Figure 1

(a) The triangle-honeycomb lattice is constructed from a honeycomb lattice by replacing each site with a triangle. (b) Topologically equivalent representation. Sites within a unit cell $\mathbf{r}$ are labeled by $\alpha =1,\dots ,6$. The dotted line indicates its vertical boundary.Reuse & Permissions

Figure 2

(a) Bulk energy gap, $\Delta$ as a function of $J^{\prime }/J$ for the symmetric case, $J_{\alpha }\equiv J$ and $J_{\alpha }^{\prime }\equiv J^{\prime }$. The quantum critical point is at $(J^{\prime }/J{)}_c=\sqrt{3}$. For small $J$, the phase is a topologically trivial CSL with Abelian excitations while for large $J$, the phase is a topologically nontrivial CSL with vortex excitations obeying non-Abelian statistics. The $J\to \infty$ spectrum is Dirac-like, with a mass gap that vanishes as $J^{\prime 2}/J$. There is a cusp at $J^{\prime }/J=2\sqrt{3}/3$, where the $\mathbf{k}$ with minimum energy jumps. (b) Energy levels calculated for $J=J^{\prime }$ on a cylinder with approximately ${10}^4$ sites and with OBC along $\widehat{x}$-direction. There is one edge state (dotted line) crossing with zero energy, corresponding to Chern number $\nu =\pm 1$.Reuse & Permissions