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

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 $π$-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.

Figures (8)

• Figure 1: The $p6$ lattice with five types of hoppings: $t_{h}$ (yellow lines), $t_{t}$ (blue lines), $\gamma_{h}$ (pink lines), $\gamma_{t}$ (purple lines), and $t_{h}'$ (green lines).
• Figure 2: Band structures of the $p6$ model with $\gamma_h$ and $\gamma_t$ varying from $0$ to $1$ for different values of the hopping parameters $t_h$ and $t_t$.
• Figure 3: Schematic illustration of discrete Wilson loops. The Brillouin zone (red parallelogram) is discretized into an $N\times N$ grid (here $N=2$). For each plaquette, the Berry flux $\Phi_m$ is evaluated by computing a Wilson loop along its boundary (blue loops). The Chern number is obtained by summing the Berry flux over all plaquettes.
• Figure 4: Topological characterization of the $p6$ lattice. (a)-(c) display the band structure for different choices of $(t_h,t_t)$ at fixed $\gamma_h=\gamma_t=1$. (d)-(f) Phase diagrams for filling $n=1$ corresponding to the above choices of $(t_h,t_t)$, for values of $\gamma_h$, $\gamma_t$ varying from $-1$ to $1$. (g)-(i) and (j)-(l) depict the same but for $2$ and $3$ filled bands, respectively. Black indicates that the system is gapless.
• Figure 5: Band structure with NN, NNNN hopping, and $\gamma_t = \gamma_h = 1$.