A Short Introduction to Anderson Localization

Hundertmark, Dirk

doi:10.1093/acprof:oso/9780199239252.003.0009

Cited by 37 publications

(39 citation statements)

References 35 publications

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“…Self-avoiding walks have also appeared in mathematical studies of large disorder localization in a survey by Hundertmark [9] and more recently in works of Tautenham [12] and Suzuki [11]. The work of Tautenhahn [12], in particular, is quite closely related to the present work.…”

Section: Introductionmentioning

confidence: 52%

How Large is Large? Estimating the Critical Disorder for the Anderson Model

Schenker

2014

Lett Math Phys

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Complete localization is shown to hold for the d-dimensional Anderson model with uniformly distributed random potentials provided the disorder strength λ > λ And where λ And satisfies λ And = μ d e ln λ And with μ d the self-avoiding walk connective constant for the lattice Z d . Notably, λ And is precisely the large disorder threshold proposed by Anderson in 1958.Mathematics Subject Classification. 82B44, 81Q10.

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Section: Introductionmentioning

confidence: 52%

How Large is Large? Estimating the Critical Disorder for the Anderson Model

Schenker

2014

Lett Math Phys

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“…This is especially important for practical applications as instantaneous readout requires effectively infinite bandwidth, which is usually unavailable. If the input and output states are identical, extending the time window yields solutions that achieve Anderson localization [4], [28], the closest equivalent to asymptotic closed-loop stability for Hamiltonian quantum networks. Sensitivity properties of time-windowed optimized controllers are analyzed from the statistical point of view of concordance between error and sensitivity as shown in Fig.…”

Section: Discussionmentioning

confidence: 99%

Design of Feedback Control Laws for Information Transfer in Spintronics Networks

Schirmer

Jonckheere

Langbein

2018

IEEE Trans. Automat. Contr.

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Information encoded in networks of stationary, interacting spin-1/2 particles is central for many applications ranging from quantum spintronics to quantum information processing. Without control, however, information transfer through such networks is generally inefficient. Currently available control methods to maximize the transfer fidelities and speeds mainly rely on dynamic control using time-varying fields and often assume instantaneous readout. We present an alternative approach to achieving efficient, high-fidelity transfer of excitations by shaping the energy landscape via the design of time-invariant feedback control laws without recourse to dynamic control. Both instantaneous readout and the more realistic case of finite readout windows are considered. The technique can also be used to freeze information by designing energy landscapes that achieve Anderson localization. Perfect state or super-optimal transfer and localization are enabled by conditions on the eigenstructure of the system and signature properties for the eigenvectors. Given the eigenstructure enabled by super-optimality, it is shown that feedback controllers that achieve perfect state transfer are, surprisingly, also the most robust with regard to uncertainties in the system and control parameters.

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“…On the contrary, the Moment Analysis is to a large extent free of this limitation: the IAD can be safely replaced by regularity of conditional distributions induced on one or two points, given the values of the potential at all remaining points. Hundertmark [15] proposed a very simple and elegant reformulation of the FMM, in the case of strong disorder on the lattice. The starting point for his analysis is a re-summation of the formal perturbation series for the inverse (H 0 + V − E) −1 via "elimination of loops" in the usual graphic expansion (alas, divergent in vicinity of the spectrum).…”

Section: Resultsmentioning

confidence: 99%

“…In [15] the latter appear in the so-called self-energy entering one-dimensional integrals over the conditional measure. Naturally, in the Gaussian case the conditional measure is either supported by a single point or analytic, with rapidly decaying tail probabilities.…”

Section: Resultsmentioning

confidence: 99%