From the Anderson Model on a Strip to the DMPK Equation and Random Matrix Theory

Bachmann, Sven; Roeck, Wojciech De

doi:10.1007/s10955-010-9947-2

Cited by 11 publications

(29 citation statements)

References 28 publications

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“…Let us comment on the limiting process of transfer matrices A(s), and compare it to the ideal ensemble M of Section 2. As already discussed in [1], the overall factor √ 4 − E 2 only corresponds to a redefinition of the mean free path and is irrelevant here. The major difference lies in the diagonal of the processes a, a generating A, which have perfectly correlated diagonal elements, whereas they are independent of each other in their cousins generating M. In the case β = 1, an additional deviation can be found in the variance of the diagonal elements of b, which are smaller here than in the ideal case by a factor √ 2 · N/(N + 1).…”

Section: Theoremmentioning

confidence: 84%

“…Moreover, we incorporate technical improvements (among other things borrowing some terminology from [20]), mostly concerning the statement of the joint scaling limit in Theorem 8. Finally, we study the convergence as 1 The physically most natural way to discuss β = 4 as well would be to consider electrons with spin, which we chose not to do for reasons of simplicity N → ∞ of a hierarchy of equations for the moments of the conductance introduced by [17]. We prove that the limit satisfies Ohm's law, see Theorem 2.…”

Section: Introductionmentioning

confidence: 98%

“…However, the localization length increases with N (roughly Date: November 6, 2018. as s ∼ N , at least in the weak disorder limit λ → 0) and we can ask how the system behaves for s N , before localization sets in. There, one can distinguish the ballistic regime s ≤ 1, where incoming electrons did not yet get scattered by the impurities, and the most interesting diffusive regime characterized by (1) 1 s, s/N 1 .…”

Section: Introductionmentioning

confidence: 99%

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Disordered Quantum Wires: Microscopic Origins of the DMPK Theory and Ohm’s Law

2012

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ABSTRACT. We study the electronic transport properties of the Anderson model on a strip, modeling a quasi one-dimensional disordered quantum wire. In the literature, the standard description of such wires is via random matrix theory (RMT). Our objective is to firmly relate this theory to a microscopic model. We correct and extend previous work [1] on the same topic. In particular, we obtain through a physically motivated scaling limit an ensemble of random matrices that is close to, but not identical to the standard transfer matrix ensembles (sometimes called TOE, TUE), corresponding to the Dyson symmetry classes β = 1, 2. In the β = 2 class, the resulting conductance is the same as the one from the ideal ensemble, i.e. from TUE. In the β = 1 class, we find a deviation from TOE. It remains to be seen whether or not this deviation vanishes in a thick-wire limit, which is the experimentally relevant regime. For the ideal ensembles, we also prove Ohm's law for all symmetry classes, making mathematically precise a moment expansion by Mello and Stone [17]. This proof bypasses the explicit but intricate solution methods that underlie most previous results.

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Section: Theoremmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Disordered Quantum Wires: Microscopic Origins of the DMPK Theory and Ohm’s Law

2012

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“…The first arxiv version of the present paper was followed by the preprint of the paper by Bachmann and De Roeck [3], who, in independent work, also study SDE limits of transfer matrices. We refer the reader to the paper by Bachmann and De Roeck [3] for a discussion of this theory. We refer the reader to the paper by Bachmann and De Roeck [3] for a discussion of this theory.…”

Section: Introductionmentioning

confidence: 99%

Random Schrödinger operators on long boxes, noise explosion and the GOE

Valkó

Virág

2014

Trans. Amer. Math. Soc.

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It is conjectured that the eigenvalues of random Schrödinger operators at the localization transition in dimensions d ≥ 2 behave like the eigenvalues of the Gaussian Orthogonal Ensemble (GOE). We show that there are sequences of n × m boxes with 1 m n so that the eigenvalues in low disorder converge to Sine 1 , the limiting eigenvalue process of the GOE. For the GOE case, this is the first example where Wigner's famous prediction is proven rigorously: we exhibit a complex system whose eigenvalues behave like those of random matrices.

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“…This is a stochastic differential equation (SDE) describing the conductance of a disordered wire with respect to its length in a macroscopic setup. Bachman and de Roeck [2] analyzed the connection of the microscopical Anderson model on a strip to DMPK theory. If the unperturbed operator describes an ideal lead, then they found an SDE describing the evolution of the transfer matrices in an appropriate scaling limit.…”

Section: Introductionmentioning

confidence: 99%

Relations between transfer and scattering matrices in the presence of hyperbolic channels

Sadel

2011

Journal of Mathematical Physics

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The generalized star product and the factorization of scattering matrices on graphs We consider a cable described by a discrete, space-homogeneous, quasi-onedimensional Schrödinger operator H 0 . We study the scattering by a finite disordered piece (the scatterer) inserted inside this cable. For energies E where H 0 has only elliptic channels we use the Lippmann-Schwinger equations to show that the scattering matrix and the transfer matrix, written in an appropriate basis, are related by a certain polar decomposition. For energies E where H 0 has hyperbolic channels we show that the scattering matrix is related to a reduced transfer matrix and both are of smaller dimension than the transfer matrix. Moreover, in this case the scattering matrix is determined from a limit of larger dimensional scattering matrices, as follows: We take a piece of the cable of length m, followed by the scatterer and another piece of the cable of length m, consider the scattering matrix of these three joined pieces inserted inside an ideal lead at energy E (ideal means only elliptic channels), and take the limit m → ∞. C

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From the Anderson Model on a Strip to the DMPK Equation and Random Matrix Theory

Cited by 11 publications

References 28 publications

Disordered Quantum Wires: Microscopic Origins of the DMPK Theory and Ohm’s Law

Disordered Quantum Wires: Microscopic Origins of the DMPK Theory and Ohm’s Law

Random Schrödinger operators on long boxes, noise explosion and the GOE

Relations between transfer and scattering matrices in the presence of hyperbolic channels

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