Topological Peierls Transitions in M\"{o}bius Molecular Devices

We study the topological properties of Peierls transitions in a monovalent M\"{o}bius ladder. Along the transverse and longitudinal directions of the ladder, there exist plenty Peierls phases corresponding to various dimerization patterns. Resulted from a special modulation, namely, staggered modulation along the longitudinal direction, the ladder system in the insulator phase behaves as a ``topological insulator'', which possesses charged solitons as the gapless edge states existing in the gap. Such solitary states promise the dispersionless propagation along the longitudinal direction of the ladder system. Intrinsically, these non-trivial edges states originates from the Peierls phases boundary, which arises from the non-trivial $\mathbb{Z}^{2}$ topological configuration.