  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.
