  We investigate the interplay between Aharonov-Bohm (AB) caging and
topological protection in a family of quasi-one-dimensional topological
insulators, which we term CSSH ladders. Hybrids of the Creutz ladder and the
SSH chain, they present a regime with completely flat bands, and a rich
topological phase diagram, with several kinds of protected zero modes. These
are reminiscent of the Creutz ladder edge states in some cases, and of the SSH
chain edge states in others. Furthermore, their high degree of tunability, and
the fact that they remain topologically protected even in small systems in the
rungless case, due to AB caging, make them suitable for quantum information
purposes. One of the ladders can belong to the BDI, AIII and D symmetry classes
depending on its parameters, the latter being unusual in a non-superconducting
model. Two of the models can also harbor topological end modes which do not
follow the usual bulk-boundary correspondence, and are instead related to a
Chern number. Finally, we propose some experimental setups to implement the
CSSH ladders with current technology, focusing on the photonic lattice case.
