Existence and Regularity Properties of the Integrated Density of States of Random Schrödinger Operators

Veselić, Ivan

doi:10.1007/978-3-540-72691-3

Cited by 14 publications

(7 citation statements)

References 244 publications

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“…1 The subscript B in the symbol C B refers to the strength of a constant magnetic field perpendicular to the plane. The C * -algebra C B is unital and is endowed with a natural finite trace T B known as the trace per unit volume (see [66] for a recent review). The NGC built in [6] is based on the spectral triple ρ, H 2 , D where H 2 := 2 Z 2 ⊗C 2 is a separable Hilbert space, ρ is the diagonal representation of C B on H 2 defined by ρ(A) := A ⊗ 1 2 and D is the Dirac operator defined by…”

Section: Introductionmentioning

confidence: 99%

The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects

Nittis

Sandoval²

2020

SIGMA

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This work provides a first step towards the construction of a noncommutative geometry for the quantum Hall effect in the continuum. Taking inspiration from the ideas developed by Bellissard during the 80's we build a spectral triple for the C∗-algebra of continuous magnetic operators based on a Dirac operator with compact resolvent. The metric aspects of this spectral triple are studied, and an important piece of Bellissard's theory (the so-called first Connes' formula) is proved.

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Section: Introductionmentioning

confidence: 99%

The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects

Nittis

Sandoval²

2020

SIGMA

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“…It is well known, see [KM82a;Ves08], that the finite volume restrictions of H ω , ω ∈ Ω, have compact resolvents, so that their spectrum is purely discrete. The eigenvalue counting functions…”

Section: General Setupmentioning

confidence: 99%

“…(1.5) Note, that in [BK13] the continuity of N on all of R is proved for d ∈ {1, 2, 3} and bounded V per and W ω . For specific types of W ω , Wegner estimates are available, implying the continuity of the IDS, see e. g. [Ves08] and references therein. For breather potentials this is discussed specifically in [TV15; Nak+18].…”

Section: By An Analogous Argument We Have For Continuity Pointsmentioning

confidence: 99%

Lifshitz asymptotics and localization for random breather models

Schumacher,

Veselic

2021

Preprint

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We prove Lifshitz behavior at the bottom of the spectrum for non-negative random potentials, i. e. show that the IDS is exponentially small at low energies. The theory is developed for the breather potential and generalized to all non-negative random potentials in a second step. Since our models need not be ergodic, we need to identify the minimum of the spectrum. We deduce an initial length scale estimate from the Lifshitz bound for the breather model and combine it with a recent Wegner estimate to establishes Anderson localization via multi-scale analysis. Finally, for ergodic models, we complement the Lifshitz behavior with a lower bound. We provide detailed proofs accessible to non-experts.

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“…The calculation uses in a crucial way the cyclicity of the trace and the translation invariance of Â ∈ A per , namely s γ ⊗1 r , Â = 0 for all γ ∈ Γ. The last term in the above equality provides the usual definition of the trace-per-unit-volume (see [Ves08] for a general review) and the equality does not depend on the particular choice of Følner sequence.…”

Section: The Algebra Of Periodic Operatorsmentioning

confidence: 99%

Topological polarization in graphene-like systems

Nittis

Lein

2013

J. Phys. A: Math. Theor.

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In this article we investigate the possibility of generating piezoelectric orbital polarization in graphene-like systems which are deformed periodically. We start with discrete two-band models which depend on control parameters; in this setting, time-dependent model hamiltonians are described by loops in parameter space. Then, the gap structure at a given Fermi energy generates a non-trivial topology on parameter space which then leads to possibly non-trivial polarizations. More precisely, we show the polarization, as given by the King-Smith-Vanderbilt formula, depends only on the homotopy class of the loop; hence, a necessary condition for non-trivial piezo effects is that the fundamental group of the gapped parameter space must not be trivial. The use of the framework of non-commutative geometry implies our results extend to systems with weak disorder. We then apply this analysis to the uniaxial strain model for graphene which includes nearest-neighbor hopping and a stagger potential, and show that it supports non-trivial piezo effects; this is in agreement with recent physics literature.

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Existence and Regularity Properties of the Integrated Density of States of Random Schrödinger Operators

Cited by 14 publications

References 244 publications

The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects

The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects

Lifshitz asymptotics and localization for random breather models

Topological polarization in graphene-like systems

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