The Fibonacci Hamiltonian

Damanik, David; Gorodetski, Anton; Yessen, William

doi:10.1007/s00222-016-0660-x

Cited by 59 publications

(90 citation statements)

References 101 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 2.5. The proof also yields the explicit formulae (38) for µ and for the "LiebRobinson velocity" v. The formula for v does not yield quantitative information however, because it involves the quantity C ′ δ , which is not determined in [4]. Our second main result says that the upper transport exponent is truly the "correct" one for the LR bound (modulo the difference between ≥ and >).…”

Section: Theorem 24 (First Main Resultsmentioning

confidence: 92%

“…Some of their tools, like the Dunford functional calculus approach, work for rather general onedimensional quantum systems. However, the crucial exponential lower bound on transfer matrix norms, which is a result of [4] quoted here as Proposition 6.5 is special to the Fibonacci case. Since the methods of [4] apply to arbitrary coupling strength λ > 0, so do our results.…”

Section: Proof Of the First Main Resultsmentioning

confidence: 95%

“…We begin the proof of the first main result, Theorem 2.4. It will be convenient for us to use the following quantity, which we will soon see agrees with α + u , introduced in the recent paper [4] to study transport exponents from a dynamical systems perspective.…”

Section: Proof Of the First Main Resultsmentioning

confidence: 99%

“…The precise definitions of these quantities are of limited relevance here, because we will use results of [4] tailor-made for the analysis of α ′ as a "black box". The limit in (10) exists, by Proposition 3.7 in [4].…”

Section: Proof Of the First Main Resultsmentioning

confidence: 99%

“…We recall that the reason why this useful formula holds is that the fermions described by the c j are non-interacting. (b) Dynamical upper bounds for H n , established by Damanik -Tcheremchantsev [8, 9, 2] and Damanik -Gorodetski -Yessen [4]. Some of their tools, like the Dunford functional calculus approach, work for rather general onedimensional quantum systems.…”

Section: Proof Of the First Main Resultsmentioning

confidence: 99%

See 4 more Smart Citations

On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

Damanik¹,

Lemm²,

Lukić³

et al. 2016

J. Spectr. Theory

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Theorem 24 (First Main Resultsmentioning

confidence: 92%

Section: Proof Of the First Main Resultsmentioning

confidence: 95%

Section: Proof Of the First Main Resultsmentioning

confidence: 99%

Section: Proof Of the First Main Resultsmentioning

confidence: 99%

Section: Proof Of the First Main Resultsmentioning

confidence: 99%

See 3 more Smart Citations

On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

Damanik¹,

Lemm²,

Lukić³

et al. 2016

J. Spectr. Theory

Self Cite

View full text Add to dashboard Cite

show abstract

On the Fibonacci Tiling and its Modern Ramifications

Baake,

Gähler,

Mazáč

2024

Israel Journal of Chemistry

View full text Add to dashboard Cite

In the last 30 years, the mathematical theory of aperiodic order has developed enormously. Many new tilings and properties have been discovered, few of which are covered or anticipated by the early papers and books. Here, we start from the well‐known Fibonacci chain to explain some of them, with pointers to various generalisations as well as to higher‐dimensional phenomena and results. This should give some entry points to the modern literature on the subject.

show abstract

Clustering resonance effects in the electronic energy spectrum of tridiagonal Fibonacci quasicrystals

Maciá

2017

Phys. Status Solidi B

View full text Add to dashboard Cite

In this work, we show that the fundamental structure of the electronic energy spectrum of binary Fibonacci quasicrystals can be decomposed in terms of two main contributions, stemming from two related characteristic symmetries. The algebraic approach, we introduce allows us for a unified and systematic description of the energy spectrum finer structure details in terms of block matrices commutators properties, within the framework of a renormalization approach based on transfer matrices. Close analytical expressions can be explicitly obtained by exploiting some algebraic properties of these blocks commutators. In particular, the overall main features of the electronic energy spectrum structure are related to the roots of a series of polynomials of the form pFjfalse(Efalse), where the subscript denotes the degree of the polynomial, which is given by a Fibonacci number Fj. These polynomials, in turn, are derived in a systematic way from commutators involving progressively longer palindromic fundamental blocks containing two types of basic renormalized matrices. In this way, we can classify the resonance energies defining the different fragmentation patterns of the energy spectrum on the basis of purely algebraic criteria. The transmission coefficient of these resonant states is always bounded below, although their related Landauer conductance values may range from highly conductive to highly resistive ones, depending on the system length. The obtained results significantly contribute to gain a better understanding of the symmetry principles governing the fragmentation process leading to the characteristic nested structure of the energy spectrum.

show abstract

The Fibonacci Hamiltonian

Cited by 59 publications

References 101 publications

On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

On the Fibonacci Tiling and its Modern Ramifications

Clustering resonance effects in the electronic energy spectrum of tridiagonal Fibonacci quasicrystals

Contact Info

Product

Resources

About