James P. Lee-Thorp

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

Significance Topological insulators (TIs) have been a topic of intense study in recent years. When appropriately interfaced with other structures, TIs possess robust edge states, which persist in the presence of localized interface perturbations. Therefore, TIs are ideal for the transfer or storage of energy or information. The prevalent analyses of TIs involve idealized discrete tight-binding models. We present a rigorous study of a class of continuum models, for which we prove the emergence of topologically protected edge states. These states are bifurcations at linear band crossings (Dirac points) of localized modes. The bifurcation is induced by the 0-energy eigenmode of a class of one-dimensional Dirac equations. 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.

Edge States in Honeycomb Structures

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

Elliptic Operators with Honeycomb Symmetry: Dirac Points, Edge States and Applications to Photonic Graphene

{\mathbb {R}}^2

Honeycomb Schrödinger Operators in the Strong Binding Regime

R2 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.

TOPOLOGICALLY PROTECTED STATES IN ONE-DIMENSIONAL SYSTEMS

We present an analytical theory of 2D topologically protected guided photonic modes for continuous periodic dielectric structures, modulated by a domain wall. We then numerically corroborate the applicability of this theory for 3D structures.

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

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 polarized waves are determined by the spectral properties of the elliptic operator

Modeling the relative dynamics of DNA-coated colloids

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