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

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

doi:10.1088/2053-1583/3/1/014008

Cited by 27 publications

(41 citation statements)

References 44 publications

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“…For large, a consequence of Theorem 6.1 and the results in [18] is the existence of spectral gaps about Dirac points (see Section 7) of C 2 V when the Hamiltonian is perturbed in such a way as to break PT symmetry. In [16] (see also [15]), we develop a theory of protected edge states for honeycomb structures, perturbed by a class of line defects (domain walls) in the direction of an element of ƒ h (rational edges). (B) Corollary 6.4: Protected edge states in honeycomb structures with line defects.…”

Section: Summary Of Resultsmentioning

confidence: 99%

Honeycomb Schrödinger Operators in the Strong Binding Regime

Fefferman

Lee-Thorp

Weinstein

2017

Comm Pure Appl Math

107

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In this article, we study the Schrödinger operator for a large class of periodic potentials with the symmetry of a hexagonal tiling of the plane. The potentials we consider are superpositions of localized potential wells, centered on the vertices of a regular honeycomb structure corresponding to the single electron model of graphene and its artificial analogues. We consider this Schrödinger operator in the regime of strong binding, where the depth of the potential wells is large. Our main result is that for sufficiently deep potentials, the lowest two Floquet-Bloch dispersion surfaces, when appropriately rescaled, converge uniformly to those of the two-band tight-binding model (Wallace, 1947 [56]). Furthermore, we establish as corollaries, in the regime of strong binding, results on (a) the existence of spectral gaps for honeycomb potentials that break PT symmetry and (b) the existence of topologically protected edge states-states that propagate parallel to and are localized transverse to a line defect or "edge"-for a large class of rational edges, and that are robust to a class of large transverse-localized perturbations of the edge. We believe that the ideas of this article may be applicable in other settings for which a tight-binding model emerges in an extreme parameter limit.We begin with a review of Floquet-Bloch theory. See, for example, [11,37,38,49] and [3,23,24,34,53]. Fourier Analysis on L 2 .R=ƒ/Let fv 1 ; v 2 g be a linearly independent set in R 2 , and introduce the following:Lattice: ƒ D Zv 1˚Z v 2 D fm 1 v 1 C m 2 v 2 W m 1 ; m 2 2 ZgI Dual lattice: ƒ D ZK 1˚Z K 2 D fm E K

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Section: Summary Of Resultsmentioning

confidence: 99%

Honeycomb Schrödinger Operators in the Strong Binding Regime

Fefferman

Lee-Thorp

Weinstein

2017

Comm Pure Appl Math

107

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“…For k ∈ B h , let ( E ε (k), ε (x; k)) denote the eigenpair associated with a lowest spectral band. In [8], we calculate that the edge state bifurcation is seeded by a discrete eigenvalue effective Schrödinger operator: 12) and…”

Section: Non-topologically Protected Bifurcations Of Edge Statesmentioning

confidence: 99%

“…If κ(ζ ) is chosen as above, then Q eff (ζ ; (1 − θ)κ + θκ ), 0 ≤ θ ≤ 1, provides a smooth homotopy from a Schrödinger Hamiltonian for which there is a bifurcation of edge states (H (ε,δ) with domain wall κ) to one for which the branch of edge states does not exist (H (ε,δ) with domain wall κ ). Therefore, this type of bifurcation is not topologically protected; see [8] for a more detailed discussion. This contrast between topologically protected states and non-protected states is explained and explored numerically, in a one-dimensional setting in [25].…”

Section: Non-topologically Protected Bifurcations Of Edge Statesmentioning

confidence: 99%

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Edge States in Honeycomb Structures

2016

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An edge state is a time-harmonic solution of a conservative wave system, e.g. Schrödinger, Maxwell, which is propagating (plane-wave-like) parallel to, and localized transverse to, a line-defect or "edge". Topologically protected edge states are edge states which are stable against spatially localized (even strong) deformations of the edge. First studied in the context of the quantum Hall effect, protected edge states have attracted huge interest due to their role in the field of topological insulators. Theoretical understanding of topological protection has mainly come from discrete (tight-binding) models and direct numerical simulation. In this paper we consider a rich family of continuum PDE models for which we rigorously study regimes where topologically protected edge states exist. Our model is a class of Schrödinger operators on R 2 with a background two-dimensional honeycomb potential perturbed by an "edge-potential". The edge potential is a domain-wall interpolation, transverse to a prescribed "rational" edge, between two distinct periodic structures. General conditions are given for the bifurcation of a branch of topologically protected edge states from Dirac points of the background honeycomb structure. The bifurcation is seeded by the zero mode of a one-dimensional effective Dirac operator. A key condition is a spectral no-fold condition for the prescribed edge. We then use this result to prove the existence of topologically protected edge states along zigzag edges of certain honeycomb structures. Our results are consistent with the physics literature and appear to be the first rigorous results on the existence of topologically protected edge states for continuum 2D PDE systems describing waves in a non-trivial periodic medium. We also show that the family of Hamiltonians we study contains cases where zigzag edge states exist, but which are not topologically protected.

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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%

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

Cited by 27 publications

References 44 publications

Honeycomb Schrödinger Operators in the Strong Binding Regime

Honeycomb Schrödinger Operators in the Strong Binding Regime

Edge States in Honeycomb Structures

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