Homogenized description of defect modes in periodic structures with localized defects

Duchêne, Vincent; Vukićević, Iva; Weinstein, Michael I.

doi:10.4310/cms.2015.v13.n3.a9

Cited by 16 publications

(20 citation statements)

References 38 publications

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“…This is in contrast to bifurcations from the edge of continuous spectra, arising from localized perturbations (for example, refs. [26][27][28][29][30][31]. A class of edge bifurcations due to a noncompact perturbation is studied in ref.…”

Section: Theorem 3 Is Illustrated Inmentioning

confidence: 99%

Topologically protected states in one-dimensional continuous systems and Dirac points

Fefferman

Lee-Thorp

Weinstein

2014

Proc. Natl. Acad. Sci. U.S.A.

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We study a class of periodic Schrödinger operators on R that have Dirac points. The introduction of an "edge" via adiabatic modulation of a periodic potential by a domain wall results in the bifurcation of spatially localized "edge states," associated with the topologically protected zero-energy mode of an asymptotic one-dimensional Dirac operator. The bound states we construct can be realized as highly robust transverse-magnetic electromagnetic modes for a class of photonic waveguides with a phase defect. Our model captures many aspects of the phenomenon of topologically protected edge states for 2D bulk structures such as the honeycomb structure of graphene.Floquet-Bloch theory | Hill's equation | surface states | multiple scale analysis | wave-packets E nergy localization in surface modes or edge states at the interface between dissimilar media has been explored, going back to the 1930s, as a vehicle for localization and transport of energy (1-8). These phenomena can be exploited in, for example, quantum, electronic, or photonic device design. An essential property for applications is robustness; the localization properties of such surface states needs to be stable with respect to distortions of or imperfections along the interface.A class of structures, which has attracted great interest since about 2005, is topological insulators (9, 10). In certain energy ranges, such structures behave as insulators in their bulk (this is associated with an energy gap in the spectrum of the bulk Hamiltonian), but have boundary conducting states with energies in the bulk energy gap; these are states that propagate along the boundary and are localized transverse to the boundary. Some of these states may be topologically protected; they persist under deformations of the interface that preserve the bulk spectral gap, e.g., localized perturbations of the interface. In honeycomb structures, e.g., graphene, where a bulk gap is opened at a "Dirac point" by breaking time-reversal symmetry (10-14), protected edge states are unidirectional. Furthermore, these edge states do not backscatter in the presence of interface perturbations (3)(4)(5)8). Chiral edge states, observed in the quantum Hall effect, are a well known instance of topological protected states in condensed matter physics. In tightbinding models, this property can be understood in terms of topological invariants associated with the band structure of the bulk periodic structure (15)(16)(17)(18)(19)(20).In this article we introduce a one-dimensional continuum model, a Schrödinger equation with periodic potential modulated by a domain wall, in which we rigorously study the bifurcation of topologically protected edge states as a parameter lifts a Dirac point degeneracy (symmetry-protected linear band crossing). This model, which has many of the features of the above examples, is motivated by the study of photonic edge states in honeycomb structures in ref.3. The bifurcation we study is governed by the existence of a topologically protected zero-energy eigenstate of a one-dim...

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Section: Theorem 3 Is Illustrated Inmentioning

confidence: 99%

Topologically protected states in one-dimensional continuous systems and Dirac points

Fefferman

Lee-Thorp

Weinstein

2014

Proc. Natl. Acad. Sci. U.S.A.

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“…In contrast to the edge state bifurcations obtained via Theorem 4.1, these bifurcations are not topologically protected; they may be destroyed by a localized perturbation of the edge. The formal multiplescale bifurcation analysis presented in Section 5.1 can be made rigorous along the lines of [31,32,33,34]; see also [35].…”

Section: Edge States Which Are Not Topologically Protectedmentioning

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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“…Since − 2 Λ eff is a small potential well, classical results [23] for the operator Hq = −∂ 2 x + σ(x) apply, and we conclude that t σ (k) and consequently t q (k) have a simple pole of order O( 2 ) on the positive imaginary axis, from which the existence of a negative discrete eigenvalue, E , of order O( 4 ) is an immediate consequence. More precisely, the asymptotic behavior of the eigenvalue corresponding to the small potential well, and therefore to the original oscillatory potential, is predicted by the Schrödinger operator with Dirac distribution potential with negative mass (see [9], consistently with [23,5]):…”

Section: Motivation Methods Of Proof and Relation To Previous Workmentioning

confidence: 99%

Oscillatory and Localized Perturbations of Periodic Structures and the Bifurcation of Defect Modes

Duchêne¹,

Vukićević²,

Weinstein³

2015

SIAM J. Math. Anal.

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Let Q(x) denote a periodic function on the real line. The Schrödinger operator, HQ = −∂ 2 x + Q(x), has L 2 (R)− spectrum equal to the union of closed real intervals separated by open spectral gaps. In this article we study the bifurcation of discrete eigenvalues (point spectrum) into the spectral gaps for the operator HQ+q , where q is spatially localized and highly oscillatory in the sense that its Fourier transform, q is concentrated at high frequencies. Our assumptions imply that q may be pointwise large but q is small in an average sense. For the special case where q (x) = q(x, x/ ) with q(x, y) smooth, real-valued, localized in x, and periodic or almost periodic in y, the bifurcating eigenvalues are at a distance of order 4 from the lower edge of the spectral gap. We obtain the leading order asymptotics of the bifurcating eigenvalues and eigenfunctions. Consider the (b * ) th spectral band (b * ≥ 1) of HQ. Underlying this bifurcation is an effective Hamiltonian associated with the lower spectral band edge: H eff = −∂xA b * ,eff ∂x − 2 B b * ,eff × δ(x) where δ(x) is the Dirac distribution, and effective-medium parameters A b * ,eff , B b * ,eff > 0 are explicit and independent of . The potentials we consider are a natural model for wave propagation in a medium with localized, high-contrast and rapid fluctuations in material parameters about a background periodic medium.

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Homogenized description of defect modes in periodic structures with localized defects

Cited by 16 publications

References 38 publications

Topologically protected states in one-dimensional continuous systems and Dirac points

Topologically protected states in one-dimensional continuous systems and Dirac points

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

Oscillatory and Localized Perturbations of Periodic Structures and the Bifurcation of Defect Modes

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