Surface electromagnetic waves in Fibonacci superlattices: Theoretical and experimental results

Hassouani, Y. El; Aynaou, H.; Boudouti, El Houssaine El; Djafari‐Rouhani, Bahram; Akjouj, Abdellatif; Velasco, V R

doi:10.1103/physrevb.74.035314

Cited by 45 publications

(30 citation statements)

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“…It would be intriguing to test this for the Fibonacci QC as well, where we expect gap-traversing boundary modes to appear. Our prediction is supported by the fact that the existence of subgap boundary states in the Fibonacci QC was noticed [30], measured [31,32], and analyzed to be quantitatively similar to those of the Harper model [33].…”

Section: (A) Note That In Hsupporting

confidence: 73%

Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model

2012

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One-dimensional quasiperiodic systems, such as the Harper model and the Fibonacci quasicrystal, have long been the focus of extensive theoretical and experimental research. Recently, the Harper model was found to be topologically nontrivial. Here, we derive a general model that embodies a continuous deformation between these seemingly unrelated models. We show that this deformation does not close any bulk gaps, and thus prove that these models are in fact topologically equivalent. Remarkably, they are equivalent regardless of whether the quasiperiodicity appears as an on-site or hopping modulation. This proves that these different models share the same boundary phenomena and explains past measurements. We generalize this equivalence to any Fibonacci-like quasicrystal, i.e. a cut and project in any irrational angle.PACS numbers: 71.23. Ft, 73.43.Nq, 05.30.Rt Recent experimental developments in photonic crystals [1,2] and ultracold atoms [3][4][5] have made the study of the dynamics of particles in one-dimensional (1D) quasiperiodic systems experimentally accessible. These fascinating systems have long been the focus of extensive research. They have been studied mainly in the context of their transport and localization properties, showing a variety of interesting transitions between metallic, localized, and critical phases [6][7][8][9]. With their recently found nontrivial connection to topological phases of matter [10] there is a growing interest in their boundary phenomena [11,12].The behavior of particles in such systems is described by 1D tight-binding models with quasiperiodic modulations. There is an abundance of quasiperiodic modulations, among which the canonical types are the Harper model (also known as the Aubry-André model) [6,13] and the Fibonacci quasicrystal (QC) [14]. The quasiperiodicity of the Harper model enters in the form of a cosine modulation incommensurate with lattice spacing, whereas the Fibonacci QC has two discrete values that appear interchangeably according to the Fibonacci sequence. Moreover, the quasiperiodicity may appear in on-site terms (diagonal), in hopping terms (off diagonal), or in both (generalized). Each of these models describes different physical phenomena. Indeed, the Harper and the Fibonacci modulations have different localization phase diagrams, depending also on their appearance as diagonal or off-diagonal terms (see e.g., Refs. [7][8][9]15]). Notably, several attempts were made to gather these models under some general framework [9,16,17], but with only partial success.These 1D quasiperiodic models play a nontrivial role in the rapidly growing field of topological phases of matter [10][11][12]. This new paradigm classifies gapped systems such as band insulators and superconductors [18,19]. Each gap in these systems is attributed an index that characterizes topological properties of the wave functions in the bands below this gap. By definition, two gapped systems belong to the same topological phase if they can be deformed continuously from one into the other witho...

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Section: (A) Note That In Hsupporting

confidence: 73%

Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model

2012

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“…In particular the topological edge states [25][26][27] of quasicrystals have been recently investigated in photonic systems [22,[28][29][30] and exploited to implement topological pumping, a key concept of topology [22]. A paradigmatic example of quasicrystal is given by the 1D Fibonacci chain.…”

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confidence: 99%

Measuring topological invariants from generalized edge states in polaritonic quasicrystals

