TY - JOUR

T1 - Honeycomb Schrödinger Operators in the Strong Binding Regime

AU - Fefferman, Charles L.

AU - Lee-Thorp, James P.

AU - Weinstein, Michael I.

N1 - Funding Information:
By going carefully through the arguments in this paper, one can check that all constants that appear, including those in Theorem 6.1, e.g., ? and c??, depend only on ˇmax, on C in (17.1), on cgap in (17.2), and on a lower bound for rcritical r0. The sole exception is in Theorem 6.2, where the constants depend on ´ 2 C n R as well. This allows us to treat atomic potentials V1.x/ not explicitly given in the Acknowledgment. The authors thank P. Deift and G. Berkolaiko for their interest and feedback concerning this work. We thank the referee for a careful reading and incisive comments that, in particular, inspired Theorem 6.2 and Section 16. This research was supported in part by National Science Foundation grants DMS-1265524 (CLF) and DMS-1412560 (MIW & JPL-T) and Simons Foundation Math + X Investigator Award #376319 (MIW).

PY - 2018/6

Y1 - 2018/6

N2 - In this article, we study the Schrödinger 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 Schrödinger 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 PT 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.

AB - In this article, we study the Schrödinger 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 Schrödinger 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 PT 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.

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U2 - 10.1002/cpa.21735

DO - 10.1002/cpa.21735

M3 - Article

AN - SCOPUS:85038092926

VL - 71

SP - 1178

EP - 1270

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 6

ER -