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We study defect modes in a one-dimensional periodic medium perturbed by an adiabatic dislocation of amplitude δ ≪ 1 . If the periodic background admits a Dirac point—a linear crossing of dispersion curves—then the dislocated operator acquires a gap in its essential spectrum. For this model (and its honeycomb analog) Fefferman et al. (Proc Natl Acad Sci USA 111(24):8759–8763, 2014, Mem Am Math Soc 247(1173):118, 2017, Ann PDE 2(2):80, 2016, 2D Mater 3:1, 2016) constructed (at leading order in δ ) defect modes with energies within the gap. These bifurcate from the eigenmodes of an effective Dirac operator. Here we address the following open problems: Do all defect modes arise as bifurcations from the Dirac operator eigenmodes? Do these modes admit expansions to all order in δ ? We respond positively to both questions. Our approach relies on (a) resolvent estimates for the bulk operators; (b) scattering theory for highly oscillatory potentials [ Dr18a , Dr18b , Dr18c ]. It has led to an understanding of the topological stability of defect states in continuous dislocated systems—in connection with the bulk-edge correspondence [ Dr18d ].

We study the single electron model of a semi-infinite graphene sheet interfaced with the vacuum and terminated along a zigzag edge. The model is a Schroedinger operator acting on L 2 ( R 2 ) : H edge λ = - Δ + λ 2 V ♯ , with a potential V ♯ given by a sum of translates an atomic potential well, V 0 , of depth λ 2 , centered on a subset of the vertices of a discrete honeycomb structure with a zigzag edge. We give a complete analysis of the low-lying energy spectrum of H edge λ in the strong binding regime ( λ large). In particular, we prove scaled resolvent convergence of H edge λ acting on L 2 ( R 2 ) , to the (appropriately conjugated) resolvent of a limiting discrete tight-binding Hamiltonian acting in l 2 ( N 0 ; C 2 ) . We also prove the existence of edge states : solutions of the eigenvalue problem for H edge λ which are localized transverse to the edge and pseudo-periodic plane-wave like parallel to the edge. These edge states arise from a “flat-band” of eigenstates of the tight-binding model.

We study a class of periodic Schrödinger operators, which in distinguished cases can be proved to have linear band-crossings or “Dirac points”. We then show that the introduction of an “edge”, via adiabatic modulation of these periodic potentials by a domain wall, results in the bifurcation of spatially localized “edge states”. These bound states are associated with the topologically protected zero-energy mode of an asymptotic one-dimensional Dirac operator. Our model captures many aspects of the phenomenon of topologically protected edge states for two-dimensional bulk structures such as the honeycomb structure of graphene. The states we construct can be realized as highly robust TM-electromagnetic modes for a class of photonic waveguides with a phase-defect.

Consider the tight binding model of graphene, sharply terminated along an edge l parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges l into those of “zigzag type” and those of “armchair type,” generalizing the classical zigzag and armchair edges.We prove that zero-energy/flat-band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulae for flat-band edge states when they exist. We produce strong evidence for the existence of dispersive (nonflat) edge state curves of nonzero energy for most l.

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.

Let E(Z, N) be the ground-state energy of N quantized electrons and a single nucleus of charge Z. For fixed Z, E(Z, N) is independent of N for N ≥ Ncritical(Z). Physically, this means that at most Ncriticalelectrons can bind to the nucleus. We prove that Ncritical≤ Z + CZawith a = 0.84.

Consider the tight binding model of graphene, sharply terminated along an edge \\( l\\) parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges \\( l\\) into those of \"zigzag type\" and those of \"armchair type\", generalizing the classical zigzag and armchair edges. We prove that zero energy/flat band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulas for flat band edge states when they exist. We produce strong evidence for the existence of dispersive (non flat) edge state curves of nonzero energy for most \\( l\\).

We study the single electron model of a semi-infinite graphene sheet interfaced with the vacuum and terminated along a zigzag edge. The model is a Schroedinger operator acting on \\(L^2(R^2)\\): \\(H^_ edge=-+^2 V_\\), with a potential \\(V_\\) given by a sum of translates an atomic potential well, \\(V_0\\), of depth \\(^2\\), centered on a subset of the vertices of a discrete honeycomb structure with a zigzag edge. We give a complete analysis of the low-lying energy spectrum of \\(H^_ edge\\) in the strong binding regime (\\(\\) large). In particular, we prove scaled resolvent convergence of \\(H^_ edge\\) acting on \\(L^2(R^2)\\), to the (appropriately conjugated) resolvent of a limiting discrete tight-binding Hamiltonian acting in \\(l^2(N_0;C^2)\\). We also prove the existence of ıt edge states: solutions of the eigenvalue problem for \\(H^_ edge\\) which are localized transverse to the edge and pseudo-periodic (propagating or plane-wave like) parallel to the edge. These edge states arise from a \"flat-band\" of eigenstates the tight-binding Hamiltonian.