  We study two coupled Su-Schrieffer-Heeger (SSH) chains system, which is shown
to contain rich quantum phases associated with topological invariants protected
by symmetries. In the weak coupling region, the system supports two non-trivial
topological insulating phases, characterized by winding number N = +1 or -1,
and two types of edge states. The boundary between the two topological phases
arises from two band closing points, which exhibit topological characteristics
in one-dimensional k space. By mapping Bloch states on a vector field in k
space, the band degenerate points correspond to a pair of kinks of the field,
with opposite topological charges. Two topological nodal points move and merge
as the inter-chain coupling strength varies. This topological invariant is
protected by the translational and inversion symmetries, rather than the
antiunitary operation. Furthermore, we find that when a pair of nodal points is
created, a second order quantum phase transition (QPT) occurs, associating with
a gap closing and spontaneously symmetry breaking. This simple model
demonstrates several central concepts in the field of quantum materials and
provides a theoretical connection between them.
