Results 1  10
of
31
Localization Bounds for an Electron Gas
, 1998
"... Mathematical analysis of the Anderson localization has been facilitated by the use of suitable fractional moments of the Green function. Related methods permit now a readily accessible derivation of a number of physical manifestations of localization, in regimes of strong disorder, extreme energies ..."
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Cited by 48 (8 self)
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Mathematical analysis of the Anderson localization has been facilitated by the use of suitable fractional moments of the Green function. Related methods permit now a readily accessible derivation of a number of physical manifestations of localization, in regimes of strong disorder, extreme energies, or weak disorder away from the unperturbed spectrum. The present work establishes on this basis exponential decay for the modulus of the two–point function, at all temperatures as well as in the ground state, for a Fermi gas within the one–particle approximation. Different implications, in particular for the Integral Quantum Hall Effect, are reviewed.
Anyons in an exactly solved model and beyond
, 2005
"... A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge f ..."
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Cited by 27 (2 self)
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A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are nonAbelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and nonAbelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.
D.: Dissipative transport: thermal contacts and tunneling junctions. Ann. Henri Poincaré 4, 897
 Goderis, D., Vets, P.: Central
, 2004
"... The general theory of simple transport processes between quantum mechanical reservoirs is reviewed and extended. We focus on thermoelectric phenomena, involving exchange of energy and particles. Entropy production and Onsager relations are relevant thermodynamic notions which are shown to emerge fro ..."
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Cited by 24 (5 self)
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The general theory of simple transport processes between quantum mechanical reservoirs is reviewed and extended. We focus on thermoelectric phenomena, involving exchange of energy and particles. Entropy production and Onsager relations are relevant thermodynamic notions which are shown to emerge from the microscopic description. The theory is illustrated on the example of two reservoirs of free fermions coupled through a local interaction. We construct a stationary state and determine energy and particle currents with the help of a convergent perturbation series. We explicitly calculate several interesting quantities to lowest order, such as the entropy production, the resistance, and the heat conductivity. Convergence of the perturbation series allows us to prove that they are strictly positive under suitable assumptions on the interaction between the reservoirs.
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 17 (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.
Topological Equivalence of Tilings
 J.MATH.PHYS., VOL
, 1997
"... We introduce a notion of equivalence on tilings which is formulated in terms of their local structure. We compare it with the known concept of locally deriving one tiling from another and show that two tilings of finite type are topologically equivalent whenever their associated groupoids are isomor ..."
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Cited by 16 (8 self)
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We introduce a notion of equivalence on tilings which is formulated in terms of their local structure. We compare it with the known concept of locally deriving one tiling from another and show that two tilings of finite type are topologically equivalent whenever their associated groupoids are isomorphic.
Intermixture of extended edge and localized bulk energy levels in macroscopic Hall systems
 J. Phys. A
"... We study the spectrum of a random Schrodinger operator for an electron submitted to a magnetic eld in a nite but macroscopic two dimensional system of linear dimensions equal to L. The y direction is periodic and in the x direction the electron is con ned by two smooth increasing boundary potent ..."
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Cited by 13 (3 self)
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We study the spectrum of a random Schrodinger operator for an electron submitted to a magnetic eld in a nite but macroscopic two dimensional system of linear dimensions equal to L. The y direction is periodic and in the x direction the electron is con ned by two smooth increasing boundary potentials. The eigenvalues of the Hamiltonian are classi ed according to their associated quantum mechanical current in the y direction. Here we look at an interval of energies inside the rst Landau band of the random operator for the in nite plane. In this energy interval, with large probability, there exist O(L) eigenvalues with positive or negative currents of O(1). Between each of these there exist O(L ) eigenvalues with in nitesimal current O(e ). We explain what is the relevance of this analysis to the integer quantum Hall eect.
Effective dynamics for Bloch electrons: Peierls substitution and beyond
, 2002
"... We reconsider the longstanding problem of an electron moving in a crystal under the influence of weak external electromagnetic fields. More precisely we analyze the dynamics generated by the Schrödinger operator H = 1 2 (−i∇x − A(εx))2 + V (x) + φ(εx), where V is a lattice periodic potential and A a ..."
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Cited by 13 (4 self)
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We reconsider the longstanding problem of an electron moving in a crystal under the influence of weak external electromagnetic fields. More precisely we analyze the dynamics generated by the Schrödinger operator H = 1 2 (−i∇x − A(εx))2 + V (x) + φ(εx), where V is a lattice periodic potential and A and φ are external potentials which vary slowly on the scale set by the lattice spacing. We study the limit ε → 0 in several steps: (i) Approximately invariant subspaces associated with isolated Bloch bands are constructed. (ii) We derive an effective quantum Hamiltonian for states inside such a decoupled subspace. The effective Hamiltonian has an asymptotic expansion in ε, starting with the term given through the Peierls substitution. Our construction allows, in principle, to compute also all higher order terms and we give the first order correction to the Peierls substitution explicitly. (iii) The semiclassical limit of the effective
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 12 (6 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.
Adiabatic Charge Transport And The Kubo Formula For 2D Hall Conductance
 Comm. Pure Appl. Math
, 2004
"... We study adiabatic charge transport in a two dimensional lattice model of electron gas at zero temperature. It is proved that if the Fermi level falls in the localization regime then, for a slowly varied weak electric eld, in the adiabatic limit the accumulated excess Hall transport is correctly ..."
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Cited by 11 (1 self)
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We study adiabatic charge transport in a two dimensional lattice model of electron gas at zero temperature. It is proved that if the Fermi level falls in the localization regime then, for a slowly varied weak electric eld, in the adiabatic limit the accumulated excess Hall transport is correctly described by the linear response Kubo Streda formula. Corrections to the leading term are given in an asymptotic series for the Hall current in powers of the adiabatic parameter. The analysis is based on an extension of an expansion of Nenciu, with the spectral gap condition replaced by localization bounds.
Cantor and band spectra for periodic quantum graphs with magnetic fields
 Comm. Math. Phys
"... ABSTRACT. We provide an exhaustive spectral analysis of the twodimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lya ..."
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Cited by 11 (3 self)
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ABSTRACT. We provide an exhaustive spectral analysis of the twodimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable KronigPenney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the BetheSommerfeld conjecture fails in this case.