Topological insulator on the kagome lattice

Itinerant electrons in a two-dimensional kagome lattice form a Dirac semimetal, similar to graphene. When lattice and spin symmetries are broken by various periodic perturbations this semimetal is shown to spawn interesting nonmagnetic insulating phases. These include a two-dimensional topological insulator with a nontrivial ${\text{Z}}_2$ invariant and robust gapless edge states, as well as dimerized and trimerized “Kekulé” insulators. The latter two are topologically trivial but the Kekulé phase possesses a complex order parameter with fractionally charged vortex excitations. A charge-density wave is shown to couple to the Dirac fermions as an effective axial gauge field.

Figure 1

(Color online) (a) Kagome lattice is a triangular Bravais lattice with a three-point basis forming a shaded triangle. (b) The first Brillouin zone with the nodal points ${\mathbf{K}}_{\pm }$ and time-reversal invariant momenta ${\mathbf{\Gamma }}_n$ marked. (c) The tight-binding band structure Eq. (3). (d) In the dimerized phase hopping amplitude along the thick (thin) bonds in ${\mathbf{a}}_1$ direction is $t+{\eta }_1$ $\left(t-{\eta }_1\right)$. In the spin-orbit phase spin-up electrons hop between second neighbor sites with amplitude $i\lambda$ when moving along the arrow, $-i\lambda$ against the arrow. For spin-down electrons the arrows are reversed. (e) Trimerized “Kekulé” phase.

Figure 2

Edge states in the lattice model for (a) spin orbit and (b) dimerized insulator. A strip of width $N_y=16$ unit cells with open boundary conditions along $y$ and infinite along $x$ is used with $\lambda /t=0.1$ and ${\eta }_1/t=0.1$ for (a) and (b), respectively.