ELLIPTIC OPERATORS WITH HONEYCOMB SYMMETRY: DIRAC POINTS, EDGE STATES AND APPLICATIONS TO PHOTONIC GRAPHENE. Lee-Thorp, J., P., Weinstein, M., I., & Zhu, Y. Paper Website abstract bibtex Consider electromagnetic waves in two-dimensional honeycomb structured media, whose constitutive laws have the symmetries of a hexagonal tiling of the plane. The properties of transverse electric (TE) polarized waves are determined by the spectral properties of the elliptic operator L A = −∇ x · A(x)∇ x , where A(x) is Λ h − periodic (Λ h denotes the equilateral triangular lattice), and such that with respect to some origin of coordinates, A(x) is inversion symmetric (A(x) = A(−x)) and 120 • rotationally invariant (A(R * x) = R * A(x)R, where R is a 120 • rotation in the plane). A summary of our results is as follows: a) For generic honeycomb structured media, the band structure of L A has Dirac points, i.e. conical intersections between two adjacent Floquet-Bloch dispersion surfaces. b) Initial data of wave-packet type, which are spectrally concentrated about a Dirac point, give rise to solutions of the time-dependent Maxwell equations whose wave-envelope, on long time scales, is governed by an effective two-dimensional massless time-dependent system of Dirac equations. c) Dirac points are unstable to arbitrary small perturbations which break either C (complex-conjugation) symmetry or P (inver-sion) symmetry. d) The introduction through small and slow variations of a domain wall across a line-defect gives rise to the bifurcation from Dirac points of highly robust (topologically protected) edge states. These are time-harmonic solutions of Maxwell's equations which are propagating parallel to the line-defect and spatially localized trans-verse to it. The transverse localization and strong robustness to perturbation of these edge states is rooted in the protected zero mode of a one-dimensional effective Dirac operator with spatially varying mass term. e) These results imply the existence of uni-directional propagating edge states for two classes of time-reversal invariant media in which C symmetry is broken: magneto-optic media and bi-anisotropic media. Our anal-ysis applies and extends the tools previously developed in the context of honeycomb Schrödinger operators.

@article{
title = {ELLIPTIC OPERATORS WITH HONEYCOMB SYMMETRY: DIRAC POINTS, EDGE STATES AND APPLICATIONS TO PHOTONIC GRAPHENE},
type = {article},
websites = {https://arxiv.org/pdf/1710.03389.pdf},
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created = {2018-04-05T19:22:52.590Z},
accessed = {2018-04-05},
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last_modified = {2018-04-23T16:00:19.211Z},
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abstract = {Consider electromagnetic waves in two-dimensional honeycomb structured media, whose constitutive laws have the symmetries of a hexagonal tiling of the plane. The properties of transverse electric (TE) polarized waves are determined by the spectral properties of the elliptic operator L A = −∇ x · A(x)∇ x , where A(x) is Λ h − periodic (Λ h denotes the equilateral triangular lattice), and such that with respect to some origin of coordinates, A(x) is inversion symmetric (A(x) = A(−x)) and 120 • rotationally invariant (A(R * x) = R * A(x)R, where R is a 120 • rotation in the plane). A summary of our results is as follows: a) For generic honeycomb structured media, the band structure of L A has Dirac points, i.e. conical intersections between two adjacent Floquet-Bloch dispersion surfaces. b) Initial data of wave-packet type, which are spectrally concentrated about a Dirac point, give rise to solutions of the time-dependent Maxwell equations whose wave-envelope, on long time scales, is governed by an effective two-dimensional massless time-dependent system of Dirac equations. c) Dirac points are unstable to arbitrary small perturbations which break either C (complex-conjugation) symmetry or P (inver-sion) symmetry. d) The introduction through small and slow variations of a domain wall across a line-defect gives rise to the bifurcation from Dirac points of highly robust (topologically protected) edge states. These are time-harmonic solutions of Maxwell's equations which are propagating parallel to the line-defect and spatially localized trans-verse to it. The transverse localization and strong robustness to perturbation of these edge states is rooted in the protected zero mode of a one-dimensional effective Dirac operator with spatially varying mass term. e) These results imply the existence of uni-directional propagating edge states for two classes of time-reversal invariant media in which C symmetry is broken: magneto-optic media and bi-anisotropic media. Our anal-ysis applies and extends the tools previously developed in the context of honeycomb Schrödinger operators.},
bibtype = {article},
author = {Lee-Thorp, J P and Weinstein, M I and Zhu, Y}
}

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The properties of transverse electric (TE) polarized waves are determined by the spectral properties of the elliptic operator L A = −∇ x · A(x)∇ x , where A(x) is Λ h − periodic (Λ h denotes the equilateral triangular lattice), and such that with respect to some origin of coordinates, A(x) is inversion symmetric (A(x) = A(−x)) and 120 • rotationally invariant (A(R * x) = R * A(x)R, where R is a 120 • rotation in the plane). A summary of our results is as follows: a) For generic honeycomb structured media, the band structure of L A has Dirac points, i.e. conical intersections between two adjacent Floquet-Bloch dispersion surfaces. b) Initial data of wave-packet type, which are spectrally concentrated about a Dirac point, give rise to solutions of the time-dependent Maxwell equations whose wave-envelope, on long time scales, is governed by an effective two-dimensional massless time-dependent system of Dirac equations. c) Dirac points are unstable to arbitrary small perturbations which break either C (complex-conjugation) symmetry or P (inver-sion) symmetry. d) The introduction through small and slow variations of a domain wall across a line-defect gives rise to the bifurcation from Dirac points of highly robust (topologically protected) edge states. These are time-harmonic solutions of Maxwell's equations which are propagating parallel to the line-defect and spatially localized trans-verse to it. The transverse localization and strong robustness to perturbation of these edge states is rooted in the protected zero mode of a one-dimensional effective Dirac operator with spatially varying mass term. e) These results imply the existence of uni-directional propagating edge states for two classes of time-reversal invariant media in which C symmetry is broken: magneto-optic media and bi-anisotropic media. 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The properties of transverse electric (TE) polarized waves are determined by the spectral properties of the elliptic operator L A = −∇ x · A(x)∇ x , where A(x) is Λ h − periodic (Λ h denotes the equilateral triangular lattice), and such that with respect to some origin of coordinates, A(x) is inversion symmetric (A(x) = A(−x)) and 120 • rotationally invariant (A(R * x) = R * A(x)R, where R is a 120 • rotation in the plane). A summary of our results is as follows: a) For generic honeycomb structured media, the band structure of L A has Dirac points, i.e. conical intersections between two adjacent Floquet-Bloch dispersion surfaces. b) Initial data of wave-packet type, which are spectrally concentrated about a Dirac point, give rise to solutions of the time-dependent Maxwell equations whose wave-envelope, on long time scales, is governed by an effective two-dimensional massless time-dependent system of Dirac equations. c) Dirac points are unstable to arbitrary small perturbations which break either C (complex-conjugation) symmetry or P (inver-sion) symmetry. d) The introduction through small and slow variations of a domain wall across a line-defect gives rise to the bifurcation from Dirac points of highly robust (topologically protected) edge states. These are time-harmonic solutions of Maxwell's equations which are propagating parallel to the line-defect and spatially localized trans-verse to it. 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