et al. 2017

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We investigate the topological properties of Fibonacci quasicrystals using cavity polaritons. Composite structures made of the concatenation of two Fibonacci sequences allow investigating generalized edge states forming in the gaps of the fractal energy spectrum. We employ these generalized edge states to determine the topological invariants of the quasicrystal. When varying a structural degree of freedom (phason) of the Fibonacci sequence, the edge states spectrally traverse the gaps, while their spatial symmetry switches: the periodicity of this spectral and spatial evolution yields direct measurements of the gap topological numbers. The topological invariants that we determine coincide with those assigned by the gap-labeling theorem, illustrating the direct connection between the fractal and topological properties of Fibonacci quasicrystals.PACS numbers: 03.65. Vf, 61.44.Br, 71.36.+c, 78.67.Pt Topology has long been recognized as a powerful tool both in mathematics and in physics. It allows identifying families of structures which cannot be related by continuous deformations and are characterized by integer numbers called topological invariants. A physical example where topological features are particularly useful is provided by quantum anomalies, i.e. classical symmetries broken at the quantum level [1], such as the chiral anomaly recently observed in condensed matter [2]. From a general viewpoint, wave or quantum systems possessing a gapped energy spectrum, such as band insulators, superconductors, or 2D conductors in a magnetic field, can be assigned topological invariants generally called Chern numbers [3]. These numbers control a variety of physical phenomena: for instance in the integer quantum Hall effect, they determine the value of the Hall conductance as a function of magnetic field [4,5] [22,[28][29][30] and exploited to implement topological pumping, a key concept of topology [22]. A paradigmatic example of quasicrystal is given by the 1D Fibonacci chain. It presents a fractal energy spectrum which consists of an infinite number of gaps [31]. A rather surprising and fascinating property is that each of theses gaps can also be assigned a topological number analogous to the aforementioned Chern numbers [32]: this constitutes the so-called gap-labeling theorem [33]. These integers can take N distinct values, N being the number of letters in the chain [34]. Despite important advances on the topological properties of quasicrystals [19][20][21][22][23][24][28][29][30]] the topological invariants have not yet been directly measured as winding numbers.The physical origin of topological numbers in a Fibonacci quasicrystal can be related to its structural properties [35]. To understand this, let us introduce a general method to generate a Fibonacci sequence: it is based on the characteristic functionproposed in [36], which takes two possible values ±1, respectively identified with two letters A and B representing e.g. two different values of a potential energy. A Fibonacci sequence of size N is a wordfor...

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“…In the case of electronic structures, it was shown recently that a finite 1D crystal made of N cells exhibits two types of confined states [91], namely, there are always N-1 states in the allowed bands, whereas there is one and only one state corresponding to each band gap [92,93]. This latter state did not depend on the width of the crystal N. This demonstration has been extended to shear horizontal and sagittal acoustic waves in continuous media made of finite solid-solid [94] and solid-fluid [95] superlattices, respectively. An experimental and theoretical verification of this rule has been given recently for electromagnetic waves in 1D coaxial cables [96].…”

Section: Surface and Confined Modes In 1d Discrete Phononic Crystalsmentioning

confidence: 78%

One-Dimensional Phononic Crystals

Boudouti¹,

Djafari‐Rouhani²

2012

Springer Series in Solid-State Sciences

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In this chapter, we discuss the vibrational properties of one-dimensional (1D) phononic crystals of both discrete and continuous media. These properties include the dispersion curves of infinite crystals as well as the confined modes and localized (surface, cavity) modes of finite and semi-infinite crystals. A general rule about the existence of localized surface modes in finite and semi-infinite superlattices with free surfaces is presented. We also present the calculations of reflection and transmission coefficients, particularly in view of selective filtering through localized modes. Most of the results presented in this chapter deal with waves propagating along the axis of the superlattice. However, in the last part of the chapter, we also discuss wave propagation out of the normal incidence and, more particularly, we demonstrate the possibility of omnidirectional transmission gap and selective filtering for any incidence angle. A comparison of the theoretical results with experimental data available in the literature is also presented and the reliability of the theoretical predictions is indicated. IntroductionThe one-dimensional (1D) phononic crystals called superlattices (SLs) are of great importance in material science. These structures are, in general, composed of two or several layers repeated periodically along the direction of growth. The layers constituting each cell of the SL can be made of a combination of solid-solid or solid-fluid-layered media. These materials enter now in the category of so-called phononic crystals (see the first chapter of this book and references therein) [1][2][3] constituted by inclusions (spheres, cylinders, etc.) arranged in a host matrix along two-dimensional (2D) and three-dimensional (3D) of the space. After the proposal of SLs by Esaki [4], the study of elementary excitations in multilayered systems has been very active. Among these excitations, acoustic phonons have received increased attention after the first observation by Colvard et al.[5] of a doublet associated to folded longitudinal acoustic phonons by means of Raman scattering. The essential property of these structures is the existence of forbidden frequency bands induced by the difference in acoustic properties of the constituents and the periodicity of these systems leading to unusual physical phenomena in these heterostructures in comparison with bulk materials [6][7][8].With regard to acoustic waves in solid-solid SLs, a number of theoretical and experimental works have been devoted to the study of the band gap structures of periodic SLs [6-10] composed of crystalline, amorphous semi-conductors, or metallic multilayers at the nanometric scale. The theoretical models used are essentially the transfer matrix [7,[11][12][13] and the Green's function methods [6,[14][15][16], whereas the experimental techniques include Raman scattering [5,17,18], ultrasonics [19][20][21][22][23][24][25][26][27][28][29], and time-resolved X-ray diffraction [30]. Besides the existence of the band-gap structures in perfe...

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Surface electromagnetic waves in Fibonacci superlattices: Theoretical and experimental results

Cited by 45 publications

References 45 publications

Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model

Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model

Measuring topological invariants from generalized edge states in polaritonic quasicrystals

One-Dimensional Phononic Crystals

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