The Su-Schrieffer-Heeger . (SSH) chain is a one-dimensional lattice that comprises two dimerized sublattices. Recently, Zhu, Prodan, and Ahn (ZPA) [Phys. Rev. B 99, 041117(R) (2019)] proposed that one-dimensional flat bands can occur at the topological domain walls of a two-dimensional array of SSH chains. Here, we suggest a two-dimensional topological insulator that is protected by inversion and time-reversal symmetries without spin-orbit coupling. It is shown that two-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 topological insulator, dubbed a two-dimensional Stiefel-Whitney insulator (2DSWI). The ZPA model is employed to study the topological phase diagrams and topological phase transitions. It is found that the phase transition occurs via the formation of massless Dirac points that wind the entire Brillouin zone. We argue that this unconventional topological phase transition is a characteristic feature of a 2DSWI, manifesting as one-dimensional domain wall states. Using first-principles calculations, we find the suggested 2DSWI should be realized in 11 known materials, such as ${\mathrm{Zn}}_2{\left(\mathrm{P}{\mathrm{S}}_3\right)}_3$. This insight from our work could help efforts to realize topological flat bands in solid-state systems.

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• Topological materials
• Two-dimensional electron system
• Inversion symmetry