Abstract
The geometric properties of a lattice can have profound consequences on its band spectrum. For example, symmetry constraints and geometric frustration can give rise to topologicially nontrivial and dispersionless bands, respectively. Line-graph lattices are a perfect example of both of these features: their lowest energy bands are perfectly flat, and here we develop a formalism to connect some of their geometric properties with the presence or absence of fragile topology in their flat bands. This theoretical work will enable experimental studies of fragile topology in several types of line-graph lattices, most naturally suited to superconducting circuits.
Highlights
Fragile topology is a property of a set of “Wannierobstructed” gapped electronic bands whose Wannier obstruction can be resolved by adding select trivial bands [1,2,3,4,5,6,7,8,9,10,11,12,13]
We have shown how to predict the representation of the energy = −2 flat bands for line-graph lattices of planar regular root-graph lattices where these bands are gapped from the rest of the spectrum
Of the line-graph lattices considered in this work, we find one D = 2 lattice with fragile topological flat bands—the line graph of the triangle lattice—and a family of D = 4 lattices with fragile topological flat bands after one of a class of specific perturbations—the 4o family
Summary
Fragile topology is a property of a set of “Wannierobstructed” gapped electronic bands whose Wannier obstruction can be resolved by adding select trivial bands [1,2,3,4,5,6,7,8,9,10,11,12,13]. We consider line-graph lattices of nonbipartite toroidal regular root-graph lattices, with flat-band degeneracy 1 < D 4 These lattices have C2, C3, or C6 symmetry, and can be further split into families based on their coordination number and the number of faces per unit cell that are bounded by an even number of edges (“even-sided faces”). This root-graph lattice has coordination number 3, zero even-sided faces, and C2 symmetry It has eight vertices and four faces per unit cell; the corresponding line-graph lattice has a D = 4-fold degeneracy of its flat bands at −2. We find a relationship between how many of each graph-element type are at a root-graph lattice’s maximal Wyckoff positions, and the lattice’s coordination number, number of even-sided faces, and symmetry These correspondences are listed, with cells pertaining to examples. Several patterns emerge across these root-graph lattices, stated and proved in Appendix C of the
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