Fragile topology for six-fold rotation symmetry indicated by the concentric Wilson loop spectrum

TL;DR

This paper explores topological phases in p6 symmetric lattice models, revealing that the six-fold symmetry-related topology is fragile and can be detected using concentric Wilson loop spectra, challenging previous assumptions about topological invariants.

Contribution

It introduces the use of Concentric Wilson Loop Spectrum to identify fragile topology in six-fold symmetric systems, expanding the understanding of topological invariants.

Findings

Topological phases vary with hopping parameters in p6 symmetric models.

The CWLS invariant detects topological properties where conventional invariants may fail.

The identified topology in six-fold symmetry systems is fragile, not robust.

Abstract

We investigate topological phase transitions for the Haldane and Kane-Mele model in a lattice with $p6$ symmetry, which consists of triangles and hexagons arranged in a two-dimensional geometry. For the Haldane model, which breaks time-reversal symmetry, we calculate the Chern number using a multi-band non-Abelian Wilson loop formalism. By varying the hopping parameters in the triangles and hexagons independently, a large variety of topological phases emerge. In the presence of a next-next-nearest neighbor hopping, the phase diagram becomes even richer, with regions exhibiting high Chern numbers. Then, we consider the Kane-Mele model, for which time-reversal symmetry is preserved, and calculate the number of $\pi$-crossings in the Concentric Wilson Loop Spectrum (CWLS). This method is appropriate to determine the topological invariant for systems hosting time-reversal and rotational symmetry, but lacking all other symmetries. According to a classification based on $K$-theory, the CWLS invariant reveals topological properties even when more conventional invariants fail to detect them. The formalism was previously successfully applied to systems with 3- and 4-fold symmetry. Here, we surprisingly find that for the 6-fold-symmetry model investigated, the topology identified by this invariant is fragile, therefore questioning the claim that this should be the strong invariant missing in a complete classification of topological insulators.