We first discuss the derivation of tight binding models from an underlying continuum Schroedinger Hamiltonians in both non-magnetic and strongly magnetic systems.
We then present recent work on the tight binding model of graphene, sharply terminated along a rational edge, a line I parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges 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. Finally, we produce strong evidence for the existence of dispersive (non flat) edge state curves of nonzero energy for most l.
This is joint work with C.L. Fefferman and S. Fliss.
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