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

TL;DR

This paper investigates the topological and geometric properties of flat band states in a one-dimensional atomic chain with s-p orbitals, using the Fubini-Study metric to distinguish states and relate topology to localized modes.

Contribution

It introduces a model mapping to a Kitaev-Creutz type system, utilizing the Fubini-Study metric to analyze flat band topology and localization, and relates these to coupled SSH chains.

Findings

Fubini-Study metric distinguishes pure states with same topology.

Flat bands linked to Compact Localized States and pseudo-Bogoliubov modes.

Model equivalent to two coupled SSH chains.

Abstract

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