  Flat band (FB) systems, i.e., models hosting dispersionless bands in their
energy spectrum, have attracted great interest recently. The eigenstates of
these FBs can be written as compact localized states (CLSs), meaning states
having finite support only over a small, non-extensive portion of the lattice.
When adjacent CLSs of a FB have spatial overlap, the subspace they span is, in
general, non-orthogonal. Here, we consider precisely this scenario, in the form
of a diamond chain with a finite magnetic flux per plaquette, which has a
gapped midspectrum FB. After developing the general framework for projecting
arbitrary operators onto a non-orthogonal subspace, we address the case of an
open chain with small local impurities in the midchain plaquette. The framework
is then applied to derive analytical expressions for the states of the FB
affected by the impurities. The results are illustrated for three
representative cases: two equal impurities, two symmetric impurities, and a
single impurity. We also address the effects of diagonal disorder on the
impurity states. Focusing on the equal impurities case, we show how these
states effectively feel an averaged disorder determined by their spatial
extension, which can be controlled by the flux. Due to this averaging effect,
these states exhibit enhanced robustness to disorder. For the single-impurity
case, we further reveal its exotic non-trivial topological phase, without
precedent in the literature on Hermitian models, characterized by a
half-integer winding number. By mapping the system onto an equivalent
two-dimensional (2D) tight-binding lattice decorated with a central impurity
line, the half-integer bulk invariants are shown to have a correspondence with
the appearance of a single in-gap edge state when open boundary conditions
(OBC) are imposed. Numerical simulations are performed to benchmark all
analytical predictions.
