Multifractal wave functions on a Fibonacci lattice

Fujiwara, Takeo; Kohmoto, Mahito; Tokihiro, Tetsuji

doi:10.1103/physrevb.40.7413

Cited by 157 publications

(119 citation statements)

References 16 publications

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“…Their intensity distributions could be further studied by using the method of the multifractal analysis. 36,37 In the multifractal analysis, the singularity spectrum f͑␣͒ characterizes the scaling properties of the multifractal, where ␣ characterizes the type of singularities and f͑␣͒ the fractal dimension of the set on which singularities of this type are defined. In Fig.…”

Section: Resultsmentioning

confidence: 99%

Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals

et al. 2007

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By using the sparse-matrix canonical-grid method, we performed large-scale multiple-scattering calculations to study band-edge states in large-sized two-dimensional photonic quasicrystals. We find that the band-edge states evolve in an abrupt and irregular way when the sample size is increased. New states with reduced symmetries can emerge at the band edge in large samples. Strong multifractal behaviors in the wave functions are also observed. Our findings unveil important differences between quasiperiodic and periodic systems at band edges.

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Section: Resultsmentioning

confidence: 99%

Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals

et al. 2007

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“…In the frequency regime outside these Fibonacci band gaps, the light waves are critically localized. In contrast with the fully disordered (Anderson) localized case, these critically localized states decay weaker than exponentially, most likely by a power law, and have a rich self-similar structure [12]. This makes these systems very interesting for light localization studies, as proposed by Kohmoto et al [13].…”

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confidence: 97%

Light Transport through the Band-Edge States of Fibonacci Quasicrystals

et al. 2003

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The propagation of light in nonperiodic quasicrystals is studied by ultrashort pulse interferometry. Samples consist of multilayer dielectric structures of the Fibonacci type and are realized from porous silicon. We observe mode beating and strong pulse stretching in the light transport through these systems, and a strongly suppressed group velocity for frequencies close to a Fibonacci band gap. A theoretical description based on transfer matrix theory allows us to interpret the results in terms of Fibonacci band-edge resonances.

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“…The latter was obtained exactly by a different technique and f (α) is obtained analytically. [15] In conclusion, we have found analytical solutions to the BAE (1) that describes the Bloch state in a magnetic field with zero energy. The flux per plaquette is φ = P/Q with coprime odd integers P and Q.…”

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Explicit Solutions of the Bethe Ansatz Equations for Bloch Electrons in a Magnetic Field

Hatsugai

Kohmoto

1994

Phys. Rev. Lett.

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For Bloch electrons in a magnetic field, explicit solutions are obtained at the center of the spectrum for the Bethe ansatz equations recently proposed by Wiegmann and Zabrodin. When the magnetic flux per plaquette is 1/Q where Q is an odd integer, distribution of the roots is uniform on the unit circle in the complex plane. For the semi-classical limit, Q → ∞, the wavefunction obeys the power low and is given by |ψ(x)| 2 = (2/ sin πx) which is critical and unnormalizable. For the golden mean flux, the distribution of roots has the exact self-similarity and the distribution function is nowhere differentiable.The corresponding wavefunction also shows a clear self-similar structure.

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Multifractal wave functions on a Fibonacci lattice

Cited by 157 publications

References 16 publications

Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals

Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals

Light Transport through the Band-Edge States of Fibonacci Quasicrystals

Explicit Solutions of the Bethe Ansatz Equations for Bloch Electrons in a Magnetic Field

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