Atomic-scale helices exist as motifs for several material lattices. We examine a tight-binding model for a single one-dimensional monatomic chain with a p-orbital basis coiled into a helix. A topologically nontrivial phase emerging from this model supports a zero-energy mode localized to a boundary, always embedded within a continuum band, regardless of termination site. We identify a topological invariant for this phase that is related to the number of zero energy end modes by means of the bulk-boundary correspondence, and give strict conditions for the existence of the bound state. Another, non-topological, gapped edge mode in the model spectrum has practical consequences for surface states in e.g. trigonal tellurium and selenium and other van der Waals-bonded one-dimensional semiconductors.
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