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The topological properties of the flat band states of a one-electronHamiltonian that describes a chain of atoms with $s-p$ orbitals are explored.This model is mapped onto a Kitaev-Creutz type model, providing a usefulframework to understand the topology through a nontrivial winding number andthe geometry introduced by the \textit{Fubini-Study (FS)} metric. This metricallows us to distinguish between pure states of systems with the same topologyand thus provides a suitable tool for obtaining the fingerprint of flat bands.Moreover, it provides an appealing geometrical picture for describing flatbands as it can be associated with a local conformal transformation overcircles in a complex plane. In addition, the presented model allows us torelate the topology with the formation of Compact Localized States (CLS) andpseudo-Bogoliubov modes. Also, the properties of the squared Hamiltonian areinvestigated in order to provide a better understanding of the localizationproperties and the spectrum. The presented model is equivalent to two coupledSSH chains under a change of basis.

Fubini-Study metric and topological properties of flat band electronic states: the case of an atomic chain with $s-p$ orbitals