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19
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 29 (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.
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 18 (6 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...
Nonabelian anyons and topological quantum computation
 Reviews of Modern Physics
"... Contents Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are partic ..."
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Cited by 12 (0 self)
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Contents Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as NonAbelian anyons, meaning that they obey nonAbelian braiding statistics. Quantum information is stored in states with multiple quasiparticles,
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.
NonAbelian Anyons and Topological Quantum Computation
, 2007
"... Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles know ..."
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Cited by 6 (1 self)
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Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as NonAbelian anyons, meaning that they obey nonAbelian braiding statistics. Quantum information is stored in states with multiple quasiparticles, which
Towards the fractional quantum Hall effect: a noncommutative geometry perspective
"... In this paper we give a survey of some models of the integer and fractional quantum Hall effect based on noncommutative geometry. We begin by recalling some classical geometry of electrons in solids and the passage to noncommutative geometry produced by the presence of a magnetic field. We recall ho ..."
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Cited by 6 (3 self)
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In this paper we give a survey of some models of the integer and fractional quantum Hall effect based on noncommutative geometry. We begin by recalling some classical geometry of electrons in solids and the passage to noncommutative geometry produced by the presence of a magnetic field. We recall how one can obtain this way a single electron model of the integer quantum Hall effect. While in the case of the integer quantum Hall effect the underlying geometry is Euclidean, we then discuss a model of the fractional quantum Hall effect, which is based on hyperbolic geometry simulating the multielectron interactions. We derive the fractional values of the Hall conductance as values of orbifold Euler characteristics. We compare the results with experimental data. 1. Electrons in solids – Bloch theory and algebraic geometry We first recall some general facts about the mathematical theory of electrons in solids. In particular, after reviewing some basic facts about Bloch theory, we recall an approach pioneered by Gieseker at al. [16] [17], which uses algebraic geometry to treat the inverse problem of determining the pseudopotential from the data of the electric and optical properties of the solid. Crystals. The Bravais lattice of a crystal is a lattice Γ ⊂ R d (where we assume d = 2, 3), which describes the symmetries of the crystal. The electron–ions interaction is described by a periodic potential (1.1) U(x) = ∑ u(x − γ), γ∈Γ namely, U is invariant under the translations in Γ, (1.2) TγU = U, ∀γ ∈ Γ. When one takes into account the mutual interaction of electrons, one obtains an Nparticles Hamiltonian of the form
Edge States For Quantum Hall Hamiltonians
"... The study of the quantum motion of a charged particle in a halfplane as well as in an in nite strip submitted to a perpendicular constant magnetic eld B reveals eigenstates propagating permanently along the edge, the socalled edge states. Moreover, in the halfplane geometry, current carried ..."
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Cited by 6 (3 self)
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The study of the quantum motion of a charged particle in a halfplane as well as in an in nite strip submitted to a perpendicular constant magnetic eld B reveals eigenstates propagating permanently along the edge, the socalled edge states. Moreover, in the halfplane geometry, current carried by edge states with energy in between the Landau levels persists in the presence of a perturbing potential small relative to B. We show here that edge states carrying current survive in an in nite strip for a long time before tunneling between the two edges has a destructive eect on it. The proof relies on HelerSjostrand functional calculus and decay properties of quantum Hall Hamiltonian resolvent.
Spectral flow and level spacing of edge states for quantum Hall Hamiltonians
"... We consider a non relativistic particle on the surface of a semiinfinite cylinder of circumference L submitted to a perpendicular magnetic field of strength B and to the potential of impurities of maximal amplitude w. This model is of importance in the context of the integer quantum Hall effect. In ..."
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Cited by 4 (2 self)
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We consider a non relativistic particle on the surface of a semiinfinite cylinder of circumference L submitted to a perpendicular magnetic field of strength B and to the potential of impurities of maximal amplitude w. This model is of importance in the context of the integer quantum Hall effect. In the regime of strong magnetic field or weak disorder B>> w it is known that there are chiral edge states, which are localised within a few magnetic lengths close to, and extended along the boundary of the cylinder, and whose energy levels lie in the gaps of the bulk system. These energy levels have a spectral flow, uniform in L, as a function of a magnetic flux which threads the cylinder along its axis. Through a detailed study of this spectral flow we prove that the spacing between two consecutive levels of edge states is bounded below by 2παL −1 with α> 0, independent of L, and of the configuration of impurities. This implies that the level repulsion of the chiral edge states is much stronger than that of extended states in the usual Anderson model and their statistics cannot obey one of the Gaussian ensembles. Our analysis uses the notion of relative index between two projections and indicates that the level repulsion is connected to topological aspects of quantum Hall systems. 1