The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" id="d1e1124" altimg="si8.gif"><mml:mi>K</mml:mi></mml:math>-theoretic bulk-boundary principle for dynamically patterned resonators

Prodan, Emil; Shmalo, Yitzchak

doi:10.1016/j.geomphys.2018.10.005

Cited by 30 publications

(35 citation statements)

References 46 publications

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“…Thus, we can investigate the spectral properties of infinite beams as a function of the projection parameter θ by discretizing its range in steps Δ θ=1/S, which leads to an infinitely dense subset of [0, 1] as  ¥ S . In this framework, the bulk spectrum for irrational θ values is not explicitly computed, but the discretization identifies all commensurate, or periodic, rings defined by rational values of θ, whose vibrational spectrum approximates the bulk spectrum evaluated for all θä[0, 1] [25,32].…”

Section: The Bulk Spectrum and Its Topologymentioning

confidence: 99%

“…Splitting of the bulk bands corresponds to a change in density of states, which leads to the topological classification of the bandgaps [32,33]. The integrated density of states (IDS) at frequency Ω is defined as: where [·] denotes the Iverson Brackets, which provide a value of 1 whenever the argument is true.…”

Section: The Bulk Spectrum and Its Topologymentioning

confidence: 99%

“…, are invariant labels of the bandgap [32,33]. In particular, the slope m gives the number of topological edge modes spanning the corresponding gap in the spectrum of a finite structure.…”

Section: The Bulk Spectrum and Its Topologymentioning

confidence: 99%

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Topological bands and localized vibration modes in quasiperiodic beams

2019

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We investigate a family of quasiperiodic continuous elastic beams, the topological properties of their vibrational spectra, and their relation to the existence of localized modes. We specifically consider beams featuring arrays of ground springs at locations determined by projecting from a circle onto an underlying periodic system. A family of periodic and quasiperiodic structures is obtained by smoothly varying a parameter defining such projection. Numerical simulations show the existence of vibration modes that first localize at a boundary, and then migrate into the bulk as the projection parameter is varied. Explicit expressions predicting the change in the density of states of the bulk define topological invariants that quantify the number of modes spanning a gap of a finite structure. We further demonstrate how modulating the phase of the ground springs distribution causes the topological states to undergo an edge-to-edge transition. The considered configurations and topological studies provide a framework for inducing localized modes in continuous elastic structural components through globally spanning, deterministic perturbations of periodic patterns defined by the considered projection operations. systems [19] by exploiting the notion that QP lattices are projections of higher dimensional manifolds onto lower dimensional lattices [20,21]. Notable examples include the investigations in [22,23], where photonic waveguides are used to achieve adiabatic pumping of waves between opposite ends of one-dimensional (1D) QP lattices. With further studies, a dynamically generated four-dimensional (4D) quantum Hall system was experimentally demonstrated using two-dimensional (2D) periodic lattices [24]. In the classic mechanics realm, localized modes at the boundary of a QP chain of magnetic spinners were shown in [25], while topological boundary and interface modes were experimentally demonstrated in acoustic waveguides with QP patterning of the walls [26]. Although open questions still remain regarding how eigenmodes may localize in any region of the domain, these studies provide insight into modes that are localized at edges or interfaces, and suggest new methodologies for wave localization and transport based on higher dimensional topological properties.Motivated by these contributions, we here investigate how localized modes arise in continuous elastic media with QP stiffness modulations. Such modulations are introduced through arrays of ground springs embedded in structural beams undergoing transverse vibrations (figure 1). The locations of the springs are determined by projecting from periodic array of circles. This pattern-generating procedure identifies families of structures ranging from periodic to QP, that are obtained through the smooth variation of the parameters defining the projection. This study contributes to the recent investigations of the dynamic response of QP continuous elastic media [27][28][29]. For example, the self-similar and invariant nature of stop and pass bands in beams and rods embedd...

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Section: The Bulk Spectrum and Its Topologymentioning

confidence: 99%

Section: The Bulk Spectrum and Its Topologymentioning

confidence: 99%

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Topological bands and localized vibration modes in quasiperiodic beams

2019

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“…For simplicity, the present study is restricted to one-dimensional patterns but generalizations to higher dimensions can be easily achieved based on [27] (see e.g. [28]).…”

Section: Introductionmentioning

confidence: 99%

Topological edge modes by smart patterning

Apigo

Qian

Prodan

et al. 2018

Phys. Rev. Materials

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103

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The research in topological materials and meta-materials has reached maturity and is gradually entering the phase of practical applications and devices. However, scaling down experimental demonstrations presents a major challenge. In this work, we study identical coupled mechanical resonators whose collective dynamics are fully determined by the pattern in which they are arranged. We call a pattern topological if boundary resonant modes fully fill all existing spectral gaps whenever the pattern is halved. This is a characteristic of the pattern and is entirely independent of the structure of the resonators and the details of the couplings. The existence of such patterns is proven using Ktheory and exemplified using a novel experimental platform based on magnetically coupled spinners. Topological meta-materials built on these principles can be easily engineered at any scale, providing a practical platform for applications and devices.

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“…Initiated in the pioneering work [33,14], where the bulk-boundary problem for the integer quantum Hall effect was solved, the K-theoretic formalism has been extended to cover the entire classification table of topological condensed matter [28,2], as well as many additional quasi-periodic and aperiodic systems [13,29]. An important development is the application of the formalism to Floquet topological insulators [34], which is related to our investigation.…”

Section: Introductionmentioning

confidence: 99%

Bulk-boundary correspondence for topological insulators with quantized magneto-electric effect

Leung

Prodan

2020

J. Phys. A: Math. Theor.

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We derive the bulk-boundary correspondence for three-dimensional topological insulators with quantized magneto-electric responses using operator algebraic methods. Both the exponential and the index connecting maps of K-theory are used to predict several surface phenomena when halved samples are driven in adiabatic cycles, such as pumping and transfer of Hall states. The surface Hall physics of time-reversal symmetric topological insulators, as well as related anomalous quantum Hall and axion insulators, are investigated using the new tools.

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The K-theoretic bulk-boundary principle for dynamically patterned resonators

Cited by 30 publications

References 46 publications

Topological bands and localized vibration modes in quasiperiodic beams

Topological bands and localized vibration modes in quasiperiodic beams

Topological edge modes by smart patterning

Bulk-boundary correspondence for topological insulators with quantized magneto-electric effect

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