Spectral Properties of Dynamical Localization for Schrödinger Operators

Germinet, François; Taarabt, Amal

doi:10.1142/s0129055x13500165

Cited by 17 publications

(15 citation statements)

References 36 publications

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“…We will simply quote a result from [1] and apply it to our models of interest. Similar results can also be found in [36,37]. We first note some notation.…”

Section: Examplesupporting

confidence: 76%

Chern Numbers, Localisation and the Bulk-edge Correspondence for Continuous Models of Topological Phases

Bourne

Rennie

2018

Math Phys Anal Geom

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In order to study continuous models of disordered topological phases, we construct an unbounded Kasparov module and a semifinite spectral triple for the crossed product of a separable C * -algebra by a twisted R d -action. The spectral triple allows us to employ the nonunital local index formula to obtain the higher Chern numbers in the continuous setting with complex observable algebra. In the case of the crossed product of a compact disorder space, the pairing can be extended to a larger algebra closely related to dynamical localisation, as in the tight-binding approximation. The Kasparov module allows us to exploit the Wiener-Hopf extension and the Kasparov product to obtain a bulk-boundary correspondence for continuous models of disordered topological phases.

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“…We will simply quote a result from [1] and apply it to our models of interest. Similar results can also be found in [36,37]. We first note some notation.…”

Section: Examplesupporting

confidence: 76%

Chern Numbers, Localisation and the Bulk-edge Correspondence for Continuous Models of Topological Phases

Bourne

Rennie

2018

Math Phys Anal Geom

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“…Note that the most studied class of operators for which (2.21) holds consists of Schrödinger operators with i.i.d. potential or more generally, potentials with sufficiently fast decay of statistical correlations, see [6,15,42] and the items (a) -(c) of the list after formula (2.21). However, the bound (2.21) also holds for one dimensional Schrödinger operators with quasiperiodic potentials (see, e.g., [21] and the item (d) of the list after (2.21)), which have, so to speak, a minimal amount of randomness.…”

Section: Resultsmentioning

confidence: 99%

“…potentials) is based on the exponential decay (4.41) of the Green function's fractional moments of the operator in question. We refer the reader to the works [3,4,6,15,21,29,42] for results and references on various aspects of the validity and applications of the bound. Although the operator P (ω) introduced in (2.9) is the Fermi projection of the ergodic Schrödinger operator H(ω) given by (2.14), a considerable amount of our results can be formulated and proved independently of the origin of P (ω).…”

Section: (215)mentioning

confidence: 99%

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Large Block Properties of the Entanglement Entropy of Free Disordered Fermions

2016

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We consider a macroscopic disordered system of free d-dimensional lattice fermions whose one-body Hamiltonian is a Schrödinger operator H with ergodic potential. We assume that the Fermi energy lies in the exponentially localized part of the spectrum of H. We prove that if S Λ is the entanglement entropy of a lattice cube Λ of side length L of the system, then for any d ≥ 1 the expectation E{L −(d−1) S Λ } has a finite limit as L → ∞ and we identify the limit. Next, we prove that for d = 1 the entanglement entropy admits a well defined asymptotic form for all typical realizations (with probability 1) as L → ∞. According to numerical results of [33] the limit is not selfaveraging even for an i.i.d. potential. On the other hand, we show that for d ≥ 2 and an i.i.d. random potential the variance of L −(d−1) S Λ decays polynomially as L → ∞, i.e., the entanglement entropy is selfaveraging.

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“…(2.25) see, for instance, the recent surveys [15] and [16]. The bound is a manifestation of complete localization, that is, the pure point character of the spectrum and the exponential decay of the eigenfunctions of the one-dimensional Schrödinger operator with i.i.d.…”

Section: )mentioning

confidence: 99%

On the analogues of Szegő's theorem for ergodic operators

Kirsch¹,

Pastur²

2015

Sb. Math.

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Szegő's theorem on the asymptotic behaviour of the determinants of large Toeplitz matrices is generalized to the class of ergodic operators. The generalization is formulated in terms of a triple consisting of an ergodic operator and two functions, the symbol and the test function. It is shown that in the case of the one-dimensional discrete Schrödinger operator with random ergodic or quasiperiodic potential and various choices of the symbol and the test function this generalization leads to asymptotic formulae which have no analogues in the situation of Toeplitz operators.Bibliography: 22 titles.

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Spectral Properties of Dynamical Localization for Schrödinger Operators

Cited by 17 publications

References 36 publications

Chern Numbers, Localisation and the Bulk-edge Correspondence for Continuous Models of Topological Phases

Chern Numbers, Localisation and the Bulk-edge Correspondence for Continuous Models of Topological Phases

Large Block Properties of the Entanglement Entropy of Free Disordered Fermions

On the analogues of Szegő's theorem for ergodic operators

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