  In flat-band systems with non-orthogonal compact localized states (CLSs),
onsite perturbations couple neighboring CLSs and generate
exponentially-decaying impurity states, whose degree of localization depends on
lattice parameters. In this work, a diamond chain with constant magnetic flux
per plaquette is decorated with several controlled onsite impurities in a
patterned arrangement, generating an effective system that emerges from the
flat band. The coupling distribution of the effective system is determined by
the relative distance between impurities and the value of the flux, which can
be chosen to engineer a wide variety of models. We employ a staggered
distribution of impurities that effectively produces the well-known
Su-Schrieffer-Heeger model, and show that the topological edge states display
an enhanced robustness to non-chiral disorder due to an averaging effect over
their extension. Finally, we provide a route to implement the system
experimentally using optical waveguides that guide orbital angular momentum
(OAM) modes. This work opens the way for the design of topologically protected
impurity states in other flat-band systems or physical platforms with
non-orthogonal bases.
