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Title : Edge states in rationally terminated honeycomb structures
Abstract
We investigate the existence of edge states of a discrete "Laplace" operator corresponding to the tight binding model and defined in a sharp termination of a honeycomb periodic lattice along a straight line ???? (i.e. the intersection between the lattice and one of the half-spaces whose boundary is ????). The answer is well known if the line runs in the so-called zigzag or armchair direction: In the first case there exists a non-dispersive edge states while in the second case there is no edge state. In this work, we consider an arbitrary line, but with rational directions: the truncated lattice is always periodic in the direction of this line, but the corresponding period can be arbitrarily large. We distinguish two classes of edges defined by the direction of the line, and we show that non-dispersive edge states exist for the first class, which we refer to as the zigzag type, while this is not true for the second class, which we refer to as the armchair type. Finally, we provide strong evidence that dispersive edge states can exist in both cases. This is a joint work with Charles L. Fefferman (Princeton University) and Michael I. Weinstein (Columbia University).
