Quantum and spectral properties of the Labyrinth model

Takahashi, Yuki

doi:10.1063/1.4953379

Cited by 13 publications

(12 citation statements)

References 38 publications

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“…(c) We expect that similar phenomena can appear also in other models, such as for example the labyrinth model, or the square off-diagonal (or tridiagonal) Fibonacci Hamiltonian, see [54,55] for the description of the models and some partial results.…”

Section: Introductionmentioning

confidence: 81%

Spectral transitions for the square Fibonacci Hamiltonian

Damanik

Gorodetski

2018

J. Spectr. Theory

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We study the spectrum and the density of states measure of the square Fibonacci Hamiltonian. We describe where the transitions from positive-measure to zero-measure spectrum and from absolutely continuous to singular density of states measure occur. This shows in particular that for almost every parameter from some open set, a positive-measure spectrum and a singular density of states measure coexist. This provides the first physically relevant example exhibiting this phenomenon.

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Section: Introductionmentioning

confidence: 81%

Spectral transitions for the square Fibonacci Hamiltonian

Damanik

Gorodetski

2018

J. Spectr. Theory

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“…This lattice was first introduced by C. Sire in 1989 obtained from a Euclidean product of two 1D aperiodic chains [185,186]. The energy spectrum of the Labyrinth tiling has been proven to be an interval if parameters λ x and λ y of the x and y direction chains, defined by λ ≡ t 2 A − t 2 B /t A t B , are sufficiently close to zero, and it is a Cantor set of zero Lebesgue measure if λ x and λ y are large enough [187,188]. The wave packet dynamics [189] and quantum diffusion [190] in the Labyrinth tiling were also analyzed using RSRM.…”

Section: Figure 3 (Color Online)mentioning

confidence: 99%

Real Space Theory for Electron and Phonon Transport in Aperiodic Lattices via Renormalization

Sánchez

Wang

2020

Symmetry

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Structural defects are inherent in solids at a finite temperature, because they diminish free energies by growing entropy. The arrangement of these defects may display long-range orders, as occurring in quasicrystals, whose hidden structural symmetry could greatly modify the transport of excitations. Moreover, the presence of such defects breaks the translational symmetry and collapses the reciprocal lattice, which has been a standard technique in solid-state physics. An alternative to address such a structural disorder is the real space theory. Nonetheless, solving 1023 coupled Schrödinger equations requires unavailable yottabytes (YB) of memory just for recording the atomic positions. In contrast, the real-space renormalization method (RSRM) uses an iterative procedure with a small number of effective sites in each step, and exponentially lessens the degrees of freedom, but keeps their participation in the final results. In this article, we review aperiodic atomic arrangements with hierarchical symmetry investigated by means of RSRM, as well as their consequences in measurable physical properties, such as electrical and thermal conductivities.

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“…where the sequence (w n ) n∈Z is called the potential. This is the off-diagonal case, which ensures that the spectrum of H is symmetric around 0; see Proposition 2.3 in [14]. See the appendix to [4] for more details on diagonal and off-diagonal operators.…”

Section: Introductionmentioning

confidence: 98%

Attractors of sequences of iterated function systems

Broderick

2018

Dynamical Systems

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If F and G are iterated function systems, then any infinite word W in the symbols F and G induces a limit set. It is natural to ask whether this Cantor set can also be realized as the limit set of a single iterated function system H. We prove that if F , G, and H consist of C 1+α diffeomorphisms, then under some additional constraints on F and G the answer is no. This problem is motivated by the spectral theory of one-dimensional quasicrystals.

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Quantum and spectral properties of the Labyrinth model

Cited by 13 publications

References 38 publications

Spectral transitions for the square Fibonacci Hamiltonian

Spectral transitions for the square Fibonacci Hamiltonian

Real Space Theory for Electron and Phonon Transport in Aperiodic Lattices via Renormalization

Attractors of sequences of iterated function systems

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