Hidden topology in flat-band topological insulators: Strong, weak, and square-root topological states

In this paper, we study a previously unexplored class of topological states protected by hidden chiral symmetries that are local, that is, that protect against any off-diagonal disorder. We derive their related topological invariant, and show that these previously unidentified symmetries can act together with standard chiral symmetries to increase the protection of the end modes, using the Creutz ladder as an example. Finally, thanks to local hidden symmetries, we show that the diamond necklace chain can have three different types of topological end modes: strong, weak, or square-root, with some of the states inheriting their topology from others.