Honeycomb lattice can support electronic states exhibiting Dirac energy dispersion, with graphene as the icon. We propose to derive nontrivial topology by grouping six neighboring sites of honeycomb lattice into hexagons and enhancing the inter-hexagon hopping energies over the intra-hexagon ones. We reveal that this manipulation opens a gap in the energy dispersion and drives the system into a...

It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more general than hexagons. They are obtained by further removing upward-pointing triangles. We call the resulting shapes triangular regions. We establish signed v...

A recently introduced model describing the folding of the triangular lattice is generalized allowing for defects in the lattice and written as an Ising model with nearest– neighbor and plaquette interactions on the honeycomb lattice. Its phase diagram is determined in the hexagon approximation of the cluster variation method and the crossover from the pure Ising to the pure folding model is inv...

A recently introduced model describing the folding of the triangular lattice is generalized allowing for defects in the lattice and written as an Ising model with nearest– neighbor and plaquette interactions on the honeycomb lattice. Its phase diagram is determined in the hexagon approximation of the cluster variation method and the crossover from the pure Ising to the pure folding model is inv...

In addition to the vertices of regular hexagon and icosahedron, there are precisely two strictly projective tight 5-designs: one constructed from short vectors Leech lattice other corresponding a generalized structure in octonion plane. This paper describes new connection between these 5-designs -- common construction using octonions. Certain involutionary matrices act on three-dimensional vect...

We study a constrained statistical-mechanical model in two dimensions that has three useful descriptions. They are (i) the Ising model on the honeycomb lattice, constrained to have three up spins and three down spins on every hexagon, (ii) the three-color and fully packed loop model on the links of the honeycomb lattice, with loops around a single hexagon forbidden, and (iii) three Ising models...

A tight binding approach based on the Bogoliubov-de Gennes approach has been used to calculate the DC Josephson current for a lattice model for S-GNR-S junctions , for short junctions with respect to superconducting coherence length. We calculate the phase, length, width and chemical potential dependence at the Josephson junction and discuss the similarities and differences with regard to the t...