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Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model

# Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model,10.1007/s00220-011-1204-2,Communications in Mathematical Physics,Lász

Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model
We consider Hermitian and symmetric random band matrices H in d ≥ 1 dimensions. The matrix elements H xy , indexed by $${x,y \in \Lambda \subset \mathbb{Z}^d}$$, are independent, uniformly distributed random variables if $${\lvert{x-y}\rvert}$$ is less than the band width W, and zero otherwise. We prove that the time evolution of a quantum particle subject to the Hamiltonian H is diffusive on time scales $${t\ll W^{d/3}}$$. We also show that the localization length of the eigenvectors of H is larger than a factor W d/6 times the band width. All results are uniform in the size $${\lvert{\Lambda}\rvert}$$ of the matrix.
Journal: Communications in Mathematical Physics - COMMUN MATH PHYS , vol. 303, no. 2, pp. 509-554, 2011
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## Citation Context (2)

• ...We proved recently [1] that the quantum time evolution e −itH/2 generated by a band matrix H with band width W is diffusive on time scales t � W d/3 ,...
• ...A key assumption in [1] was that the matrix entries Hxy satisfy...
• ...with the extended states conjecture for random Schr¨ odinger operators,see the introduction of [1], where we also presented an overview of related results and references...
• ...Under these assumptions we show that all results of [1] remain valid...
• ...4 we briefly summarize the Chebyshev expansion of the propagator from [1]...
• ...The arguments in this section are similar to those of [1], except that we also need to analyse higher order cumulants...
• ...11 of [1], this involves a refined classification of all skeleton graphs in terms of how much they deviate from the 2/3 rule (Lemma 7.7 in [1])...
• ...11 of [1], this involves a refined classification of all skeleton graphs in terms of how much they deviate from the 2/3 rule (Lemma 7.7 in [1])...
• ...The quantity (t, x) has the interpretation of the probability of finding a quantum particle at the lattice site x at time t, provided it started from the origin at time 0. Here the time evolution of the quantum particle is governed by the Hamiltonian H. See [1] for more details...
• ...Our main result generalizes Theorem 3.1 of [1] to the class of band matrices with general distribution and covariance introduced in Sect...
• ...Remark 3.2. The number λ ∈ [0, 1] in (3.4) represents the fraction of the macroscopic time T that the particle spends moving effectively; the remaining fraction 1 − λ of T represents time the particle “wastes” in backtracking...
• ...√ 1−λ2 1(0 λ 1) of the particle moving a fraction λ of the total macroscopic time T . See Sect. 3 of [1] for a more detailed discussion...
• ...Remark 3.3. As a corollary of Theorem 3.1 ,w e getdelocalization of eigenvectors of H on scales W 1+dκ/2 . Indeed, the methods of [1], Sect...
• ...10, imply that the localization length of the eigenvectors of H is with high probability larger than the band width times W dκ/2 . See [1], Theorem 3.3 and Corollary 3.4, for a precise statement as well as a proof...
• ...<{[SECTION]}>4. Summary of the Chebyshev Expansion from [1]...
• ...The starting point of our proof is the same as in [1], i.e...
• ...The Chebyshev transform αn(t) of the propagator e −itξ was computed in [1 ]( see [1], Lemma 5.1),...
• ...The Chebyshev transform αn(t) of the propagator e −itξ was computed in [1 ]( see [1], Lemma 5.1),...
• ...We shall need the following basic estimates on αn(t); see [1] Eqs...
• ...As observed in [1], the coefficient αn(t) is very small for n � t .T hus, we choose a cutoff exponent μ satisfying...
• ...In order to prove the claim (iii), pick ζ 0 such that 1 + ξ =c oshζ. Using (11.1) we get for ξ ∈ [0, 1]...
• ...H .T hen we get forξ ∈ [0, 1], using Lemma 11.2 (ii) and (iii),...

### László Erdos, et al. Quantum Diffusion and Delocalization for Band Matrices with General Di...

• ...For � W −1/3 , the asymptotics (6 )f ollows from the recent result of Erd˝ os and Knowles [4 ]( see also [5] for an extension to more general random band matrices)...
• ...On the other hand, the methods of [2 ]a nd [4, 5] allow to handle the (more difficult and physically more interesting) quantity � |(H − E 0 − i�) −1 (0, 0)| 2 , (8)...

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