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Honeycomb Schrodinger Operators in the Strong Binding Regime

Author(s): Fefferman, Charles L.; Lee-Thorp, James P; Weinstein, Michael I

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Abstract: In this article, we study the Schrodinger 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 Schrodinger 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 P T 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. (C) 2017 Wiley Periodicals, Inc.
Publication Date: Jun-2018
Electronic Publication Date: 18-Dec-2017
Citation: Fefferman, Charles L, Lee-Thorp, James P, Weinstein, Michael I. (2018). Honeycomb Schrodinger Operators in the Strong Binding Regime. COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 71 (1178 - 1270
ISSN: 0010-3640
EISSN: 1097-0312
Pages: 1178 - 1270
Type of Material: Journal Article
Version: Author's manuscript

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