Topological tight binding models on some non-trivial lattices: Union of geometry, flat bands and topology.

We introduce a topological tight binding model based on certain rules that we have formulated to study systems with certain non-trivial bulks (abbreviated as SAB). These rules allows us to study bulks that have twists and branching. We discuss certain cases in the SAB model with different number of bands, exhibiting several interesting physical properties. For every bulk there can be two sets of configurations: the orientable and the non-orientable configuration. The later exhibits several non-trivial physical properties like exact flat bands (exactly at particle hole symmetry level), zero energy states localised in the bulk, topological edge states etc. We then discuss a three band non-orientable SAB model which is easy to visualise and hence can be realized in experiments first. We also investigate the effects of disorder (both chiral symmetry preserving and breaking) in the non-orientable configurations hosting flat bands. We find for chiral symmetry preserving disorders, some of them (non-degenerate flat band) are robust to large disorders while others (degenerate flat band) exhibit an insulator to metal transition beyond certain disorder strength due to band gap closing as a result of the broadening of the zero energy states. For chiral symmetry breaking disorders, in both the cases the zero energy bulk states broaden and close the gap beyond certain critical disorder strength.