Topological Modes in Composite Nanohoops

1D polymers admit a band structure according to Bloch's theorem. In the case where the polymer has inversion symmetry, for each band one can compute a quantized topological index proportional to the Berry phase integrated along the first Brillouin zone. By combining polymers of finite length at their ends to form a closed ring, one obtains a composite nanohoop. When the polymers are topologically distinct in the bulk, topologically protected states appear in the nanohoop at the boundary between the different polymers. Using quantum chemical methods, we compute the topological indices for the polymers polyparaphenylene and polyisothianaphthene and explore the existence of boundary states in the composite nanohoops of these polymers. We hope that these topologically protected modes may eventually find applications in error-resistant quantum computing.