Chiral topological phases and fractional domain wall excitations in one-dimensional chains and wires

According to the general classification of topological insulators, there exist one-dimensional chirally (sublattice) symmetric systems that can support any number of topological phases. We introduce a zigzag fermion chain with spin-orbit coupling in magnetic field and identify three distinct topological phases. Zero-mode excitations, localized at the phase boundaries, are fractionalized: two of the phase boundaries support $\pm e/2$ charge states while one of the boundaries support $\pm e$ and neutral excitations. In addition, a finite chain exhibits $\pm e/2$ edge states for two of the three phases. We explain how the studied system generalizes the Peierls-distorted polyacetylene model and discuss possible realizations in atomic chains and quantum spin Hall wires.