The bulk-edge correspondence for continuous honeycomb lattices

Drouot, Alexis

doi:10.1080/03605302.2019.1643362

Cited by 33 publications

(37 citation statements)

References 75 publications

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“…In this section we explain the sense in which (a) the family of edge states of Figure 1 is not topologically protected, and (b) the family displayed in Figure 2 is topologically protected. These conclusions follow from the detailed arguments in [13,14], applied to the present context. Consider the family of Fredholm operators…”

Section: Introductionsupporting

confidence: 77%

“…Such models are of interest in the field of metamaterials; see, for example, [30,36,45] in the physics literature, and the mathematical study [38]. Our methods also apply to honeycomb magnetic Schrödinger operators [13,14].…”

Section: Introductionmentioning

confidence: 99%

“…Edge state diagrams are plots of E(k ) vs. k . The global character of these diagrams depends strongly on the type of edge and on the manner in which the symmetries are broken by the perturbation; see Figures 1 and 2. In this paper we advance the spectral analysis of H δ edge initiated in [22,23] and continued in [13,14]. Edge states which bifurcate from Dirac points were first constructed for zigzagtype edges in [14,23] and in a related 1D model in [21,24].…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by [21,23,24] we use a Lyapunov-Schmidt / Schur reduction strategy to obtain resolvent expansions. • We interpret the robustness of edge states, following the analysis of [13,14].…”

Section: Introductionmentioning

confidence: 99%

“…The spectral flow (signed count of eigenvalues crossing the gap) is 2. It equals the difference of Chern numbers of low-lying eigenbundles at either side of the edge; see [13,14] and §1.5.…”

Section: Introductionmentioning

confidence: 99%

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Edge states and the valley Hall effect

Drouot

Weinstein

2020

Advances in Mathematics

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We study energy propagation along line-defects (edges) in two dimensional continuous, energy preserving periodic media. The unperturbed medium (bulk) is modeled by a honeycomb Schroedinger operator, which is periodic with respect to the triangular lattice, invariant under parity, P, and complex-conjugation, C . A honeycomb operator has Dirac points in its band structure: two dispersion surfaces touch conically at an energy level, E D [25,27]. Periodic perturbations which break P or C open a gap in the essential spectrum about energy E D . Such operators model an insulator near energy E D .Our edge operator is a small perturbation of the bulk and models a transition (via a domain wall) between distinct periodic, P or C breaking perturbations. The edge operator permits energy transport along the line-defect. The associated energy channels are called edge states. They are time-harmonic solutions of the underlying wave equation, which are localized near and propagating along the line-defect. They are of great scientific interest due to their remarkable stability, and are a key property of topological insulators.We completely characterize the edge state spectrum within the bulk spectral gap about E D . At the center of our analysis is an expansion of the edge operator resolvent for energies near E D . The leading term features the resolvent of an effective Dirac operator. Edge state eigenvalues are poles of the resolvent, which bifurcate from the Dirac point. The corresponding eigenstates have the multiscale structure identified in [23].We extend earlier work on zigzag-type edges [14] to all rational edges. We elucidate the role in edge state formation played by the type of symmetry-breaking and the orientation of the edge. We prove the resolvent expansion by a new direct and transparent strategy. Our results also provide a rigorous explanation of the numerical observations in [22,38]; see also the photonic experimental study in [42]. Finally we discuss implications for the Valley Hall Effect, which concerns quantum Hall-like energy transport in honeycomb structures.

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Section: Introductionsupporting

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Edge states and the valley Hall effect

Drouot

Weinstein

2020

Advances in Mathematics

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Magnetic slowdown of topological edge states

Bal,

Becker,

Drouot

2023

Comm Pure Appl Math

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We study the propagation of wavepackets along curved interfaces between topological, magnetic materials. Our Hamiltonian is a massive Dirac operator with a magnetic potential. We construct semiclassical wavepackets propagating along the curved interface as adiabatic modulations of straight edge states under constant magnetic fields. While in the magnetic‐free case, the wavepackets propagate coherently at speed one, here they experience slowdown, dispersion, and Aharonov–Bohm effects. Several numerical simulations illustrate our results.

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Topology in Shallow-Water Waves: A Spectral Flow Perspective

Tauber

Thiang

2022

Ann. Henri Poincaré

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The bulk-edge correspondence for continuous honeycomb lattices

Cited by 33 publications

References 75 publications

Edge states and the valley Hall effect

Edge states and the valley Hall effect

Magnetic slowdown of topological edge states

Topology in Shallow-Water Waves: A Spectral Flow Perspective

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