Extended States in the Anderson Model on the Bethe Lattice

Klein, Abel

doi:10.1006/aima.1997.1688

Cited by 132 publications

(144 citation statements)

References 25 publications

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“…Before closing this subsection, it is worth mentioning that stability conditions corresponding to minimization of the velocity [i.e., analogous to our Eq. (27)] are known to emerge in a broad class of related non-linear problems describing propagation of a front between an unstable and stable phases (here ψ = 1 and ψ = 0, respectively (42) and (48). On the other hand, the singularities II and IV in ψR(t), which develop into curved parts of ln[1 − ξR,r(t)] (as indicated by arrows), give rise to the non-linear segments (47) and (50) of the spectrum τq.…”

Section: A ψR(t) and Wave Function Moments At The Rootmentioning

confidence: 99%

Multifractality of wave functions on a Cayley tree: From root to leaves

2017

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We explore the evolution of wave-function statistics on a finite Bethe lattice (Cayley tree) from the central site ("root") to the boundary ("leaves"). We show that the eigenfunction moments Pq = N |ψ| 2q (i) exhibit a multifractal scaling Pq ∝ N −τq with the volume (number of sites) N at N → ∞. The multifractality spectrum τq depends on the strength of disorder and on the parameter s characterizing the position of the observation point i on the lattice. Specifically, s = r/R, where r is the distance from the observation point to the root, and R is the "radius" of the lattice. We demonstrate that the exponents τq depend linearly on s and determine the evolution of the spectrum with increasing disorder, from delocalized to the localized phase. Analytical results are obtained for the n-orbital model with n 1 that can be mapped onto a supersymmetric σ model. These results are supported by numerical simulations (exact diagonalization) of the conventional (n = 1) Anderson tight-binding model.

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Section: A ψR(t) and Wave Function Moments At The Rootmentioning

confidence: 99%

Multifractality of wave functions on a Cayley tree: From root to leaves

2017

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“…There are many articles on the multi-scale analysis approach, for a readable book see Stollmann (2001), which, however, does not contain any of the new developments in the multi-scale approach. We will also not discuss at all the results on delocalization in the Anderson model on trees instead of Z d , see , Froese et al (2007), and Klein (1998). For a nice and readable introduction to the physics of random Schrödinger operators, see, for example, Lifshits et al (1988).…”

Section: Disclaimermentioning

confidence: 99%

A Short Introduction to Anderson Localization

Hundertmark¹

2008

Analysis and Stochastics of Growth Processes and Interface Models

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“…The first rigorous proof of absolutely continuous spectrum for the Anderson model in the Bethe lattice was obtained by Klein, [22], [25], and [23], using a supersymmetric transfer matrix method. These methods were extended to the Bethe strip in our previous work [28], where we proved the existence of absolutely continuous spectrum in the Bethe strip.…”

Section: Introductionmentioning

confidence: 99%

Ballistic behavior for random Schrödinger operators on the Bethe strip

Klein¹,

Sadel²

2011

J. Spectr. Theory

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Abstract. The Bethe strip of width m is the Cartesian product B f1; : : : ; mg, where B is the Bethe lattice (Cayley tree). We consider Anderson-like Hamiltonians H D 1 2 ˝1 C 1˝A C V on a Bethe strip with connectivity K 2, where A is an m m symmetric matrix, V is a random matrix potential, and is the disorder parameter. Under certain conditions on A and K, for which we previously proved the existence of absolutely continuous spectrum for small , we now obtain ballistic behavior for the spreading of wave packets evolving under H for small . Mathematics Subject Classification (2010). Primary 82B44; Secondary 47B80, 60H25.

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Extended States in the Anderson Model on the Bethe Lattice

Cited by 132 publications

References 25 publications

Multifractality of wave functions on a Cayley tree: From root to leaves

Multifractality of wave functions on a Cayley tree: From root to leaves

A Short Introduction to Anderson Localization

Ballistic behavior for random Schrödinger operators on the Bethe strip

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