Impurity flat band states in the diamond chain

Flat band (FB) systems, featuring dispersionless energy bands, have garnered significant interest due to their compact localized states (CLSs). However, a detailed account on how local impurities affect the physical properties of overlapping CLSs is still missing. Here we study a diamond chain with a finite magnetic flux per plaquette that exhibits a gapped midspectrum FB with non-orthogonal CLSs, and develop a framework for projecting operators onto such non-orthogonal bases. This framework is applied to the case of an open diamond chain with small local impurities in the midchain plaquette, and analytical expressions are derived for FB states influenced by these impurities. For equal impurities in top and bottom sites under diagonal disorder, we show how the impurity states experience an averaged disorder dependent on their spatial extension, leading to enhanced robustness against disorder. For a single impurity, an exotic topological phase with a half-integer winding number is discovered, which is linked to a single in-gap edge state under open boundary conditions. Numerical simulations validate the analytical predictions.