Symmetry and Dirac points in graphene spectrum

Berkolaiko, Gregory; Comech, Andrew

doi:10.4171/jst/223

Cited by 53 publications

(55 citation statements)

References 47 publications

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“…These are quasi-momentum / energy pairs, (K , E ), in the band structure of H (0) at which neighboring dispersion surfaces touch conically at a point [11,21,30]. The existence of Dirac points, located at the six vertices of the Brillouin zone, B h (regular hexagonal dual period cell) for generic honeycomb structures was proved in [10,11]; see also [2,16]. The quasi-momenta of Dirac points partition into two equivalence classes; the K− points consisting of K, RK and R 2 K, where R is a rotation by 2π/3 and K − points consisting of K = −K, RK and R 2 K .…”

Section: Detailed Discussion Of Main Resultsmentioning

confidence: 93%

“…It is also shown in [11] that a h − periodic perturbation of V (x), which breaks inversion or time-reversal symmetry lifts the eigenvalue degeneracy; a 4 Lowest three dispersion surfaces k ≡ (k (1) , k (2) ) ∈ B h → E(k) of the band structure of H (0) ≡ − + V (x), where V is the honeycomb potential: V (x) = 10 (cos(k 1 · x) + cos(k 2 · x) + cos((k 1 + k 2 ) · x)). Dirac points occur at the intersection of the lower two dispersion surfaces, at the six vertices of the Brillouin zone, B h .…”

Section: Detailed Discussion Of Main Resultsmentioning

confidence: 99%

“…an L 2 k ( )− spectral gap exists; see Theorem 8.3 and Section 1.3. Figure 5 and Figure 6 are illustrative of Cases (1) and (2). The simulations were done for the Hamiltonian H (ε,δ) with ε = ±10 and 0 ≤ δ ≤ 10:…”

Section: Remark 13 (Directional Versus Omnidirectional Spectral Gaps)mentioning

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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Section: Detailed Discussion Of Main Resultsmentioning

confidence: 93%

Section: Detailed Discussion Of Main Resultsmentioning

confidence: 99%

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Edge States in 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%

“…So far, there is no complete understanding of this phenomenon (analogous to the one provided in [26,135,186] for the honeycomb case).…”

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

confidence: 99%

An overview of periodic elliptic operators

Kuchment

2016

Bull. Amer. Math. Soc.

189

186

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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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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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Symmetry and Dirac points in graphene spectrum

Cited by 53 publications

References 47 publications

Edge States in Honeycomb Structures

Edge States in Honeycomb Structures

An overview of periodic elliptic operators

Honeycomb Schrödinger Operators in the Strong Binding Regime

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