Honeycomb Schrödinger Operators in the Strong Binding Regime

Abstract

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.We begin with a review of Floquet-Bloch theory. See, for example, [11,37,38,49] and [3,23,24,34,53]. Fourier Analysis on L 2 .R=ƒ/Let fv 1 ; v 2 g be a linearly independent set in R 2 , and introduce the following:Lattice: ƒ D Zv 1˚Z v 2 D fm 1 v 1 C m 2 v 2 W m 1 ; m 2 2 ZgI Dual lattice: ƒ D ZK 1˚Z K 2 D fm E K

Edge States in Honeycomb Structures

“…A corollary of this result is that the spectral no-fold condition, as stated in the present article, is satisfied for sufficiently deep potentials (high contrast) for a very large classes of edge directions in h (including the zigzag edge). In fact, we believe that the analysis of the present article can be extended and together with [9] will yield the existence of edge states which are localized, transverse to arbitrary edge directions v 1 ∈ h . This is work in progress.…”

“…In a forthcoming article [9], we study the strong binding regime (deep potentials) for a large class of honeycomb Schrödinger operators. We prove that the two lowest energy dispersion surfaces, after a rescaling by the potential well's depth, converge uniformly to those of the celebrated Wallace (1947) [40] tight-binding model of graphite.…”

Rigorous derivation of nonlinear Dirac equations for wave propagation in honeycomb structures

“…In these models, however, the resulting mode equations are of transport type, and any coupling between the amplitudes stems purely from the nonlinearity, in contrast to the Dirac model. We finally remark that discrete mode equations, valid in the tight binding regime, have recently been studied in [2,15]. This paper is now organized as follows: In Section 2 we recall some basic properties of honeycomb lattice structures and the associated lattice potentials.…”

Edge states and the valley Hall effect

“…Since the density of electronic states is zero at energy E D , graphene is often called a semi-metal. The condition (H2) holds in the (explicitly solvable) tight binding model of graphene [41] and by [27,Corollary 6.4] the condition (H2) holds in the strong binding regime. Honeycomb structures that satisfy (H1) but fail to satisfy (H2) can be thought as metallic at energy E D .…”