states. A minimal rate of transport is exhibited in the region of delocalization. To do so, we exploit the a priori quantization of the Hall conductance and extend recent Wegner estimates to the case of unbounded random variables.