Abstract.We investigate the energy dispersion of the edge states in zigzag silicene, germanene and stanene nanoribbons with and without hydrogen termination based on a multiorbital tight-binding model. Since the low buckled structures are crucial for these materials, both the π and σ orbitals have a strong influence on the edge states, different from the case for graphene nanoribbons. The obtained dispersion of helical edge states is nonlinear, similar to that obtained by first-principles calculations. On the other hand, the dispersion derived from the single-orbital tight-binding model is always linear. Therefore, we find that the non-linearity comes from the multi-orbital effects, and accurate results cannot be obtained by the single-orbital model but can be obtained by the multi-orbital tight-binding model. We show that the multi-orbital model is essential for correctly understanding the dispersion of the edge states in tetragen nanoribbons with a low buckled geometry.

We introduce the quantum dimer-pentamer model (QDPM) on the square lattice. This model is a generalization of the square lattice quantum dimer model as its configuration space comprises fully-packed hard-core dimer coverings as well as dimer configurations containing pentamers, where four dimers touch a vertex. Thus in the QDPM, the fully-packed, hard-core constraint of the quantum dimer model is relaxed such that the local dimer number at each vertex is fixed modulo 3, resulting in an exact local Z3 gauge symmetry. We construct a local Hamiltonian for which the Rokhsar-Kivelson (RK) equal superposition state is the exact ground state and has a 9-fold topological degeneracy on the torus. Using Monte Carlo calculations, we find no spontaneous symmetry breaking in the RK wavefunction and that its dimer-dimer correlation function decays exponentially. By doping the QDPM RK state with a pair of monomers, we demonstrate that Z3 electric charges are deconfined. Additionally, we introduce a Z3 magnetic string operator that we find decays exponentially and shows no signatures of magnetic vortex condensation and with correlations. These numerical results suggest that the ground state of the QDPM is a dimer liquid with Z3 topological order.