A new proof of localization in the Anderson tight binding model

Dreifus, Henrique von; Klein, Abel

doi:10.1007/bf01219198

Cited by 256 publications

(319 citation statements)

References 17 publications

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“…The existing results on localization in the continuum of dimension larger than one have been based on the multiscale analysis, first obtained for discrete operators [30,25,29,69,74] and then extended to the continuum [50,18,32]. We do not attempt to give an exhaustive survey of the related literature.…”

Section: (3) Decay Of the Fermi-projection Kernelmentioning

confidence: 99%

“…With the bounds provided by Lemmas 3.3 and 3.4, the proof of Theorem 1.2 proceeds according to a set of arguments familiar from the fractional moment method for discrete operators [6], and related to the multi-scale analysis of random Schrödinger operators, e.g. [74,18], as well as the analysis of a number of lattice models in statistical mechanics, e.g. [37,66,52,7,5,26].…”

Section: Finite-volume Criteriamentioning

confidence: 99%

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Moment analysis for localization in random Schrödinger operators

et al. 2005

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Abstract. We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weak-L 1 estimate concerning the boundary-value distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.

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Section: (3) Decay Of the Fermi-projection Kernelmentioning

confidence: 99%

Section: Finite-volume Criteriamentioning

confidence: 99%

Moment analysis for localization in random Schrödinger operators

et al. 2005

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“…What one usually proves is the so-called exponential localization [1,6,28], i.e., pure point spectrum and exponentially decaying eigenfunctions. On the other hand, it is also known that exponential localization does not imply dynamical localization [12]; it is usually needed a precise control of the decay of the eigenfunctions, called SULE [12,16], that can be obtained through the method of multiscale analysis, a technique set out by Fröhlich and Spencer [14,15].…”

Section: Introductionmentioning

confidence: 99%

“…For all energies E for which γ m (E) > 0, a initial estimate for localization (Lemma 4) and the Wegner's estimate (Lemma 3) will be checked; by adapting the method multiscale analysis [28,15,16] to this model, it will be shown (see Theorems 2 and 3) that for typical realizations the spectrum of ID(m, c) is pure point and the corresponding eigenfunctions are semi-uniformly exponentially localized (SULE) [12,16]. This and the results of [16] (properly adapted to ID(m, c)) imply dynamical localization.…”

Section: Introductionmentioning

confidence: 99%

Spectral and localization properties for the one-dimensional Bernoulli discrete Dirac operator

Oliveira

Prado

2005

Journal of Mathematical Physics

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Abstract. A 1D Dirac tight-binding model is considered and it is shown that its nonrelativistic limit is the 1D discrete Schrödinger model. For random Bernoulli potentials taking two values (without correlations), for typical realizations and for all values of the mass, it is shown that its spectrum is pure point, whereas the zero mass case presents dynamical delocalization for specific values of the energy. The massive case presents dynamical localization (excluding some particular values of the energy). Finally, for general potentials the dynamical moments for distinct masses are compared, especially the massless and massive Bernoulli cases.

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“…Critical reworkings of multiscale analysis were developed by von Dreifus in his thesis [18] and with Klein [19]. Among extensions to settings beyond the lattice models that [36] consider, I would mention [14,15,16,40,46,48,53]; see the review articles mentioned next for more.…”

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confidence: 99%

A Celebration of Jürg and Tom

Simon

2008

J Stat Phys

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It is both a pleasure and an honor to write the introduction of this issue in honor of the (recent) sixtieth birthdays of Jürg Fröhlich and Tom Spencer and to be able to place their joint work in some perspective.Tom and Jürg have about twenty-five joint papers, several with additional authors including two with me. I want to focus here on two sets of methods: infrared bounds and multiscale analysis, which are surely among the most significant developments in rigorous statistical physics in the last quarter of the last century.Infrared bounds ([29, 30]), discovered in 1975 and proven using reflection positivity, provide upper bounds on the Fourier transform of the spin-spin correlation at nonzero momentum and force a macroscopic occupation of zero momentum at low temperature (aka Bose-Einstein condensation of spin waves). This implies long-range order, and so, a phase transition.The method was used for quantum spin antiferromagnets by DysonLieb-Simon [20,21]. Remarkably, after more than thirty years, it remains the only method known to rigorously prove breaking of nonabelian symmetry-even for the abelian case, there is only one other approach to the short-range case using multiscale analysis (see below). For slow decay two-dimensional plane rotors, there are also results of .Among later applications of infrared bounds are Sokal's specific heat bounds [59], Aizenman's [1] and Fröhlich's [24] proofs of the triviality of φ 4 theories in five or more dimensions, the Aizenman-Fernández analysis of long-range models [4], Helffer's estimates of eigenvalue splitting for certain Schrödinger operators in the thermodynamic limit [43,44], and the work of Biskup-Chayes on mean-field driven phase transitions [7,8].Reflection positivity was introduced in Euclidean field theory by Osterwalder-Schrader [54] and was a key element, albeit implicitly, Date: September 15, 2008. Mathematics 253-37,

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A new proof of localization in the Anderson tight binding model

Abstract: We give a new proof of exponential localization in the Anderson tight binding model which uses many ideas of the Frohlich, Martinelli, Scoppola and Spencer proof, but is technically simpler-particularly the probabilistic estimates.

Cited by 256 publications

References 17 publications

Moment analysis for localization in random Schrödinger operators

Moment analysis for localization in random Schrödinger operators

Spectral and localization properties for the one-dimensional Bernoulli discrete Dirac operator

A Celebration of Jürg and Tom

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