  The Su-Schrieffer-Heeger (SSH) model describes a one-dimensional $Z_{2}$
topological insulator, which has two topological distinct phases corresponding
to two different dimerizations. When spin-orbit coupling is introduced into the
SSH model, we find the structure of the Bloch bands can be greatly changed, and
most interestingly, a new topological phase with single zero-energy bound state
which exhibits non-Abelian statistics at each end emerges, which suggests that
a new topological invariant is needed to fully classify all phases. In a
comparatively large range of parameters, we find that spin-orbit coupling
induces completely flat band with nontrivial topology. For the case with
non-uniform dimerizaton, we find that spin-orbit coupling changes the
symmetrical structure of topological excitations known as solitons and
antisolitons and when spin-orbit coupling is strong enough to induce a
topological phase transition, the whole system is topologically nontrivial but
with the disappearance of solitons and antisolitons, consequently, the system
is a real topological insulator with well-protected end states.
