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

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.

Flat bands without twists: periodic holey graphene

“…Additional plots are provided in the appendix, showing that such flat bands are also seen for bigger sizes of the system. Recent research suggests that the formation of flat bands is intricately linked with compact localized states (CLS) [10], also known as confined states [68,69], signifying the existence of a non-trivial localization behavior. In order to ascertain whether there exists differences in the localization properties of states within the flat bands and those outside of them, we employ the inverse participation ratio (IPR) [44,[70][71][72][73] IPR…”

“…The reason to define such quantities is that sublattice site imbalance in bipartite lattices is known to induce flat bands of confined states in other systems [76]. Such confinement is due to destructive interference [10]. Bipartite sublattice site imbalance can only occur in our periodic HG at hole edges.…”

“…Therefore, we conclude that flat bands are produced by the sublattice imbalance and thus we can generate flat bands at will. Moreover, it has been demonstrated that flat bands appear as a consequence of brakening the path-exchange symmetry [10,74]. Upon breaking, this symmetry introduces a complex phase [10,74], leading to the lifting of the degeneracy while simultaneously preserving the flatness of the band.…”

“…Moreover, it has been demonstrated that flat bands appear as a consequence of brakening the path-exchange symmetry [10,74]. Upon breaking, this symmetry introduces a complex phase [10,74], leading to the lifting of the degeneracy while simultaneously preserving the flatness of the band. In the periodic holey system presented here, it can be demonstrated that the imbalance number is related to the path-exchange symmetry, and consequently the number of flat bands is equal to the imbalance number.…”

“…In recent years, the concept of electronic flat bands has gained prominence in the fields of materials science and condensed matter physics [1][2][3][4][5][6][7][8][9][10]. This heightened interest can be attributed to the electronic, geometrical and topological properties inherent to flat bands [10][11][12][13] and their potential to give rise to novel phenomena [7,[14][15][16][17][18][19][20]. Specifically, the discovery of superconductivity in twisted bilayer graphene in 2018 fueled considerable attention in the field [7].…”