Disconnected Elementary Band Representations, Fragile Topology, and Wilson Loops as Topological Indices: An Example on the Triangular Lattice

In this work, we examine the topological phases that can arise in triangular lattices with disconnected elementary band representations. We show that, although these phases may be "fragile" with respect to the addition of extra bands, their topological properties are manifest in certain nontrivial holonomies (Wilson loops) in the space of nontrivial bands. We introduce an eigenvalue index for fragile topology, and we show how a nontrivial value of this index manifests as the winding of a hexagonal Wilson loop; this remains true even in the absence of time-reversal or sixfold rotational symmetry. Additionally, when time-reversal and twofold rotational symmetry are present, we show directly that there is a protected nontrivial winding in more conventional Wilson loops. Crucially, we emphasize that these Wilson loops cannot change without closing a gap to the nontrivial bands. By studying the entanglement spectrum for the fragile bands, we comment on the relationship between fragile topology and the "obstructed atomic limit" of B. Bradlyn et al., Nature 547, 298--305 (2017). We conclude with some perspectives on topological matter beyond the K-theory classification.Comment: 13 pages, 10 figures v2. accepted versio

Symmetry indicators in commensurate magnetic flux

We derive a framework to apply topological quantum chemistry in systems subject to magnetic flux. We start by deriving the action of spatial symmetry operators in a uniform magnetic field, which extends Zak's magnetic translation groups to all crystal symmetry groups. Ultimately, the magnetic symmetries form a projective representation of the crystal symmetry group. As a consequence, band representations acquire an extra gauge invariant phase compared to the non-magnetic theory. Thus, the theory of symmetry indicators is distinct from the non-magnetic case. We give examples of new symmetry indicators that appear at $\pi$ flux. Finally, we apply our results to an obstructed atomic insulator with corner states in a magnetic field. The symmetry indicators reveal a topological-to-trivial phase transition at finite flux, which is confirmed by a Hofstadter butterfly calculation. The bulk phase transition provides a new probe of higher order topology in certain obstructed atomic insulators.Comment: 24 pages, 7 figure

Intrinsically-multilayer moir\'e heterostructures

We introduce trilayer and multilayer moir\'e heterostructures that cannot be viewed from the ``moir\'e-of-moir\'e" perspective of helically-twisted trilayer graphene. These ``intrinsically trilayer" moir\'e systems feature periodic modulation of a local quasicrystalline structure. They open the door to realizing moir\'e heterostructures with vastly more material constituents because they do not constrain the lattice constants of the layers. In this manuscript, we define intrinsically multilayer patterns, provide a recipe for their construction, derive their local configuration space, and connect the visual patterns to physical observables in material systems.Comment: Fixed missing figur