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Register a new userThe Zak phase and the existence of edge states in graphene
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We develop a method to predict the existence of edge states in graphene ribbons for a large class of boundaries. This approach is based on the bulk-edge correspondence between the quantized value of the Zak phase Z(k), which is a Berry phase across an appropriately chosen one-dimensional Brillouin zone, and the existence of a localized state of momentum k at the boundary of the ribbon. This bulk-edge correspondence is rigorously demonstrated for a one dimensional toy model as well as for graphene ribbons with zigzag edges. The range of k for which edge states exist in a graphene ribbon is then calculated for arbitrary orientations of the edges. Finally, we show that the introduction of an anisotropy leads to a topological transition in terms of the Zak phase, which modifies the localization properties at the edges. Our approach gives a new geometrical understanding of edge states, it confirms and generalizes the results of several previous works.
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Topological phases of matter have attracted much attention over the years. Motivated by analogy with photonic lattices, here we examine the edge states of a one-dimensional trimer lattice in the phases with and without inversion symmetry protection. In contrast to the Su-Schrieffer-Heeger model, we show that the edge states in the inversion-symmetry broken phase of the trimer model turn out to be chiral, i.e., instead of appearing in pairs localized at opposite edges they can appear at a $textit{single}$ edge. Interestingly, these chiral edge states remain robust to large amounts of disorder. In addition, we use the Zak phase to characterize the emergence of degenerate edge states in the inversion-symmetric phase of the trimer model. Furthermore, we capture the essentials of the whole family of trimers through a mapping onto the commensurate off-diagonal Aubry-Andre-Harper model, which allow us to establish a direct connection between chiral edge modes in the two models, including the calculation of Chern numbers. We thus suggest that the chiral edge modes of the trimer lattice have a topological origin inherited from this effective mapping. Also, we find a nontrivial connection between the topological phase transition point in the trimer lattice and the one in its associated two-dimensional parent system, in agreement with results in the context of Thouless pumping in photonic lattices.
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