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

TL;DR

This paper investigates fragile topological phases in triangular lattices, demonstrating how Wilson loops serve as topological indices and revealing their robustness even without certain symmetries.

Contribution

It introduces an eigenvalue index for fragile topology and links Wilson loop windings to topological properties in disconnected band representations.

Findings

Wilson loops reveal fragile topology in triangular lattices.

Nontrivial Wilson loop windings are protected without time-reversal symmetry.

Fragile topology relates to obstructed atomic limits and persists without gap closing.

Abstract

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.