The bulk-edge correspondence for continuous dislocated systems

Drouot, Alexis

doi:10.5802/aif.3420

Cited by 9 publications

(15 citation statements)

References 95 publications

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“…Furthermore, we study the resonances for finite topological structures, for which the eigenvalues are complex-valued and the eigenmodes increase exponentially at infinity. We mention several closely related work [20,14,12], where one-dimensional Schrödinger equations with periodic potentials are studied. It is shown in [20] that for a class of background periodic Schrödinger operators with Dirac points, localized edge states can be induced via small and adiabatic modulation of the periodic potentials with a domain wall, and the bifurcation of these states are associated with the discrete eigenmodes of an effective Dirac operator.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, full asymptotic expansions of the eigenpairs are derived. In [12], the bulk-edge correspondence is rigorously established for a family of operators, wherein each operator corresponds to a dislocation of the background periodic Schrödinger operator. It is proved that certain edge index is equal to the bulk index given by the Chern number of the Bloch eigenbundle below the band gap.…”

Section: Introductionmentioning

confidence: 99%

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Mathematical theory for topological photonic materials in one dimension

Lin,

Zhang

2021

Preprint

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This work presents a rigorous theory for topological photonic materials in one dimension. The main focus is on the existence and stability of interface modes that are induced by topological properties of the bulk structure. For a general 1D photonic structure with time-reversal symmetry, the associated Zak phase (or Berry phase) may not be quantized. We investigate the existence of an interface mode which is induced by a Dirac point upon perturbation. Specifically, we establish conditions on the perturbation which guarantee the opening of a band gap around the Dirac point and the existence of an interface mode. For a periodic photonic structure with both timereversal and inversion symmetry, the Zak phase is quantized, taking only two values 0, π. We show that the Zak phase is determined by the parity (even or odd) of the Bloch modes at the band edges. For a photonic structure consisting of two semi-infinite systems on the two sides of an interface with distinct topological indices, we show the existence of an interface mode inside the common gap. The stability of the mode under perturbations is also investigated. Finally, we study resonances for finite topological structures. Our results are based on the transfer matrix method and the oscillation theory for Sturm-Liouville operators. The methods and results can be extended to general topological Sturm-Liouville systems in one dimension.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mathematical theory for topological photonic materials in one dimension

Lin,

Zhang

2021

Preprint

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“…An example from the setting of the Schrödinger operator is to introduce dislocations to periodic potentials. This has been widely studied in both one [21, 22, 25, 43, 44] and two dimensions [36–39].…”

Section: Introductionmentioning

confidence: 99%

“…Most notably, when a periodic potential is dislocated, the original configuration will be recovered periodically. Then, a quantity of interest is the edge index , which can be defined as the net number of eigenvalues which cross a band gap over a period of dislocation (see, for example, [17, 22]). If the edge index is non‐zero, it means that a mid‐gap frequency can be placed at any given position within the band gap (which, we said, is our goal).…”

Section: Introductionmentioning

confidence: 99%

Robust edge modes in dislocated systems of subwavelength resonators

Ammari

Davies

Hiltunen

2022

Journal of London Math Soc

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Robustly manipulating waves on subwavelength scales can be achieved by, first, designing a structure with a subwavelength band gap and, second, introducing a defect so that eigenfrequencies fall within the band gap. Such frequencies are well known to correspond to localized modes. We study a one-dimensional array of subwavelength resonators, prove that there is a subwavelength band gap, and show that by introducing a dislocation we can place localized modes at any point within the band gap. We complement this analysis by studying the stability properties of the corresponding finite array of resonators, demonstrating the value of being able to customize the position of eigenvalues within the band gap.

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“…The quantum analogue of high-contrast subwavelength metamaterials (in condensed matter physics) is graphene, which is a single atomic layer of carbon atoms arranged in a honeycomb structure. While Fefferman, Weinstein and their collaborators [15][16][17][18][19][20][21][22] proved fundamental mathematical results on the graphene model, little was known on the classical analogue (in wave physics) of this. This is due to many fundamental differences in the mathematical treatments of the classical and quantum problems.…”

Section: Introductionmentioning

confidence: 99%

Unidirectional edge modes in time-modulated metamaterials

Ammari

Jinghao

2022

Proc. R. Soc. A.

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We prove the possibility of achieving unidirectional edge modes in time-modulated supercell structures. Such finite structures consist of two trimers repeated periodically. Because of their symmetry, they admit degenerate edge eigenspaces. When the trimers are time-modulated with two opposite orientations, the degenerate eigenspace splits into two one-dimensional eigenspaces described by an analytical formula, each corresponding to a mode which is localized at one edge of the structure. Our results on the localization and stability of these edge modes with respect to fluctuations in the time-modulation amplitude are illustrated by several numerical simulations.

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The bulk-edge correspondence for continuous dislocated systems

Abstract: The bulk-edge correspondence for continuous dislocated systems Article à paraître, mis en ligne le août , p.Ann. Inst. Fourier, Grenoble Article à paraître Mis en ligne le 6 août 2021.

Cited by 9 publications

References 95 publications

Mathematical theory for topological photonic materials in one dimension

Mathematical theory for topological photonic materials in one dimension

Robust edge modes in dislocated systems of subwavelength resonators

Unidirectional edge modes in time-modulated metamaterials

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