We study a 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 nontrivial topological insulating phases, characterized by winding number $\mathcal{N}=\pm 1$, 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 interchain 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.

• Received 18 April 2017
• Revised 14 August 2017

DOI:https://doi.org/10.1103/PhysRevB.96.125418