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The Su-Schrieffer-Heeger (SSH) chain is an one-dimensional lattice thatcomprises two dimerized sublattices. Recently, Zhu, Prodan, and Ahn (ZPA)proposed in [L. Zhu, E. Prodan, and K. H. Ahn, Phys. Rev. B \textbf{99}, 041117(2019)] that one-dimensional flat bands can occur at topological domain wallsof a two-dimensional array of the SSH chains. Here, we newly suggest atwo-dimensional topological insulator that is protected by inversion andtime-reversal symmetries without spin-orbit coupling. It is shown that thetwo-dimensional SSH chains realize the proposed topological insulator.Utilizing the first Stiefel-Whitney numbers, a weak type of $\mathbb{Z}_2$topological indices are developed, which identify the proposed topologicalinsulator, dubbed a two-dimensional Stiefel-Whitney insulator (2DSWI). The ZPAmodel is employed to study the topological phase diagrams and topological phasetransitions. It is found that the phase transition occurs via the formation ofthe massless Dirac points that wind the entire Brillouin zone. We argue thatthis unconventional topological phase transition is a characteristic feature ofthe 2DSWI, manifested as the one-dimensional domain wall states. The newinsight from our work could help efforts to realize topological flat bands insolid-state systems.

Two-dimensional weak-type topological insulators in inversion symmetric crystals