Topological boundary states in 1D: An effective Fabry-Perot model

Levy, E.; Akkermans, Éric

doi:10.1140/epjst/e2016-60341-8

Cited by 8 publications

(9 citation statements)

References 39 publications

(95 reference statements)

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“…Critical modes feature highly fragmented multi-fractal envelopes with a power-law decay that found recent applications in aperiodic lasing, optical sensing, photo-detection, and nonlinear optical devices [10,16,[26][27][28]. Moreover, topologically protected edge-states were recently discovered in the pseudo-gap spectra of quasicrystals [29][30][31], significantly broadening our understanding of topological phases in optical media. However, the vast majority of previous studies focused on quasicrystalline structures that are constructed by local matching rules, such as the Penrose lattice, or on deterministic scattering arrays generated by binary inflation rules, of which the Fibonacci, Thue-Morse, and Rudin-Shapiro sequences are the primary examples [19,26,[29][30][31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

Spectral statistics and scattering resonances of complex primes arrays

2018

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We introduce a novel class of aperiodic arrays of electric dipoles generated from the distribution of prime numbers in complex quadratic fields (Eisenstein and Gaussian primes) as well as quaternion primes (Hurwitz and Lifschitz primes), and study the nature of their scattering resonances using the vectorial Green's matrix method. In these systems we demonstrate several distinctive spectral properties, such as the absence of level repulsion in the strongly scattering regime, critical statistics of level spacings, and the existence of critical modes, which are extended fractal modes with long lifetimes not supported by either random or periodic systems. Moreover, we show that one can predict important physical properties, such as the existence spectral gaps, by analyzing the eigenvalue distribution of the Green's matrix of the arrays in the complex plane. Our results unveil the importance of aperiodic correlations in prime number arrays for the engineering of novel gapped photonic media that support far richer mode localization and spectral properties compared to usual periodic and random media.

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Section: Introductionmentioning

confidence: 99%

Spectral statistics and scattering resonances of complex primes arrays

2018

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“…Figure 4 The states either show a maximum or a minimum intensity at the interface (x = 0), corresponding to either a node or an antinode of the wave function. In the frame of a Fabry-Perot model [46] we will denote these two cases as symmetric (S) and antisymmetric (AS) respectively. Because of experimental imperfections the states are not perfectly S or AS but we shall employ this convenient terminology.…”

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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 this section, we provide a simplified model for stratifications with interface states as well as general arguments based on scattering matrices for their existence adapted from Refs. [31][32][33][37][38][39][40].…”

Section: Simplified Model Of Interface Statesmentioning

confidence: 99%

“…It is also not necessary that the gapped stratifications are spatially periodic: they can also be quasiperiodic (see Ref. [33] and references therein for examples in photonic crystals) or disordered.…”

Section: Surface Modesmentioning

confidence: 99%

“…Although it is tempting to expect that a surface state will still exist in these intermediate situations, this cannot be determined from considerations of topology alone. Instead, the existence of surface states can be phrased as a general condition of constructive interference [31][32][33]. In our experiments, the upper layer can be viewed as a cavity bounded by two mirrors with reflection coefficients r(ω) and r wall , as represented in Fig.…”

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Internal wave crystals

Ghaemsaidi¹,

Fruchart²,

Atis³

2021

Preprint

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Geophysical fluids such as the ocean and atmosphere can be stratified: their density depends on the depth. As a consequence, they can host internal gravity waves that propagate in the bulk of the fluid, far from the surface [1]. These waves can transport energy and momentum over large distances, thereby affecting large-scale circulation patterns [2], as well as the transport of heat, sediments, nutrients and pollutants in the ocean [3]. When the density stratification is not uniform, internal waves can exhibit wave phenomena such as resonances, tunneling, and frequencydependent transmissions [4][5][6]. Spatially periodic density profiles formed by thermohaline staircases are commonly found in stratified fluids ranging from the Arctic Ocean [7] to giant planet interiors [8,9], and can produce extended regions with periodically stratified fluid. Here, we report on the experimental observation of band gaps for internal gravity waves, ranges of frequencies over which the wave propagation is prohibited in the presence of a periodic stratification. We show the existence of surface states controlled by boundary conditions and discuss their topological origin. Our results suggest that energy transport can be profoundly affected by the presence of periodic stratifications in geophysical fluids ranging from Earth's oceans to gas giants.

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Topological boundary states in 1D: An effective Fabry-Perot model

Cited by 8 publications

References 39 publications

Spectral statistics and scattering resonances of complex primes arrays

Spectral statistics and scattering resonances of complex primes arrays

Measuring topological invariants from generalized edge states in polaritonic quasicrystals

Internal wave crystals

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