Transfer matrices and transport for Schrödinger operators

Germinet, François; Kiselev, Alexander; Tcheremchantsev, Serguei

doi:10.5802/aif.2034

Cited by 56 publications

(80 citation statements)

References 36 publications

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“…See the appendix in [13] for a few-line proof, which also directly extends to the case considered here where the discrete Laplacian is restricted to a box (this extension was already explicitly observed in [22] for real z).…”

Section: Proof Of the First Main Resultssupporting

confidence: 54%

On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

Damanik¹,

Lemm²,

Lukić³

et al. 2016

J. Spectr. Theory

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Abstract. We rigorously prove a new kind of anomalous (or sub-ballistic) Lieb-Robinson bound for the isotropic XY chain with Fibonacci external magnetic field at arbitrary coupling. It is anomalous in that the usual exponential decay in |x|−v|t| is replaced by exponential decay in |x|−v|t| α with 0 < α < 1. In fact, we can characterize the values of α for which such a bound holds as those exceeding α + u , the upper transport exponent of the one-body Fibonacci Hamiltonian. Following the approach of [14], we relate Lieb-Robinson bounds to dynamical bounds for the one-body Hamiltonian corresponding to the XY chain via the Jordan-Wigner transformation; in our case the one-body Hamiltonian with Fibonacci potential. We can bound its dynamics by adapting techniques developed in [8, 9, 2, 4] to our purposes. To our knowledge, this is the first rigorous derivation of anomalous quantum many-body transport.Along the way, we prove a new result about the one-body Fibonacci Hamiltonian: the upper transport exponent agrees with the time-averaged upper transport exponent, see Corollary 2.9. We also explain why our method does not extend to yield anomalous Lieb-Robinson bounds of power-law type for the random dimer model.

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Section: Proof Of the First Main Resultssupporting

confidence: 54%

On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

Damanik¹,

Lemm²,

Lukić³

et al. 2016

J. Spectr. Theory

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show abstract

“…Another method was recently developed by Germinet, Kiselev, and Tcheremchantsev [12]. While their approach is similar in spirit to ours, namely that upper bounds on transfer matrix norms imply lower bounds for diffusion exponents, our results give better bounds for the applications we have in mind.…”

Section: Introductionmentioning

confidence: 58%

Lower Transport Bounds for One-dimensional Continuum Schrödinger Operators

2006

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Dedicated to Joachim Weidmann on the occasion of his 65th birthday.Abstract. We prove quantum dynamical lower bounds for one-dimensional continuum Schrödinger operators that possess critical energies for which there is slow growth of transfer matrix norms and a large class of compactly supported initial states. This general result is applied to a number of models, including the Bernoulli-Anderson model with a constant single-site potential.

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“…u as the rates of propagation of the fastest (polynomially small) part of the wavepacket; compare [15]. In particular, if α > α + u , then P (T α , T ) goes to 0 faster than any inverse power of T , and if α > α − u , then there is a sequence of times T k → ∞ such that P (T α k , T k ) goes to 0 faster than any inverse power of T k .…”

Section: The Number Dim H (S) ∈ [0 1] Is Called the Hausdorff Dimensmentioning

confidence: 99%

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

Damanik

Embree

Gorodetski

et al. 2008

Commun. Math. Phys.

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Abstract. We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as λ → ∞, dim(σ(H λ )) · log λ converges to an explicit constant (≈ 0.88137). We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schrödinger dynamics generated by the Fibonacci Hamiltonian.

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Transfer matrices and transport for Schrödinger operators

Cited by 56 publications

References 36 publications

On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

Lower Transport Bounds for One-dimensional Continuum Schrödinger Operators

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

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