Multiparameter perturbation theory of Fredholm operators applied to bloch functions

Grushin, V. V.

doi:10.1134/s0001434609110194

Cited by 21 publications

(25 citation statements)

References 2 publications

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“…Section 7 reviews the notion of Dirac points and results on the existence of Dirac points for generic honeycomb lattice potentials [17,18]; see also [6,7,28]. sufficiently large.…”

Section: Outlinementioning

confidence: 99%

“…We also state and prove consequences for Schrödinger operators on R 2 with perturbed honeycomb structures in the regime of strong binding: Corollary 6.3 on spectral gaps and Corollary 6.4 on protected edge states. Section 7 reviews the notion of Dirac points and results on the existence of Dirac points for generic honeycomb lattice potentials [17,18]; see also [6,7,28]. These are energy / quasi-momentum pairs, which occur at quasi-momenta located at the vertices of the Brillouin zone, B h , and at which neighboring dispersion surfaces touch conically.…”

Section: Outlinementioning

confidence: 99%

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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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“…Section 7 reviews the notion of Dirac points and results on the existence of Dirac points for generic honeycomb lattice potentials [17,18]; see also [6,7,28]. sufficiently large.…”

Section: Outlinementioning

confidence: 99%

Section: Outlinementioning

confidence: 99%

Honeycomb Schrödinger Operators in the Strong Binding Regime

Fefferman

Lee-Thorp

Weinstein

2017

Comm Pure Appl Math

107

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“…The k− pseudo-periodic Floquet-Bloch modes can be expressed in the form (x; k) = e ik·x p(x; k), where p(x; k) is h − periodic. For k = K + λK 2 , consider the family of eigenvalue problems, parametrized by |λ| ≤ 1/2: 16) where…”

Section: Proof Of Proposition 45mentioning

confidence: 99%

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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“…The Schrödinger operator H ε = −∆ + εq( x) in R 2 with the real-valued potential q( x) that has honeycomb symmetry was considered by Grushin [20]. A condition for a multiple eigenvalue to be a conical point was established and checked in the perturbative regime of a weak potential (small ε).…”

Section: Introductionmentioning

confidence: 99%

Symmetry and Dirac points in graphene spectrum

Berkolaiko

Comech

2018

J. Spectr. Theory

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Existence and stability of Dirac points in the dispersion relation of operators periodic with respect to the hexagonal lattice is investigated for different sets of additional symmetries. The following symmetries are considered: rotation by 2π/3 and inversion, rotation by 2π/3 and horizontal reflection, inversion or reflection with weakly broken rotation symmetry, and the case where no Dirac points arise: rotation by 2π/3 and vertical reflection.All proofs are based on symmetry considerations. In particular, existence of degeneracies in the spectrum is deduced from the (co)representation of the relevant symmetry group. The conical shape of the dispersion relation is obtained from its invariance under rotation by 2π/3. Persistence of conical points when the rotation symmetry is weakly broken is proved using a geometric phase in one case and parity of the eigenfunctions in the other.

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Multiparameter perturbation theory of Fredholm operators applied to bloch functions

Cited by 21 publications

References 2 publications

Honeycomb Schrödinger Operators in the Strong Binding Regime

Honeycomb Schrödinger Operators in the Strong Binding Regime

Edge States in Honeycomb Structures

Symmetry and Dirac points in graphene spectrum

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