Edge States in Honeycomb Structures

Fefferman, Charles; Lee-Thorp, James P.; Weinstein, Michael I.

doi:10.1007/s40818-016-0015-3

Cited by 78 publications

(154 citation statements)

References 50 publications

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“…As k varies over B h , each map k → E b (k) is Lipschitz continuous and sweeps out a closed interval in R. The union of these intervals is the L 2 (R 2 )− spectrum of H (ε,0) . More detail on the general theory of periodic operators and the specific context of honeycomb structures appears in [22,23,24] and [25,19], respectively.…”

Section: Honeycomb Potentialsmentioning

confidence: 99%

Bifurcations of edge states—topologically protected and non-protected—in continuous 2D honeycomb structures

2016

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Abstract. Edge states are time-harmonic solutions to energy-conserving wave equations, which are propagating parallel to a line-defect or "edge" and are localized transverse to it. This paper summarizes and extends the authors' work on the bifurcation of topologically protected edge states in continuous twodimensional honeycomb structures.We consider a family of Schrödinger Hamiltonians consisting of a bulk honeycomb potential and a perturbing edge potential. The edge potential interpolates between two different periodic structures via a domain wall. We begin by reviewing our recent bifurcation theory of edge states for continuous two-dimensional honeycomb structures (http://arxiv.org/abs/1506.06111). The topologically protected edge state bifurcation is seeded by the zero-energy eigenstate of a one-dimensional Dirac operator. We contrast these protected bifurcations with (more common) non-protected bifurcations from spectral band edges, which are induced by bound states of an effective Schrödinger operator.Numerical simulations for honeycomb structures of varying contrasts and "rational edges" (zigzag, armchair and others), support the following scenario: (a) For low contrast, under a sign condition on a distinguished Fourier coefficient of the bulk honeycomb potential, there exist topologically protected edge states localized transverse to zigzag edges. Otherwise, and for general edges, we expect long lived edge quasi-modes which slowly leak energy into the bulk. (b) For an arbitrary rational edge, there is a threshold in the medium-contrast (depending on the choice of edge) above which there exist topologically protected edge states. In the special case of the armchair edge, there are two families of protected edge states; for each parallel quasimomentum (the quantum number associated with translation invariance) there are edge states which propagate in opposite directions along the armchair edge.

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Section: Honeycomb Potentialsmentioning

confidence: 99%

Bifurcations of edge states—topologically protected and non-protected—in continuous 2D honeycomb structures

2016

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“…13 For 2D Schrödinger operators with honeycombsymmetric potentials, existence and stability of Dirac cones was established, explained, and studied [26,[132][133][134][135][136]186]. The Schrödinger operator with honeycomb lattice of point scatterers was considered in [263].…”

Section: On a Z 2 -Periodic Graph γ Having Just Two Vertices (Atoms)mentioning

confidence: 99%

“…5 in the book [213] and references therein (e.g., to Chuburin's work). See also somewhat related works [97,134,155,156,361]. It seems to the author that this topic has not been sufficiently studied yet.…”

Section: Inhomogeneous Equationsmentioning

confidence: 99%

Untitled

2016

Bull. Amer. Math. Soc.

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Abstract. The article surveys the main topics, techniques, and results of the theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which significantly influence most spectral features of such operators.The approaches described are applicable not only to the standard model example of Schrödinger operator with periodic electric potential −Δ + V (x), but to a wide variety of elliptic periodic equations and systems, equations on graphs, ∂-operator, and other operators on abelian coverings of compact bases.Important applications are mentioned. However, due to the size restrictions, they are not dealt with in detail.

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“…The existence of the Dirac cone is attributed to the symmetry of the lattice; it is not affected by the geometric size of the inclusions in the crystal [24]. Because of the linear dispersion of the Dirac cone, breaking the T symmetry may open a nontrivial band gap and introduce edge states [25]. This process can be described by a two-band model [2].…”

Section: Introductionmentioning

confidence: 99%

Tunable Topological Phononic Crystals

2016

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Topological insulators first observed in electronic systems have inspired many analogues in photonic and phononic crystals in which remarkable one-way propagation edge states are supported by topologically nontrivial band gaps. Such band gaps can be achieved by breaking the time-reversal symmetry to lift the degeneracy associated with Dirac cones at the corners of the Brillouin zone. Here, we report on our construction of a phononic crystal exhibiting a Dirac-like cone in the Brillouin zone center. We demonstrate that simultaneously breaking the time-reversal symmetry and altering the geometric size of the unit cell result in a topological transition that we verify by the Chern number calculation and edge-mode analysis. We develop a complete model based on the tight binding to uncover the physical mechanisms of the topological transition. Both the model and numerical simulations show that the topology of the band gap is tunable by varying both the velocity field and the geometric size; such tunability may dramatically enrich the design and use of acoustic topological insulators.

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Edge States in Honeycomb Structures

Cited by 78 publications

References 50 publications

Bifurcations of edge states—topologically protected and non-protected—in continuous 2D honeycomb structures

Bifurcations of edge states—topologically protected and non-protected—in continuous 2D honeycomb structures

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Tunable Topological Phononic Crystals

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